Ising model - Knowledge

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account for this phenomenon has been cited by numerous investigators (Ricciiardi and Umezawa, 1967, Hokkyo 1972, Stuart, Takahashi and Umezawa 1978, 1979) as significant enough to warrant radically new assumptions on a very fundamental level and have suggested the existence of quantum cooperative modes within the brain…In addition, it is interesting to note that the (modeling) of…Goldstone particles or bosons (as per Umezawa, et al)…within the brain, demonstrates the long-range correlation of quantum numbers preserved in the ground state…In the closed Cayley tree model ground states of pairs of sites, as well as the state variable of individual sites, (can) exhibit long-range correlation.

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correlations are unconstrained by the multipliers. An activity pattern sampled from this distribution requires the largest number of bits to store in a computer, in the most efficient coding scheme imaginable, as compared with any other distribution with the same average activity and pairwise correlations. This means that Ising models are relevant to any system which is described by bits which are as random as possible, with constraints on the pairwise correlations and the average number of 1s, which frequently occurs in both the physical and social sciences.

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exhibiting intermediate sites with zero correlation, which indeed was a relevant matter for large neural networks at the time of its consideration. The model's behavior is also of relevance for any other divergent-convergent tree physical (or biological) system exhibiting a closed Cayley tree topology with an Ising-type of interaction. This topology should not be ignored since its behavior for Ising models has been solved exactly, and presumably nature will have found a way of taking advantage of such simple symmetries at many levels of its designs.

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increases exponentially (between the extreme vertices) and so even though the contribution to spin correlations decrease exponentially, the correlation between sites such as the extreme vertex (A) in one tree and the extreme vertex in the joined tree (Ā) remains finite (above the critical temperature.) In addition, A and B also exhibit a non-vanishing correlation (as do their reflections) thus lending itself to, for B level sites (with A level), being considered “clusters” which tend to exhibit synchronization of firing.

8083: 31: 5659: 7714: 21579:, a fact which may have direct implications associated with neural structure vs. its function (in that it relates the energies of interaction and branching ratio to its transitional behavior.) For example, a relationship between the transition behavior of activities of neural networks between sleeping and wakeful states (which may correlate with a spin-spin type of phase transition) in terms of changes in neural interconnectivity ( 11648:, so the free energy and critical temperature are the same for the antiferromagnetic case. For the triangular lattice, which is not bi-partite, the ferromagnetic and antiferromagnetic Ising model behave notably differently. Specifically, around a triangle, it is impossible to make all 3 spin-pairs antiparallel, so the antiferromagnetic Ising model cannot reach the minimal energy state. This is an example of 3395: 7440: 5287: 5244:. It is important to note that the Metropolis–Hastings algorithm does not perform well around the critical point due to critical slowing down. Other techniques such as multigrid methods, Niedermayer's algorithm, Swendsen–Wang algorithm, or the Wolff algorithm are required in order to resolve the model near the critical point; a requirement for determining the critical exponents of the system. 3007: 8078:{\displaystyle Z(\beta )=\sum _{\sigma _{1},\ldots ,\sigma _{L}}e^{\beta h\sigma _{1}}e^{\beta J\sigma _{1}\sigma _{2}}e^{\beta h\sigma _{2}}e^{\beta J\sigma _{2}\sigma _{3}}\cdots e^{\beta h\sigma _{L}}e^{\beta J\sigma _{L}\sigma _{1}}=\sum _{\sigma _{1},\ldots ,\sigma _{L}}V_{\sigma _{1},\sigma _{2}}V_{\sigma _{2},\sigma _{3}}\cdots V_{\sigma _{L},\sigma _{1}}.} 370:(where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of 6775: 20462: = 0 we have the original Ising model. This system deserves interest in its own; particularly one has "non-ergodic" properties leading to strange relaxation behaviour. Much attention has been also attracted by the related bond and site dilute Ising model, especially in two dimensions, leading to intriguing critical behavior. 7132: 12723: 15476: 20754: 2704: 5654:{\displaystyle {\frac {P(\mu ,\nu )}{P(\nu ,\mu )}}={\frac {g(\mu ,\nu )A(\mu ,\nu )}{g(\nu ,\mu )A(\nu ,\mu )}}={\frac {A(\mu ,\nu )}{A(\nu ,\mu )}}={\frac {P_{\beta }(\nu )}{P_{\beta }(\mu )}}={\frac {{\frac {1}{Z}}e^{-\beta (H_{\nu })}}{{\frac {1}{Z}}e^{-\beta (H_{\mu })}}}=e^{-\beta (H_{\nu }-H_{\mu })}.} 13941: 21662:

For this topology, the spin-spin correlation was found to be zero between the extreme vertices and the central sites at which the two trees (or branches) are joined (i.e. between A and individually C, D, or E.) This behavior is explained to be due to the fact that, as k increases, the number of links

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The process of continuing away from dimension 4 is not completely well defined without a prescription for how to do it. The prescription is only well defined on diagrams. It replaces the Schwinger representation in dimension 4 with the Schwinger representation in dimension 4 − ε defined by:

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can be thought of as the square of the density of the random walkers at any point. In order for such a term to alter the finite order correlation functions, which only introduce a few new random walks into the fluctuating environment, the new paths must intersect. Otherwise, the square of the density

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so that the Gaussian at time τ is diminished in height by a factor that decreases steadily exponentially. In the quantum field theory context, these are the paths of relativistically localized quanta in a formalism that follows the paths of individual particles. In the pure statistical context, these

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At low temperatures (β ≫ 1) the configurations are near the lowest-energy configuration, the one where all the spins are plus or all the spins are minus. Peierls asked whether it is statistically possible at low temperature, starting with all the spins minus, to fluctuate to a state where most of the

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This argument works for a finite sum of exponentials, and correctly establishes that there are no singularities in the free energy of a system of a finite size. For systems which are in the thermodynamic limit (that is, for infinite systems) the infinite sum can lead to singularities. The convergence

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It was a natural and common belief among early neurophysicists (e.g. Umezawa, Krizan, Barth, etc.) that classical neural models (including those with statistical mechanical aspects) will one day have to be integrated with quantum physics (with quantum statistical aspects), similar perhaps to how the

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are not restricted to neighbors. Note that this generalization of Ising model is sometimes called the quadratic exponential binary distribution in statistics. This energy function only introduces probability biases for a spin having a value and for a pair of spins having the same value. Higher order

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Starting from the Ising model and repeating this iteration eventually changes all the couplings. When the temperature is higher than the critical temperature, the couplings will converge to zero, since the spins at large distances are uncorrelated. But when the temperature is critical, there will be

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But the spontaneous magnetization in magnetic systems and the density in gasses near the critical point are measured very accurately. The density and the magnetization in three dimensions have the same power-law dependence on the temperature near the critical point, but the behavior from experiments

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To investigate three dimensions starting from the four-dimensional theory should be possible, because the intersection probabilities of random walks depend continuously on the dimensionality of the space. In the language of Feynman graphs, the coupling does not change very much when the dimension is

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at wavelengths near the cutoff can affect the longer-wavelength fluctuations. If the system is scaled along with the cutoff, the parameters will scale by dimensional analysis, but then comparing parameters doesn't compare behavior because the rescaled system has more modes. If the system is rescaled

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In three as in two dimensions, the most studied case of the Ising model is the translation-invariant model on a cubic lattice with nearest-neighbor coupling in the zero magnetic field. Many theoreticians searched for an analytical three-dimensional solution for many decades, which would be analogous

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The contribution to the partition function for each past/future pair of configurations on a slice is the sum of two terms. There is the number of spin flips in the past slice and there is the number of spin flips between the past and future slice. Define an operator on configurations which flips the

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is satisfied. If the new state ν is accepted, then we move to that state and repeat with selecting a new state and deciding to accept it. If ν is not accepted then we stay in μ. This process is repeated until some stopping criterion is met, which for the Ising model is often when the lattice becomes

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For very high temperatures, the magnetization is zero, as it is at infinite temperature. To see this, note that if spin A has only a small correlation ε with spin B, and B is only weakly correlated with C, but C is otherwise independent of A, the amount of correlation of A and C goes like ε. For two

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was that atoms naturally explain the sharp phase boundaries observed in materials, as when ice melts to water or water turns to steam. His idea was that small changes in atomic-scale properties would lead to big changes in the aggregate behavior. Others believed that matter is inherently continuous,

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Several additional statistical mechanical problems of interest remain to be solved for the closed Cayley tree, including the time-dependent case and the external field situation, as well as theoretical efforts aimed at understanding interrelationships with underlying quantum constituents and their

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early on noted the possibility of interrelationships between (1) the classical large neural network model (with similar coupled divergent-convergent topologies) with (2) an underlying statistical quantum mechanical model (independent of topology and with persistence in fundamental quantum states):

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The spin-spin correlation between sites (in general, m and n) on the tree was found to have a transition point when considered at the vertices (e.g. A and Ā, its reflection), their respective neighboring sites (such as B and its reflection), and between sites adjacent to the top and bottom extreme

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The Ising model can be reinterpreted as a statistical model for the motion of atoms. Since the kinetic energy depends only on momentum and not on position, while the statistics of the positions only depends on the potential energy, the thermodynamics of the gas only depends on the potential energy

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In three dimensions, the perturbative series from the field theory is an expansion in a coupling constant λ which is not particularly small. The effective size of the coupling at the fixed point is one over the branching factor of the particle paths, so the expansion parameter is about 1/3. In two

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The instability means that increasing the mean field above zero a little bit produces a statistical fraction of spins which are + which is bigger than the value of the mean field. So a mean field which fluctuates above zero will produce an even greater mean field, and will eventually settle at the

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is defined as the long wavelength Fourier components of the spin variable, in the limit that the wavelengths are long. There are many ways to take the long wavelength average, depending on the details of how high wavelengths are cut off. The details are not too important, since the goal is to find

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In any dimension, the Ising model can be productively described by a locally varying mean field. The field is defined as the average spin value over a large region, but not so large so as to include the entire system. The field still has slow variations from point to point, as the averaging volume

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Kramers and Wannier were able to show that the high-temperature expansion and the low-temperature expansion of the model are equal up to an overall rescaling of the free energy. This allowed the phase-transition point in the two-dimensional model to be determined exactly (under the assumption that

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The most significant result obtained from the closed Cayley tree model involves the occurrence of long-range correlation in the absence of intermediate-range correlation. This result has not been demonstrated by other classical models. The failure of the classical view of impulse transmission to

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representing the nearest-neighbor interaction energy) and there are k (→ ∞ in the thermodynamic limit) generations in each of the tree branches (forming the closed tree architecture as shown in the given closed Cayley tree diagram.) The sum in the last term can be shown to converge uniformly and

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simulations, exact diagonalization results in quantum models, and quantum field theoretical arguments. Although it is an open problem to establish rigorously the renormalization group picture or the conformal field theory picture, theoretical physicists have used these two methods to compute the

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Once the electron's spin was discovered, it was clear that the magnetism should be due to a large number of electron spins all oriented in the same direction. It was natural to ask how the electrons' spins all know which direction to point in, because the electrons on one side of a magnet don't

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The idea is to integrate out lattice spins iteratively, generating a flow in couplings. But now the couplings are lattice energy coefficients. The fact that a continuum description exists guarantees that this iteration will converge to a fixed point when the temperature is tuned to criticality.

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model for closed Cayley trees of arbitrary branching ratio, and thereby, arbitrarily large dimensionality within tree branches. The solution to this model exhibited a new, unusual phase transition behavior, along with non-vanishing long-range and nearest-neighbor spin-spin correlations, deemed

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Based upon a review of other classical network models as a comparison, the Ising model on a closed Cayley tree was determined to be the first classical statistical mechanical model to demonstrate both local and long-range sites with non-vanishing spin-spin correlations, while at the same time

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The two possible values of the odd spins will be integrated out, by summing over both possible values. This will produce a new free energy function for the remaining even spins, with new adjusted couplings. The even spins are again in a lattice, with axes tilted at 45 degrees to the old ones.

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From the description in terms of independent tosses, the statistics of the model for long lines can be understood. The line splits into domains. Each domain is of average length exp(2β). The length of a domain is distributed exponentially, since there is a constant probability at any step of

4301: 12999:. It is believed that the critical point can be described by a renormalization group fixed point of the Wilson-Kadanoff renormalization group transformation. It is also believed that the phase transition can be described by a three-dimensional unitary conformal field theory, as evidenced by 6392:

interactions, E. Ising provided an exact solution of the model. At any positive temperature (i.e. finite β) the free energy is analytic in the thermodynamics parameters, and the truncated two-point spin correlation decays exponentially fast. At zero temperature (i.e. infinite β), there is a

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To do this, he compared the high-temperature and low-temperature limits. At infinite temperature (β = 0) all configurations have equal probability. Each spin is completely independent of any other, and if typical configurations at infinite temperature are plotted so that plus/minus are

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had given a complete formalism to reproduce the laws of thermodynamics from the laws of mechanics. But many faulty arguments survived from the 19th century, when statistical mechanics was considered dubious. The lapses in intuition mostly stemmed from the fact that the limit of an infinite

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This includes nearest neighbor and next-nearest neighbor interactions, as expected, but also a four-spin interaction which is to be discarded. To truncate to nearest neighbor interactions, consider that the difference in energy between all spins the same and equal numbers + and – is:

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of an ordinary random walk is 2. The number of balls of size ε required to cover the path increase as ε. Two objects of fractal dimension 2 will intersect with reasonable probability only in a space of dimension 4 or less, the same condition as for a generic pair of planes.

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From this expression for the free energy, all thermodynamic functions of the model can be calculated by using an appropriate derivative. The 2D Ising model was the first model to exhibit a continuous phase transition at a positive temperature. It occurs at the temperature

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are constants which are to be determined so as to reproduce the partition function. The interpretation is that the statistical configuration at this slice contributes according to both the number of spin flips in the slice, and whether or not the spin at position

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This term is the most significant, but it gives trivial behavior. This form of the free energy is ultralocal, meaning that it is a sum of an independent contribution from each point. This is like the spin-flips in the one-dimensional Ising model. Every value of

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The Ising model in four dimensions is described by a fluctuating field, but now the fluctuations are interacting. In the polymer representation, intersections of random walks are marginally possible. In the quantum field continuation, the quanta interact.

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The system is called ferromagnetic or antiferromagnetic if all interactions are ferromagnetic or all are antiferromagnetic. The original Ising models were ferromagnetic, and it is still often assumed that "Ising model" means a ferromagnetic Ising model.

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directly interact with the electrons on the other side. They can only influence their neighbors. The Ising model was designed to investigate whether a large fraction of the electron spins could be oriented in the same direction using only local forces.

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In three as in two dimensions, Peierls' argument shows that there is a phase transition. This phase transition is rigorously known to be continuous (in the sense that correlation length diverges and the magnetization goes to zero), and is called the

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Following the general approach of Jaynes, a later interpretation of Schneidman, Berry, Segev and Bialek, is that the Ising model is useful for any model of neural function, because a statistical model for neural activity should be chosen using the

11390: 7435:{\displaystyle Z(\beta )=\sum _{\sigma _{1},\ldots ,\sigma _{L}}e^{\beta J\sigma _{1}\sigma _{2}}e^{\beta J\sigma _{2}\sigma _{3}}\cdots e^{\beta J\sigma _{L-1}\sigma _{L}}=2\prod _{j=2}^{L}\sum _{\sigma '_{j}}e^{\beta J\sigma '_{j}}=2\left^{L-1}.} 20548:

obtained the exact analytical expression for the free energy of the Ising model on the closed Cayley tree (with an arbitrarily large branching ratio) for a zero-external magnetic field (in the thermodynamic limit) by applying the methodologies of

11802: 3566: 21619:), over time, is just one possible avenue suggested for further experimental investigation into such a phenomenon. In any case, for this Ising model it was established, that “the stability of the long-range correlation increases with increasing 21107: 5058: 17746: 19256:

nonzero coefficients linking spins at all orders. The flow can be approximated by only considering the first few terms. This truncated flow will produce better and better approximations to the critical exponents when more terms are included.

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The renormalization iteration divides the lattice into two parts – even spins and odd spins. The odd spins live on the odd-checkerboard lattice positions, and the even ones on the even-checkerboard. When the spins are indexed by the position

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cases, so solutions for dimensions higher than two are also intractable. Istrail's result only concerns the spin glass model with spatially varying couplings, and tells nothing about Ising's original ferromagnetic model with equal couplings.

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is negative, the fluctuations about the new minimum are described by a new positive quadratic coefficient. Since this term always dominates, at temperatures below the transition the fluctuations again become ultralocal at long distances.

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go up because the fluctuations are more correlated. This means that the average of a large number of spins does not become small as quickly as if they were uncorrelated, because they tend to be the same. This corresponds to decreasing

19872: 19725:, where each bit is either 0 or 1 depending on whether the position is occupied or not. An attractive interaction reduces the energy of two nearby atoms. If the attraction is only between nearest neighbors, the energy is reduced by −4 9604: 4544:

As before, this only proves that the average magnetization is zero at any finite volume. For an infinite system, fluctuations might not be able to push the system from a mostly plus state to a mostly minus with a nonzero probability.

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only determines the overall scale of fluctuations. The ultralocal model describes the long wavelength high temperature behavior of the Ising model, since in this limit the fluctuation averages are independent from point to point.

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Bai, Fan; Branch, Richard W.; Nicolau, Dan V.; Pilizota, Teuta; Steel, Bradley C.; Maini, Philip K.; Berry, Richard M. (2010-02-05). "Conformational Spread as a Mechanism for Cooperativity in the Bacterial Flagellar Switch".

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El-Showk, Sheer; Paulos, Miguel F.; Poland, David; Rychkov, Slava; Simmons-Duffin, David; Vichi, Alessandro (2014). "Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents".

14424: 6780: 4433:. For high, but not infinite temperature, there are small correlations between neighboring positions, the snow tends to clump a little bit, but the screen stays randomly looking, and there is no net excess of black or white. 3002:{\displaystyle {\begin{aligned}H(\sigma )&=-\sum _{ij\in E(V^{+})}J_{ij}-\sum _{ij\in E(V^{-})}J_{ij}+\sum _{ij\in \delta (V^{+})}J_{ij}\\&=-\sum _{ij\in E(G)}J_{ij}+2\sum _{ij\in \delta (V^{+})}J_{ij},\end{aligned}}} 3351: 1639:

In a ferromagnetic Ising model, spins desire to be aligned: the configurations in which adjacent spins are of the same sign have higher probability. In an antiferromagnetic model, adjacent spins tend to have opposite signs.

20313: 19749:, which is a multiplicative probability cost for adding one more atom. A multiplicative factor in probability can be reinterpreted as an additive term in the logarithm – the energy. The extra energy of a configuration with 18571: 1151: 9747: 5145: 1994: 14828:

term is less and less important at longer and longer wavelengths. The dimension at which nonquadratic contributions begin to contribute is known as the critical dimension. In the Ising model, the critical dimension is 4.

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greater than or equal to 2, the free energy is a continuous function of temperature T. Further analysis of the free energy indicates that it exhibits an unusual discontinuous first derivative at the critical temperature

14841: 19278:, consider the four neighbors of an odd site. These are the only spins which interact with it. The multiplicative contribution to the partition function from the sum over the two values of the spin at the odd site is: 18034: 12410: 9978: 20197:

But the activity of each neuron in this model is statistically independent. To allow for pair correlations, when one neuron tends to fire (or not to fire) along with another, introduce pair-wise lagrange multipliers:

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Zhu, Wei; Han, Chao; Huffman, Emilie; Hofmann, Johannes S.; He, Yin-Chen (2023). "Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization".

10341: 16246: 15054:, since the free energy is the analytic continuation of the quantum field action for a free scalar field. For dimensions 5 and higher, all the other correlation functions at long distances are then determined by 8126: 20088:

In biological systems, modified versions of the lattice gas model have been used to understand a range of binding behaviors. These include the binding of ligands to receptors in the cell surface, the binding of

18239: 1033:. Note that the sign in the second term of the Hamiltonian above should actually be positive because the electron's magnetic moment is antiparallel to its spin, but the negative term is used conventionally. The 10838:

In the nearest-neighbor case, the free energy was exactly computed by Onsager, through the equivalence of the model with free fermions on lattice. The spin-spin correlation functions were computed by McCoy and

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between the spin we choose to "flip" to move to the new state and that spin's neighbor. Thus, in a 1D Ising model, where each site has two neighbors (left and right), the maximum difference in energy would be

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To restore the old cutoff, perform a partial integration over all the wavenumbers which used to be forbidden, but are now fluctuating. In Feynman diagrams, integrating over a fluctuating mode at wavenumber

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The numerical factors are there to simplify the equations of motion. The goal is to understand the statistical fluctuations. Like any other non-quadratic path integral, the correlation functions have a

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approximations can be created using the Ising model; sea ice topography data bears rather heavily on the results. The state variable is binary for a simple 2D approximation, being either water or ice.

12862: 10942: 9389: 7546: 6770:{\displaystyle f(\beta ,h)=-\lim _{L\to \infty }{\frac {1}{\beta L}}\ln Z(\beta )=-{\frac {1}{\beta }}\ln \left(e^{\beta J}\cosh \beta h+{\sqrt {e^{2\beta J}(\sinh \beta h)^{2}+e^{-2\beta J}}}\right),} 21535: 16833: 3797: 655: 11972: 6061:

only depends on the value of the spin and its nearest graph neighbors. So if the graph is not too connected, the algorithm is fast. This process will eventually produce a pick from the distribution.

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While the laws of chemical bonding made it clear to nineteenth century chemists that atoms were real, among physicists the debate continued well into the early twentieth century. Atomists, notably

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The energy of a droplet of plus spins in a minus background is proportional to the perimeter of the droplet L, where plus spins and minus spins neighbor each other. For a droplet with perimeter

2709: 24134:"Statistical mechanics, three-dimensionality and NP-completeness. I. Universality of intractability for the partition function of the Ising model across non-planar surfaces (extended abstract)" 15823: 12120: 4837: 4686: 3143: 4359:

of the atoms correctly describes room temperature gases. But classical statistical mechanics did not account for all of the properties of liquids and solids, nor of gases at low temperature.

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with wavenumbers larger than λ are not allowed to fluctuate. A rescaling of length that make the whole system smaller increases all wavenumbers, and moves some fluctuations above the cutoff.

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to the thermodynamic limit is fast, so that the phase behavior is apparent already on a relatively small lattice, even though the singularities are smoothed out by the system's finite size.

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are dimensionless, this is misleading. The long wavelength statistical fluctuations are not exactly scale invariant, and only become scale invariant when the interaction strength vanishes.

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encountering a flip. The domains never become infinite, so a long system is never magnetized. Each step reduces the correlation between a spin and its neighbor by an amount proportional to

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are damped because the mean field restores the fluctuation to zero field. Below the critical temperature, the mean field is driven to a new equilibrium value, which is either the positive

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The idea is that if each spin is connected to a large number of spins, only the average ratio of + spins to − spins is important, since the fluctuations about this mean will be small. The

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In dimensions above 4, the critical fluctuations are described by a purely quadratic free energy at long wavelengths. This means that the correlation functions are all computable from as

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which goes to zero at large β. For β sufficiently large, this exponentially suppresses long loops, so that they cannot occur, and the magnetization never fluctuates too far from −1.

420:. The Ising model for greater dimensions was also explored with respect to various tree topologies in the late 1970s, culminating in an exact solution of the zero-field, time-independent 18365: 17621: 6379: 6245: 6175: 4625:

A bound on the total correlation is given by the contribution to the correlation by summing over all paths linking two points, which is bounded above by the sum over all paths of length

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In the usual Ising basis, acting on any linear combination of past configurations, it produces the same linear combination but with the spin at position i of each basis vector flipped.

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The exponent is also universal, since it is the same in the Ising model as in the experimental magnet and gas, but it is not equal to the mean-field value. This was a great surprise.

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have been derived rigorously for the Ising spin correlations (for general lattice structures), which have enabled mathematicians to study the Ising model both on and off criticality.

3435: = 1 case, which can be thought of as a linear horizontal lattice where each site only interacts with its left and right neighbor. In one dimension, the solution admits no 1203: 597: 20878: 16354: 12211: 4886: 20195: 16521:

argued that this implies that the critical Ising fluctuations in dimensions higher than 4 should be described by a free field. This argument eventually became a mathematical proof.

15787: 14183: 10763: 9019: 18628: 12718:{\displaystyle \sigma _{i}^{x}=D{\psi ^{\dagger }}_{i}\psi _{i+1}+D^{*}{\psi ^{\dagger }}_{i}\psi _{i-1}+C\psi _{i}\psi _{i+1}+C^{*}{\psi ^{\dagger }}_{i}{\psi ^{\dagger }}_{i+1}.} 9680: 8890:

is the volume of configurations, each configuration weighted by its Boltzmann weight. Since each configuration is described by the sign-changes, the partition function factorizes:

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Barahona, Francisco; Grötschel, Martin; Jünger, Michael; Reinelt, Gerhard (1988). "An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design".

19655: 19577: 16679: 2488: 15720: 14338: 1436: 19934: 19767: 9473: 967: 20849: 17037: 16395: 5064:, since many questions that are posed to be solved using the model can be answered in absence of an external field. This leads us to the following energy equation for state σ: 2159: 22894:

El-Showk, Sheer; Paulos, Miguel F.; Poland, David; Rychkov, Slava; Simmons-Duffin, David; Vichi, Alessandro (2012). "Solving the 3D Ising Model with the Conformal Bootstrap".

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postulated that permanent magnets are caused by permanent atomic currents. The motion of classical charged particles could not explain permanent currents though, as shown by

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which is .333 in 3 dimensions (ε = 1) and .166 in 2 dimensions (ε = 2). This is not so far off from the measured exponent .308 and the Onsager two dimensional exponent .125.

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alone in his 1924 thesis; it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by

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A coarse model is to make space-time a lattice and imagine that each position either contains an atom or it doesn't. The space of configuration is that of independent bits

19042: 15471:{\displaystyle {\begin{aligned}&&\left(-\nabla _{x}^{2}+t\right)\langle H(x)H(y)\rangle &=0\\\rightarrow {}&&\nabla ^{2}G(x)+tG(x)&=0\end{aligned}}} 11983:

Similarly, one can divide the sum over all partition function configurations into slices, where each slice is the one-dimensional configuration at time 1. This defines the

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In the nearest neighbor case (with periodic or free boundary conditions) an exact solution is available. The Hamiltonian of the one-dimensional Ising model on a lattice of

2694: 749: 519: 20749:{\displaystyle -\beta f=\ln 2+{\frac {2\gamma }{(\gamma +1)}}\ln(\cosh J)+{\frac {\gamma (\gamma -1)}{(\gamma +1)}}\sum _{i=2}^{z}{\frac {1}{\gamma ^{i}}}\ln J_{i}(\tau )} 17283:

But λ also changes. The change in λ requires considering the lines splitting and then quickly rejoining. The lowest order process is one where one of the three lines from

8895: 3972: 1068: 21369: 16004: 5666: 3790: 3763: 1789: 1582: 1540: 546: 15532: 13143:, only even powers contribute. By reflection symmetry on a square lattice, only even powers of gradients contribute. Writing out the first few terms in the free energy: 12150: 9927: 5171:(μ, ν), which represent the probability that state ν is selected by the algorithm out of all states, given that one is in state μ. It then uses acceptance probabilities 4366:

was formulated, atomism was no longer in conflict with experiment, but this did not lead to a universal acceptance of statistical mechanics, which went beyond atomism.

3595: 3058: 1725: 1686: 1493: 21657: 21617: 21577: 20898: 10792: 805: 19250: 16688:

as particles travelling along random walks, splitting and rejoining at vertices. The interaction strength is parametrized by the classically dimensionless quantity λ.

3245: 1624: 469: 21637: 21597: 21557: 21239: 20919: 20784: 20201: 18450: 11980:

is the number of time slices. The sum over all paths is given by a product of matrices, each matrix element is the transition probability from one slice to the next.

3078: 3029: 2338: 2228: 2179: 1764: 1456: 1075: 20542: 20515: 20343: 13936:{\displaystyle \langle H(x_{1})H(x_{2})\cdots H(x_{n})\rangle ={\int DH\,e^{-\int d^{d}x\left}H(x_{1})H(x_{2})\cdots H(x_{n}) \over \int DH\,e^{-\int d^{d}x\left}}.} 10722: 8725:

of having a spin-flip at each position is independent. The ratio of the probability of finding a flip to the probability of not finding one is the Boltzmann factor:

5067: 4877:

After Onsager's solution, Yang and Lee investigated the way in which the partition function becomes singular as the temperature approaches the critical temperature.

2625: 1916: 723: 21419: 21392: 21342: 21315: 21210: 21134: 19039:

Write the two-dimensional Ising model with an infinite number of possible higher order interactions. To keep spin reflection symmetry, only even powers contribute:

16509:

coefficient by a constant. But the intersection probability of random walks depends on the dimension, and random walks in dimension higher than 4 do not intersect.

12916: 12889: 11620: 11593: 11465: 11269: 10935: 10908: 4074: 2392: 2365: 2282: 2255: 779: 24542: 18926:

are, so long as neither of them is tuned to zero, the spontaneous magnetization will grow as the square root of ε. This argument can only fail if the free energy β

4375:

which are absent in finite systems: an infinitesimal change in a parameter can lead to big differences in the overall, aggregate behavior, as Democritus expected.

2208: 1031: 20109:

down the axon in any given time window, and the inactive ones are those that do not. Because the neural activity at any one time is modelled by independent bits,

11646: 10881: 12328: 11622:

are both negative, the Ising model becomes an antiferromagnet. Since the square lattice is bi-partite, it is invariant under this change when the magnetic field

2648: 12155:

Like the Hamiltonian, the transfer matrix acts on all linear combinations of states. The partition function is a matrix function of T, which is defined by the

10812: 9742: 4553:, the amount of correlation goes as ε, but if there is more than one path by which the correlations can travel, this amount is enhanced by the number of paths. 4182: 4162: 4142: 4122: 4102: 1397: 1296: 1007: 987: 493: 18271:

is the average fraction of spins which are + minus the average fraction of spins which are −. The energy cost of flipping a single spin in the mean field

13400:. Rotational symmetry emerges spontaneously at large distances just because there aren't very many low order terms. At higher order multicritical points, this 14116:

can be used to predict which terms are most important by dimensional analysis. Dimensional analysis is not completely straightforward, because the scaling of

16484:

This is not an exact form, except in three dimensions, where interactions between paths become important. The exact forms in high dimensions are variants of

17903:

of the closed loop with a single vertex is a closed loop with two vertices. This means that the only difference between the scaling of the coupling and the

3880: 17605:{\displaystyle {\begin{aligned}{dt \over t}&=\left(2-{B\lambda \over 2}\right)b\\{d\lambda \over \lambda }&={-3B\lambda \over 2}b\end{aligned}}} 16147: 18189: 7058: 6381:

with α > 2 (which includes the case of finite-range interactions), there is no phase transition at any positive temperature (i.e. finite β), since the

22833:

Delamotte, Bertrand; Tissier, Matthieu; Wschebor, Nicolás (2016). "Scale invariance implies conformal invariance for the three-dimensional Ising model".

19670:

Variants of this method produce good numerical approximations for the critical exponents when many terms are included, in both two and three dimensions.

16496:

The interpretation of the correlations as fixed size quanta travelling along random walks gives a way of understanding why the critical dimension of the

12979: 7555: 5953:{\displaystyle A(\mu ,\nu )={\begin{cases}e^{-\beta (H_{\nu }-H_{\mu })},&{\text{if }}H_{\nu }-H_{\mu }>0,\\1&{\text{otherwise}}.\end{cases}}} 4016:

This means that spins are positively correlated on the Ising ferromagnet. An immediate application of this is that the magnetization of any set of spins

19188:

is only a function of i-j. By the accidental rotational symmetry, at large i and j its size only depends on the magnitude of the two-dimensional vector

3570:

and the system is disordered. On the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension.

25077: 19023:

But renormalization can also be productively applied to the spins directly, without passing to an average field. Historically, this approach is due to

6393:

second-order phase transition: the free energy is infinite, and the truncated two-point spin correlation does not decay (remains constant). Therefore,

10145: 5147:

Given this Hamiltonian, quantities of interest such as the specific heat or the magnetization of the magnet at a given temperature can be calculated.

4296:{\displaystyle \langle \sigma _{x}\sigma _{y}\rangle \leq \sum _{z\in S}\langle \sigma _{x}\sigma _{z}\rangle \langle \sigma _{z}\sigma _{y}\rangle .} 20105:

in the brain can be modelled statistically. Each neuron at any time is either active + or inactive −. The active neurons are those that send an

19281: 15488:

obeys classical equations of motion for the same reason that quantum mechanical operators obey them—its fluctuations are defined by a path integral.

13416: 1210: 19228:

Unrotating the system restores the old configuration, but with new parameters. These parameters describe the interaction between spins at distances

12727:

Ignore the constant coefficients, and focus attention on the form. They are all quadratic. Since the coefficients are constant, this means that the

8806: 24901: 23577: 17210: 16413: 1885: = 0, the Ising model is symmetric under switching the value of the spin in all the lattice sites; a nonzero field breaks this symmetry. 21185:

is the number of graphs counted for odd vertices with even intermediate sites (see cited methodologies and references for detailed calculations),

20762:

A Closed Cayley Tree with Branching Ratio = 4. (Only sites for generations k, k-1, and k=1(overlapping as one row) are shown for the joined trees)

14797:

To find the behavior of fluctuations, rescale the field to fix the gradient term. Then the length scaling dimension of the field is 1 −

12925:

using a limiting process of transfer matrix eigenvalues. The proof was subsequently greatly simplified in 1963 by Montroll, Potts, and Ward using

18261:. This type of description is appropriate to very-high-dimensional square lattices, because then each site has a very large number of neighbors. 13028: 12040:

has matrix elements between two configurations of spins, one in the immediate future and one in the immediate past. These two configurations are

4445: 317: 15818:

is fluctuating at a temperature slightly away from critical, the two point function decays at long distances. The equation it obeys is altered:

12752: 9320: 7449: 4425:

Shortly after Lenz and Ising constructed the Ising model, Peierls was able to explicitly show that a phase transition occurs in two dimensions.

2029: 25504: 24497: 24405: 21424: 16770: 17427:

The numerical factor is three times bigger because there is an extra factor of three in choosing which of the three new lines to contract. So

16009:

To see how, it is convenient to represent the two point function as an integral, introduced by Schwinger in the quantum field theory context:

11890: 11829:

A path integral can be rewritten as a Hamiltonian evolution. The Hamiltonian steps through time by performing a unitary rotation between time

10857:

obtained the following analytical expression for the free energy of the Ising model on the anisotropic square lattice when the magnetic field

25034: 25014: 24448: 6088:

other states, where each transition corresponds to flipping a single spin site to the opposite value. Furthermore, since the energy equation

15597: 13324: 12444: 10513:{\displaystyle {\frac {1}{N}}\ln Z_{N}(K)={\frac {1}{2}}\ln \left(2{\sqrt {\cosh(2K)}}\right)+{\frac {1}{2}}{\frac {1}{N/2}}\ln Z_{N/2}(K')} 14605:{\displaystyle H(x)^{4}=-\langle H(x)^{2}\rangle ^{2}+2\langle H(x)^{2}\rangle H(x)^{2}+\left(H(x)^{2}-\langle H(x)^{2}\rangle \right)^{2}} 9395: 6907:{\displaystyle \langle \sigma _{i}\sigma _{j}\rangle -\langle \sigma _{i}\rangle \langle \sigma _{j}\rangle =C(\beta )e^{-c(\beta )|i-j|},} 4767:

So that the total spin contribution from droplets, even overcounting by allowing each site to have a separate droplet, is bounded above by

22734:

Cosme, Catarina; Lopes, J. M. Viana Parente; Penedones, Joao (2015). "Conformal symmetry of the critical 3D Ising model inside a sphere".

14781:

argument is correct in dimensions larger than 5. The magnetization exponent in dimensions higher than 5 is equal to the mean-field value.

25418: 13387:

the metric for space. So any translationally invariant Ising model is rotationally invariant at long distances, in coordinates that make

6247:

with 1 < α < 2, Dyson proved, by comparison with the hierarchical case, that there is phase transition at small enough temperature.

6084:

simulation, and since we use single-spin-flip dynamics in the Metropolis algorithm, every state can be viewed as having links to exactly

4511:

A bogus argument analogous to the argument in the last section now establishes that the magnetization in the Ising model is always zero.

111: 21917: 13011:

This conformal field theory describing the three-dimensional Ising critical point is under active investigation using the method of the

12061: 9893:{\displaystyle Z_{N}(\beta ,J)=\sum _{\sigma }e^{K\sigma _{2}(\sigma _{1}+\sigma _{3})}e^{K\sigma _{4}(\sigma _{3}+\sigma _{5})}\cdots } 5262:

total sites on the lattice, using single-spin-flip as the only way we transition to another state, we can see that there are a total of

4770: 4632: 24459: 24444: 22643: 10250: 8887: 4384: 1275: 20458:(also known as the random-bond Ising model). These bonds stay fixed or "quenched" even in the presence of thermal fluctuations. When 18140: 17432: 16837:

This equation is an identity inside any correlation function away from other insertions. After integrating out the modes with Λ <

15021:{\displaystyle \langle S(x)S(y)\rangle \propto \langle H(x)H(y)\rangle =G(x-y)=\int {dk \over (2\pi )^{d}}{e^{ik(x-y)} \over k^{2}+t}} 14824:

This is an essential difference. In dimensions higher than 4, fixing the scale of the gradient term means that the coefficient of the

13952: 11885:

The product of the U matrices, one after the other, is the total time evolution operator, which is the path integral we started with.

9159:

To express the Ising Hamiltonian using a quantum mechanical description of spins, we replace the spin variables with their respective

25335: 9163:. However, depending on the direction of the magnetic field, we can create a transverse-field or longitudinal-field Hamiltonian. The 8730: 1495:

is conventional. Using this sign convention, Ising models can be classified according to the sign of the interaction: if, for a pair

18124:{\displaystyle {\begin{aligned}{d\lambda \over \lambda }&=\varepsilon -3B\lambda \\{dt \over t}&=2-\lambda B\end{aligned}}} 10135:{\displaystyle Z_{N}(\beta ,J)=\sum _{\sigma }(2\cosh(K(\sigma _{1}+\sigma _{3})))\cdot (2\cosh(K(\sigma _{3}+\sigma _{5})))\cdots } 8123:

can be seen as the entries of a matrix. There are different possible choices: a convenient one (because the matrix is symmetric) is

4902:

The Ising model can often be difficult to evaluate numerically if there are many states in the system. Consider an Ising model with

3590:

in 2 dimensions or more. Namely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order:

25019: 18975: 18937: 16848:

Since the form of the equation will be preserved, to find the change in coefficients it is sufficient to analyze the change in the

18372: 15284:{\displaystyle \langle S(x_{1})S(x_{2})\cdots S(x_{2n})\rangle =C^{n}\sum G(x_{i1},x_{j1})G(x_{i2},x_{j2})\ldots G(x_{in},x_{jn})} 6316:, Fröhlich and Spencer proved that there is phase transition at small enough temperature (in contrast with the hierarchical case). 6080:(ν) of transitioning to a future state ν only depends on the present state μ. The Metropolis algorithm is actually a version of a 25345: 25029: 18313: 23232:

Vtyurina, Natalia N.; Dulin, David; Docter, Margreet W.; Meyer, Anne S.; Dekker, Nynke H.; Abbondanzieri, Elio A. (2016-04-18).

20450:

are taken from a random distribution. For spin glasses a typical distribution chooses antiferromagnetic bonds with probability

20361: 18814:

on the right hand side. Changing β should only smoothly change the coefficients. The transition happens when the coefficient of

16262:-space, which is the Fourier transform of a delta function source localized at the origin. So it satisfies the same equation as 13411:

is a function of a slowly spatially varying field, the probability of any field configuration is (omitting higher-order terms):

10636: 2024:

For the Ising model without an external field on a graph G, the Hamiltonian becomes the following sum over the graph edges E(G)

25387: 25284: 22667: 18787:

can be calculated from the solution of an analytic equation which is symmetric between positive and negative values, which led

18747: 16286:

paths still appear by the mathematical correspondence with quantum fields, but their interpretation is less directly physical.

13148: 12952: 5240:; the number of nearest neighbors that any lattice site has. We assume that all sites have the same number of neighbors due to 606: 20137:. Given a collection of neurons, a statistical model which can reproduce the average firing rate for each neuron introduces a 8236:{\displaystyle V_{\sigma ,\sigma '}=e^{{\frac {\beta h}{2}}\sigma }e^{\beta J\sigma \sigma '}e^{{\frac {\beta h}{2}}\sigma '}} 7711:≠ 0 we need the transfer matrix method. For the periodic boundary conditions case is the following. The partition function is 25574: 25564: 25410: 25102: 25087: 24440:

Interactive Monte Carlo simulation of the Ising, XY and Heisenberg models with 3D graphics(requires WebGL compatible browser)

24125: 23860: 23842: 23784: 23755: 23726: 23646: 22587: 22534: 22409: 12166: 23946: 20144: 15737: 15058:. All the odd moments are zero, by ± symmetry. The even moments are the sum over all partition into pairs of the product of 14132: 25474: 25438: 21798: 12946: 19523: 16550: 25742: 25479: 22524: 17287:

splits into three, which quickly joins with one of the other lines from the same vertex. The correction to the vertex is

15679: 15313:, consider that the fields in a path integral obey the classical equations of motion derived by varying the free energy: 14237: 13124:

within one averaging volume from that region, the free energy contribution from each region only depends on the value of

12974:

In three dimensions, the Ising model was shown to have a representation in terms of non-interacting fermionic strings by

6104: 310: 23914: 12749:

of a two-dimensional Ising ferromagnet on the square lattice at two different conferences in 1948, though without proof

8539:{\displaystyle Z(\beta )=\operatorname {Tr} \left(V^{L}\right)=\lambda _{1}^{L}+\lambda _{2}^{L}=\lambda _{1}^{L}\left,} 3083: 25391: 24589: 24490: 24156: 23063: 19667:

s flow to large couplings. The magnetization exponent is determined from the slope of the equation at the fixed point.

14614:

The first term is a constant contribution to the free energy, and can be ignored. The second term is a finite shift in

3977: 145: 19435:

is the number of neighbors which are ±. Ignoring the factor of 2, the free energy contribution from this odd site is:

16856:

term in a correlation function inside a correlation has three dangling lines. Joining two of them at large wavenumber

15541: 13086:

are known, the long-distance correlations between the spins will be proportional to the long-distance correlations in

12286: 12240: 11844: 25544: 23991: 23954: 23922: 23900: 23880: 22247: 19596: 17750:

and is proportional to the area of a three-dimensional sphere of radius λ, times the width of the integration region

14404:≤ 4. Above four dimensions, at long wavelengths, the overall magnetization is only affected by the ultralocal terms. 13000: 11990: 9932: 5982: 5161: 4722: 4323:, of which includes a representation of the Ising model. It is used to determine the critical temperatures of planar 22681:

Billó, M.; Caselle, M.; Gaiotto, D.; Gliozzi, F.; Meineri, M.; others (2013). "Line defects in the 3d Ising model".

21916:

El-Showk, Sheer; Paulos, Miguel F.; Poland, David; Rychkov, Slava; Simmons-Duffin, David; Vichi, Alessandro (2014).

19438: 17882:{\displaystyle B=(2\pi ^{2}\Lambda ^{3}){1 \over (2\pi )^{4}}{b\Lambda }{1 \over b\Lambda ^{4}}={1 \over 8\pi ^{2}}} 17418:{\displaystyle \delta \lambda =-{3\lambda ^{2} \over 2}\int _{k}dk{1 \over (k^{2}+t)^{2}}=-{3\lambda ^{2} \over 2}b} 17020:{\displaystyle \delta H^{3}=3H\int _{\Lambda <|k|<(1+b)\Lambda }{d^{4}k \over (2\pi )^{4}}{1 \over (k^{2}+t)}} 14618:. The third term is a quantity that scales to zero at long distances. This means that when analyzing the scaling of 13109:) is defined to be the sum over all Ising configurations which are consistent with the long wavelength field. Since 8371:{\displaystyle V={\begin{bmatrix}e^{\beta (h+J)}&e^{-\beta J}\\e^{-\beta J}&e^{-\beta (h-J)}\end{bmatrix}}.} 4567: 25773: 25589: 25395: 25379: 25294: 25122: 25092: 24514: 24417:

by Dirk Brockman, is an interactive simulation that allows users to export the working code to a presentation slide

23591:

Yi-Ping Ma; Ivan Sudakov; Courtenay Strong; Kenneth Golden (2017). "Ising model for melt ponds on Arctic sea ice".

19593:. Since each link will contribute to two odd spins, the right value to compare with the previous one is half that: 18795: 14100:, the difference being that all the derivative terms enter with a positive sign, and there is no overall factor of 11234:{\displaystyle -\beta f=\ln 2+{\frac {1}{8\pi ^{2}}}\int _{0}^{2\pi }d\theta _{1}\int _{0}^{2\pi }d\theta _{2}\ln.} 153: 73: 24382: 23354:

Amari, Shun-Ichi (1972). "Learning patterns and pattern sequences by self-organizing nets of threshold elements".

20901:

rapidly (i.e. for z → ∞, it remains finite) yielding a continuous and monotonous function, establishing that, for

16006:, the solution diverges exactly the same way as in the critical case, but the long distance behavior is modified. 14649:

The magnetization is at the minimum of the free energy, and this is an analytic equation. In terms of the shifted

11660:

Start with an analogy with quantum mechanics. The Ising model on a long periodic lattice has a partition function

5164:

is the most commonly used Monte Carlo algorithm to calculate Ising model estimations. The algorithm first chooses

3868:{\displaystyle \langle \sigma _{A}\sigma _{B}\rangle \geq \langle \sigma _{A}\rangle \langle \sigma _{B}\rangle ,} 3360:

A significant number of statistical questions to ask about this model are in the limit of large numbers of spins:

25494: 25459: 25428: 25423: 24859: 24776: 24007: 21845: 21723: 18029:

In dimension 4 − ε, the coupling λ has positive scale dimension ε, and this must be added to the flow.

12996: 6322: 6188: 6118: 4019: 3197: 2400: 931:{\displaystyle H(\sigma )=-\sum _{\langle ij\rangle }J_{ij}\sigma _{i}\sigma _{j}-\mu \sum _{j}h_{j}\sigma _{j},} 24002: 23959: 23299:

J. J. Hopfield (1982), "Neural networks and physical systems with emergent collective computational abilities",

15508: 25778: 25433: 25062: 25057: 24864: 24761: 24424: 23913:, Vol. I, "Superflow and Vortex Lines", pp. 1–742, Vol. II, "Stresses and Defects", pp. 743–1456, 21244: 16121:{\displaystyle G(x)=\int d\tau {1 \over \left({\sqrt {2\pi \tau }}\right)^{d}}e^{-{x^{2} \over 4\tau }-t\tau }} 13065:

moves. These fluctuations in the field are described by a continuum field theory in the infinite system limit.

12439:

counts the number of spin flips, which we can write in terms of spin-flip creation and annihilation operators:

11822:. This is an independent sum over all the values that the spins can take at each time slice. This is a type of 5266:

new states ν from our present state μ. The algorithm assumes that the selection probabilities are equal to the

347: 303: 20069:

For lattices where every site has an equal number of neighbors, this is the Ising model with a magnetic field

14801:/2. Now the field has constant quadratic spatial fluctuations at all temperatures. The scale dimension of the 8679:{\displaystyle \lambda _{1}=e^{\beta J}\cosh \beta h+{\sqrt {e^{2\beta J}(\sinh \beta h)^{2}+e^{-2\beta J}}},} 1158: 551: 25788: 25747: 25524: 25360: 25259: 25244: 24783: 24656: 24572: 24483: 23677: 20854: 20060:{\displaystyle E=-{\frac {1}{2}}\sum _{\langle i,j\rangle }JS_{i}S_{j}-{\frac {1}{2}}\sum _{i}(4J-\mu )S_{i}} 16707:

in such a way that the short wavelength cutoff remains fixed, the long-wavelength fluctuations are modified.

16300: 12435:

Just as in the one-dimensional case, we will shift attention from the spins to the spin-flips. The σ term in

10883:

in the thermodynamic limit as a function of temperature and the horizontal and vertical interaction energies

9298:{\displaystyle H(\sigma )=-J\sum _{i=1,\ldots ,L}\sigma _{i}^{z}\sigma _{i+1}^{z}-h\sum _{i}\sigma _{i}^{x}.} 4889:

Quench of an Ising system on a two-dimensional square lattice (500 × 500) with inverse temperature

3692:

of the Ising model are determined by a noninteracting lattice fermion. Onsager announced the formula for the

3194:

thus turns the Ising problem without an external field into a graph Max-Cut problem maximizing the cut size

25519: 25399: 21918:"Solving the 3d Ising Model with the Conformal Bootstrap II. C -Minimization and Precise Critical Exponents" 13015:. This method currently yields the most precise information about the structure of the critical theory (see 10727: 25529: 24414: 21728: 20134: 19687:. Iron is magnetic; once it is magnetized it stays magnetized for a long time compared to any atomic time. 15969:{\displaystyle \nabla ^{2}G+tG=0\to {1 \over r^{d-1}}{d \over dr}\left(r^{d-1}{dG \over dr}\right)+tG(r)=0} 13032: 12975: 9635: 9164: 6257: 5241: 4711:, but this contributes to the partition function multiplied by the total number of droplets with perimeter 2000: 472: 433: 78: 25534: 25170: 19581:

From nearest neighbor couplings, the difference in energy between all spins equal and staggered spins is 8

12277:

Define a second operator which multiplies the basis vector by +1 and −1 according to the spin at position

8710:

times the number of sign changes that are encountered when scanning the configuration from left to right.

8088: 4692:

spins are plus. For this to happen, droplets of plus spin must be able to congeal to make the plus state.

3402:

The most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a

25132: 24716: 24661: 24577: 23672: 23543:, VB Andreichenko, and VS Dotsenko (1990), "The critical behaviour of the two-dimensional dilute model", 21788: 11385:{\displaystyle \sinh \left({\frac {2J_{1}}{kT_{c}}}\right)\sinh \left({\frac {2J_{2}}{kT_{c}}}\right)=1.} 4857: 1405: 68: 63: 25464: 19879: 16273:τ is that the two point function is the sum over all random walk paths that link position 0 to position 11797:{\displaystyle \sum _{\{S\}}\exp {\biggl (}\sum _{ij}S_{i,j}\left(S_{i,j+1}+S_{i+1,j}\right){\biggr )}.} 4940:

different states that are possible. This motivates the reason for the Ising model to be simulated using

4344:

not atomic, and that the large-scale properties of matter are not reducible to basic atomic properties.

943: 25469: 25454: 25097: 25067: 24634: 24532: 23931: 23927: 21783: 21733: 20821: 17202: 16363: 12215:

Since this is a matrix equation, it can be evaluated in any basis. So if we can diagonalize the matrix

10849: 5255:

Specifically for the Ising model and using single-spin-flip dynamics, one can establish the following.

3561:{\displaystyle \langle \sigma _{i}\sigma _{j}\rangle _{\beta }\leq C\exp \left(-c(\beta )|i-j|\right),} 2128: 1651:

interacts with the external field. Namely, the spin site wants to line up with the external field. If:

440: 375: 83: 21102:{\displaystyle \langle s_{m}s_{n}\rangle ={Z_{N}}^{-1}(0,T)^{N_{b}}2^{N}\sum _{l=1}^{z}g_{mn}(l)t^{l}} 10626:{\displaystyle f(\beta )={\frac {1}{2}}\ln \left(2{\sqrt {\cosh(2K)}}\right)+{\frac {1}{2}}f(\beta ')} 9307:

The transverse-field model experiences a phase transition between an ordered and disordered regime at

3676:

The Ising model on a two-dimensional square lattice with no magnetic field was analytically solved by

393:), who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by 25549: 25350: 25264: 25249: 25180: 24756: 24639: 24537: 20789: 18783:

grows near the critical point as the square root of the change in temperature. This is true whenever

16542: 14207:

To find the critical point, lower the temperature. As the temperature goes down, the fluctuations in

12743: 11823: 9685: 6554: 6382: 5199:

a system's energy only fluctuates within a small range. This is the motivation behind the concept of

5053:{\displaystyle H(\sigma )=-J\sum _{\langle i~j\rangle }\sigma _{i}\sigma _{j}-h\sum _{j}\sigma _{j}.} 3693: 3689: 663: 24402: 23853:

Statistical field theory, Volume 1: From Brownian motion to renormalization and lattice gauge theory

22107:"A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model" 21139: 19589:. Ignoring four-spin interactions, a reasonable truncation is the average of these two energies or 6 17741:{\displaystyle Bb=\int _{\Lambda <|k|<(1+b)\Lambda }{d^{4}k \over (2\pi )^{4}}{1 \over k^{4}}} 5839: 3148: 2446: 2287: 25383: 25269: 24771: 24746: 24691: 22668:"Ken Wilson recalls how Murray Gell-Mann suggested that he solve the three-dimensional Ising model" 21743: 21684:

domain of chemistry has historically integrated itself into quantum physics via quantum chemistry.

20122: 16134:, since the Fourier transform of this integral is easy. Each fixed τ contribution is a Gaussian in 13016: 11397: 6530:{\displaystyle H(\sigma )=-J\sum _{i=1,\ldots ,L-1}\sigma _{i}\sigma _{i+1}-h\sum _{i}\sigma _{i},} 6081: 2653: 728: 498: 137: 24397: 22264: 14756:{\displaystyle {\partial \over \partial H}\left(tH^{2}+\lambda H^{4}\right)=2tH+4\lambda H^{3}=0} 7048:{\displaystyle H(\sigma )=-J\left(\sigma _{1}\sigma _{2}+\cdots +\sigma _{L-1}\sigma _{L}\right).} 3951: 1047: 25783: 25684: 25674: 25489: 25365: 25147: 25072: 24886: 24751: 24607: 24562: 24415:“I sing well-tempered” The Ising Model: A simple model for critical behavior in a system of spins 24280:; Glasser, M.L. (1983), "Exact Phase Transitions for the Ising Model on the Closed Cayley Tree", 23667: 21772: 21347: 19691: 15985: 14821:

term has positive scale dimension. In dimensions higher than 4 it has negative scale dimensions.

12956: 6949:= 0, it is very easy to obtain the free energy in the case of free boundary condition, i.e. when 3768: 3741: 3398:

Visualization of the translation-invariant probability measure of the one-dimensional Ising model

1776:

Ising models are often examined without an external field interacting with the lattice, that is,

1551: 1509: 524: 363: 199: 121: 45: 20758: 20482: 14353:

is vanishing, fixing the scale of the field using this term makes the other terms blow up. Once

13008:

of the phase transition, which agree with the experiments and with the Monte Carlo simulations.

12971:

Such a solution has not been found until now, although there is no proof that it may not exist.

12930: 12127: 9903: 4529:

The system should therefore spend equal amounts of time in the configuration with magnetization

4305:

This inequality can be used to establish the sharpness of phase transition for the Ising model.

3701: 3034: 1697: 1658: 1469: 25626: 25554: 24979: 24969: 24813: 21642: 21602: 21562: 20883: 12156: 11984: 10768: 9134:{\displaystyle \beta f=\log(1+p)=\log \left(1+{\frac {e^{-2\beta J}}{1+e^{-2\beta J}}}\right),} 3721: 1038: 940:

where the first sum is over pairs of adjacent spins (every pair is counted once). The notation

788: 406: 18727:{\displaystyle H=\tanh(\beta JH)\approx (1+\varepsilon )H-{(1+\varepsilon )^{3}H^{3} \over 3}} 18137:

is dimension dependent, but it will cancel. The fixed point for λ is no longer zero, but at:

12054:, and they are all one-dimensional spin configurations. We can think of the vector space that 12036:

The configuration in each slice is a one-dimensional collection of spins. At each time slice,

9151:

is a non-analytic free energy, so the one-dimensional model does not have a phase transition.

8706:, when all the spins are the same. For any other configuration, the extra energy is equal to 2 1593: 454: 25649: 25631: 25611: 25606: 25325: 25157: 25137: 24984: 24927: 24766: 24676: 22604: 21966: 21622: 21582: 21542: 21215: 20904: 20769: 19231: 15496: 14833: 11649: 4951:

that is commonly used to represent the energy of the model when using Monte Carlo methods is

4948: 4845: 4356: 3063: 3014: 2588:{\displaystyle \left|\delta (V^{+})\right|={\frac {1}{2}}\sum _{ij\in \delta (V^{+})}W_{ij},} 2323: 2213: 2164: 2010: 1736: 1441: 355: 216: 22: 24065:(1944), "Crystal statistics. I. A two-dimensional model with an order-disorder transition", 23474:"Weak pairwise correlations imply strongly correlated network states in a neural population" 22399: 22348:"General transfer matrix formalism to calculate DNA-protein-drug binding in gene regulation" 21212:

is the multiplicity resulting from two-valued spin possibilities and the partition function

20520: 20493: 20318: 19690:

In the 19th century, it was thought that magnetic fields are due to currents in matter, and

18798:

because changes in the character of solutions of analytic equations are always described by

17479:

These two equations together define the renormalization group equations in four dimensions:

10707: 2600: 2017:

subset G\S. The size of the cut is the sum of the weights of the edges between S and G\S. A

1438:

represent the probability that (in equilibrium) the system is in a state with configuration

698: 25768: 25724: 25679: 25669: 25355: 25330: 25299: 25279: 25117: 25039: 25024: 24891: 24309: 24257: 24229: 24213: 24193: 24166: 24094: 24074: 24044: 24016: 23870: 23802: 23765: 23693: 23656: 23552: 23495: 23446: 23403: 23308: 23245: 23182: 23138: 23020: 23001:

Simmons-Duffin, David (2015). "A semidefinite program solver for the conformal bootstrap".

22967: 22913: 22852: 22753: 22700: 22544: 22489: 22438: 22128: 22063: 21999: 21942: 21397: 21374: 21320: 21293: 21188: 21112: 19867:{\displaystyle E=-{\frac {1}{2}}\sum _{\langle i,j\rangle }4JB_{i}B_{j}+\sum _{i}\mu B_{i}} 18791:

to suspect that all Ising type phase transitions in all dimensions should follow this law.

16765:

The lowest order effect of integrating out can be calculated from the equations of motion:

14097: 13318:

are isotropic in a coordinate system where the different directions of space are rescaled.

13113:

is a coarse description, there are many Ising configurations consistent with each value of

12934: 12894: 12867: 11598: 11571: 11443: 11247: 10913: 10886: 10828: 9599:{\displaystyle H(\sigma )=-h\sum _{i=1,\ldots ,L}T_{i}^{z}T_{i+1}^{z}-J\sum _{i}T_{i}^{x}.} 4872: 4367: 4320: 4052: 3733: 3705: 3685: 2370: 2343: 2260: 2233: 808: 757: 410: 259: 24387: 19585:. The difference in energy between all spins equal and nonstaggered but net zero spin is 4 18257:

The behavior of an Ising model on a fully connected graph may be completely understood by

17157:{\displaystyle \int dk{1 \over k^{2}}-t\int dk{1 \over k^{2}(k^{2}+t)}=A\Lambda ^{2}b+Bbt} 13304:

of the derivative terms are all equal. But even for an anisotropic Ising model, where the

13120:

Since the allowed range of values of the spin in any region only depends on the values of

4515:

Every configuration of spins has equal energy to the configuration with all spins flipped.

3385:

On a lattice Λ, what is the fractal dimension of the shape of a large cluster of +1 spins?

2184: 1016: 366:

that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a

8: 25719: 25559: 25484: 25289: 25049: 24959: 24849: 24453: 23735: 21713: 20138: 18909:{\displaystyle H={\partial (\beta F) \over \partial h}=(1+A\varepsilon )H+BH^{3}+\cdots } 17277: 17029:

The factor of 3 comes from the fact that the loop can be closed in three different ways.

13401: 13012: 11625: 10860: 5196: 4941: 4348: 2013:

set V(G) of a weighted undirected graph G determines a cut of the graph G into S and its

1042: 416:

In dimensions greater than four, the phase transition of the Ising model is described by

254: 244: 56: 24313: 24261: 24233: 24197: 24078: 24020: 23806: 23697: 23556: 23499: 23471: 23450: 23407: 23312: 23249: 23186: 23142: 23024: 22971: 22917: 22856: 22757: 22704: 22493: 22442: 22132: 22067: 22003: 21946: 18020:{\displaystyle G(x-y)=\int d\tau {1 \over t^{d \over 2}}e^{{x^{2} \over 2\tau }+t\tau }} 12058:

acts on as all complex linear combinations of these. Using quantum mechanical notation:

2630: 25689: 25654: 25569: 25539: 25370: 25309: 25304: 25127: 24964: 24629: 24567: 24506: 24434: 24370: 24337: 24220:

Glasser, M. L. (1970), "Exact Partition Function for the Two-Dimensional Ising Model",

24170: 23818: 23592: 23516: 23485: 23419: 23374: 23276: 23233: 23214: 23154: 23128: 23044: 23010: 22983: 22957: 22929: 22903: 22876: 22842: 22815: 22797: 22769: 22743: 22716: 22690: 22505: 22454: 22372: 22347: 22302: 22276: 22201: 22172:"The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1" 22152: 22118: 22087: 22015: 21958: 21932: 21898: 20126: 19746: 18799: 14778: 13005: 11559:{\displaystyle T_{c}={\frac {2J}{k\ln(1+{\sqrt {2}})}}=(2.269185\cdots ){\frac {J}{k}}} 10797: 9727: 4355:, applied Hamilton's formulation of Newton's laws to large systems, and found that the 4167: 4147: 4127: 4107: 4087: 1382: 1370:{\displaystyle \langle f\rangle _{\beta }=\sum _{\sigma }f(\sigma )P_{\beta }(\sigma )} 1281: 992: 972: 478: 359: 129: 24133: 23636: 23331: 19172:{\displaystyle E=\sum _{ij}J_{ij}S_{i}S_{j}+\sum J_{ijkl}S_{i}S_{j}S_{k}S_{l}\ldots .} 18443: − 1. But this average is the same for all spins, and is therefore equal to 16266:

with the same boundary conditions that determine the strength of the divergence at 0.

15055: 14228:

can contribute, but since the first term dominates at long distances, the coefficient

11394:

In the isotropic case when the horizontal and vertical interaction energies are equal

6549:

is a constant representing the interaction strength between the nearest neighbors and

3938:{\displaystyle \langle \sigma _{A}\rangle =\langle \prod _{j\in A}\sigma _{j}\rangle } 3364:

In a typical configuration, are most of the spins +1 or −1, or are they split equally?

25709: 24922: 24839: 24808: 24701: 24681: 24671: 24527: 24522: 24365: 24321: 24293: 24269: 24152: 24121: 24032: 23987: 23970:(1920), "Beiträge zum Verständnis der magnetischen Eigenschaften in festen Körpern", 23950: 23918: 23896: 23876: 23856: 23838: 23822: 23780: 23751: 23722: 23642: 23624: 23571: 23564: 23521: 23473: 23336: 23281: 23263: 23218: 23206: 23198: 23036: 22987: 22868: 22819: 22720: 22624: 22583: 22530: 22509: 22458: 22405: 22377: 22306: 22294: 22243: 22193: 22156: 22144: 22079: 22019: 21962: 21890: 21708: 20490:

In order to investigate an Ising model with potential relevance for large (e.g. with

18258: 18252: 16513: 10832: 10831:, also when longer range interactions are added. Meanwhile, at high temperature, the 9144: 9001:{\displaystyle Z=\sum _{\text{configs}}e^{\sum _{k}S_{k}}=\prod _{k}(1+p)=(1+p)^{L}.} 4399: 4363: 2650:

and the scaling 1/2 is introduced to compensate for double counting the same weights

1585: 417: 279: 221: 25514: 25165: 23590: 23423: 23158: 23048: 22933: 22880: 22773: 22205: 22091: 13022: 9468:

Upon rewriting the Hamiltonian in terms of this change-of-basis matrices, we obtain

5765:{\displaystyle {\frac {A(\mu ,\nu )}{A(\nu ,\mu )}}=e^{-\beta (H_{\nu }-H_{\mu })}.} 4844:

So Peierls established that the magnetization in the Ising model eventually defines

4707:/4) (the square box). The probability cost for introducing a droplet has the factor 1872:{\displaystyle H(\sigma )=-\sum _{\langle i~j\rangle }J_{ij}\sigma _{i}\sigma _{j}.} 25729: 25616: 25499: 25375: 25112: 24869: 24844: 24793: 24644: 24597: 24347: 24317: 24289: 24265: 24237: 24201: 24174: 24144: 24082: 24024: 23810: 23743: 23715: 23701: 23560: 23538: 23511: 23503: 23454: 23411: 23326: 23316: 23271: 23253: 23190: 23146: 23028: 22975: 22921: 22860: 22807: 22761: 22708: 22616: 22575: 22497: 22446: 22367: 22359: 22286: 22183: 22136: 22071: 22007: 21950: 21882: 21758: 21703: 20106: 16485: 15500: 12926: 10824: 9148: 5279: 5176: 5060:

Furthermore, the Hamiltonian is further simplified by assuming zero external field

4372: 4352: 3709: 3656:{\displaystyle \langle \sigma _{i}\sigma _{j}\rangle _{\beta }\geq c(\beta )>0.} 3587: 3579: 3436: 2014: 379: 371: 274: 249: 98: 24721: 24392: 21539:

The critical temperature for this model is only determined by the branching ratio

21317:

is consistent with the referenced literature in this section and is equivalent to

18930:

is either non-analytic or non-generic at the exact β where the transition occurs.

18595:= 1 the mean-field Ising model undergoes a phase transition in the limit of large 14192:

at any point fluctuates completely independently of the value at any other point.

25694: 25594: 25579: 25340: 25274: 24952: 24896: 24879: 24624: 24463: 24409: 24209: 24162: 24090: 24048: 24040: 23998: 23761: 23652: 23373:

Schmidhuber, Juergen (2022). "Annotated History of Modern AI and Deep Learning".

22540: 22473: 22426: 21803: 21748: 19699: 17899:

flow and in the coupling flow. The reason is that the derivative with respect to

16685: 14357:

is small, the scale of the field can either be set to fix the coefficient of the

13040: 12983: 3346:{\displaystyle H(\sigma )=\sum _{ij\in E(G)}W_{ij}-4\left|\delta (V^{+})\right|.} 2480: 1010: 367: 264: 226: 206: 25509: 24741: 24470: 24420: 24300:

Glasser, M. L.; Goldberg, M. (1983), "The Ising model on a closed Cayley tree",

23032: 22765: 22712: 22326: 20308:{\displaystyle E=-{\tfrac {1}{2}}\sum _{ij}J_{ij}S_{i}S_{j}-\sum _{i}h_{i}S_{i}} 18566:{\displaystyle H=2p-1={1-e^{-2\beta JH} \over 1+e^{-2\beta JH}}=\tanh(\beta JH)} 10823:

In the ferromagnetic case there is a phase transition. At low temperature, the

10240:{\displaystyle Ae^{K'\sigma _{1}\sigma _{3}}=2\cosh(K(\sigma _{1}+\sigma _{3}))} 5663:

Thus, we want to select the acceptance probability for our algorithm to satisfy

4327:

using percolation arguments (which includes the Ising model as a special case).

1146:{\displaystyle P_{\beta }(\sigma )={\frac {e^{-\beta H(\sigma )}}{Z_{\beta }}},} 432:

The Ising problem without an external field can be equivalently formulated as a

25699: 25664: 25584: 25190: 24937: 24854: 24823: 24818: 24798: 24788: 24731: 24706: 24686: 24651: 24619: 24602: 24429: 24351: 24277: 24003:"Correlations and spontaneous magnetization of the two-dimensional Ising model" 23938: 23906: 23616: 22925: 22864: 22811: 22290: 21763: 21753: 20110: 19684: 16281:

is Gaussian, but the random walkers disappear at a steady rate proportional to

14411:

is fluctuating statistically, and the fluctuations can shift the zero point of

14400:/2. In this scaling, the gradient term is only important at long distances for 13380: 9160: 5140:{\displaystyle H(\sigma )=-J\sum _{\langle i~j\rangle }\sigma _{i}\sigma _{j}.} 4414: 4314: 3666: 1989:{\displaystyle H(\sigma )=-J\sum _{\langle i~j\rangle }\sigma _{i}\sigma _{j}.} 351: 269: 176: 88: 40: 24726: 23872:

Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction

23747: 23705: 22979: 22579: 22548: 22188: 22171: 22140: 22011: 21954: 14626:

that is important. This was historically very confusing, because the shift in

1888:

Another common simplification is to assume that all of the nearest neighbors ⟨

25762: 25601: 25142: 24974: 24932: 24874: 24696: 24612: 24552: 24036: 23267: 23202: 23150: 23040: 22628: 22197: 22148: 22083: 21894: 19695: 18803: 18575:

The solutions to this equation are the possible consistent mean fields. For β

16735:

in a correlation function in pairs, with a factor of the inverse propagator.

16518: 10827:

proves positive magnetization for the nearest neighbor case and then, by the

6397:= 0 is the critical temperature of this case. Scaling formulas are satisfied. 5970:(μ, ν) and calculate the contribution to the energy involving this spin. 5181: 4430: 3583: 1543: 24435:

Three-dimensional proof for Ising Model impossible, Sandia researcher claims

24141:

Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing

23458: 23437:

Jaynes, Edwin T. (1957), "Information Theory and Statistical Mechanics II",

23415: 23258: 23194: 23094: 23071: 22106: 18591:

stable solution. This means that for temperatures below the critical value β

17189:

This new linear term adds to the first term on the left hand side, changing

25659: 25621: 25175: 25107: 24996: 24991: 24803: 24736: 24711: 24547: 24328:

Süzen, Mehmet (2014), "Effective ergodicity in single-spin-flip dynamics",

24205: 24101: 24086: 24062: 23967: 23772: 23628: 23540: 23525: 23321: 23285: 23210: 22872: 22381: 22298: 21738: 20355: 20114: 19024: 19008: 18583:= 0. For bigger values of β there are three solutions, and the solution at 18431:

The mean value of the spin is given by averaging 1 and −1 with the weights

17201:

is the sum of the term from dimensional analysis and this second term from

13052: 12405:{\displaystyle \sum _{i}A\sigma _{i}^{x}+B\sigma _{i}^{z}\sigma _{i+1}^{z}} 6553:

is the constant external magnetic field applied to lattice sites. Then the

6100: 6070: 3677: 398: 386: 343: 289: 24184:(1952), "The spontaneous magnetization of a two-dimensional Ising model", 24148: 23619:(1981), "Cooperativity and the Transition Behavior of Large Neural Nets", 23340: 21886: 17170:, and in the equation of motion it can be absorbed by a constant shift in 9632:

When there is no external field, we can derive a functional equation that

4387:

could never describe a phase transition, based on the following argument:

2340:-depended set of edges that connects the two complementary vertex subsets 2115:{\displaystyle H(\sigma )=-\sum _{ij\in E(G)}J_{ij}\sigma _{i}\sigma _{j}} 25704: 25239: 25223: 25218: 25213: 25203: 25006: 24947: 24906: 24666: 24557: 24374: 23663: 23119:

Shi, Y.; Duke, T. (1998-11-01). "Cooperative model of bacteril sensing".

22363: 21768: 21698: 20544:

interactions per node) neural nets, at the suggestion of Krizan in 1979,

16762:. The other two coefficients are dimensionless and do not change at all. 13132:

is a sum over all regions of a local contribution, which only depends on

4324: 2018: 2003: 436: 339: 284: 186: 171: 23891: 23507: 23133: 22427:"Existence of a phase-transition in a one-dimensional Ising ferromagnet" 22052:"Correlation inequalities and the decay of correlations in ferromagnets" 20486:

An Open Cayley Tree or Branch with Branching Ratio = 2 and k Generations

12495:

The first term flips a spin, so depending on the basis state it either:

3573: 25714: 25254: 25198: 25082: 25035:

Generalized autoregressive conditional heteroskedasticity (GARCH) model

24475: 24181: 23814: 23472:

Elad Schneidman; Michael J. Berry; Ronen Segev; William Bialek (2006),

22501: 22450: 22075: 21778: 20090: 18788: 18625:

can be calculated from the Taylor expansion of the hyperbolic tangent:

18265: 15301:

is enough. It determines all the multipoint correlations of the field.

15051: 12159:

over all histories which come back to the original configuration after

8721:. Since the energy is additive in the number of flips, the probability 5192: 4336: 2125:

Here each vertex i of the graph is a spin site that takes a spin value

24241: 24028: 23490: 23394:

Jaynes, E. T. (1957), "Information Theory and Statistical Mechanics",

21902: 15480:

This is valid at noncoincident points only, since the correlations of

5206:

The maximum amount of change between the energy of the present state,

30: 25208: 24378: 22569: 22051: 21813: 21793: 20471: 16537:

The negative logarithm of the probability of any field configuration

16241:{\displaystyle G(k)=\int d\tau e^{-(k^{2}-t)\tau }={1 \over k^{2}-t}} 16138:, whose Fourier transform is another Gaussian of reciprocal width in 13036: 12937:

by treating the magnetization as the limit of correlation functions.

8790: 1784:

in the lattice Λ. Using this simplification, the Hamiltonian becomes

23597: 23234:"Hysteresis in DNA compaction by Dps is described by an Ising model" 22620: 21990:

Peierls, R.; Born, M. (1936). "On Ising's model of ferromagnetism".

20934:

vertices of the two trees (e.g. A and B), as may be determined from

18234:{\displaystyle {\tfrac {1}{2}}\left(1-{\varepsilon \over 3}\right)} 14224:. The phase transition can only happen when the subleading terms in 13097:, the free energy (log-probability) is a local analytic function of 7120:{\displaystyle \sigma '_{j}=\sigma _{j}\sigma _{j-1},\quad j\geq 2.} 5804:(ν, μ) to be 1. By this reasoning the acceptance algorithm is: 23379: 23015: 22847: 22802: 22748: 22123: 21808: 20477: 20118: 14232:

must be tuned to zero. This is the location of the critical point:

12734:

Carrying out the diagonalization produces the Onsager free energy.

7687:{\displaystyle \langle \sigma _{j}\sigma _{j+N}\rangle =\left^{N},} 6115:

The thermodynamic limit exists as long as the interaction decay is

4885: 4383:

In the early part of the twentieth century, some believed that the

3696:

for the 2-dimensional model in 1949 but did not give a derivation.

211: 24342: 24248:

Jellito, R. J. (1979), "The Ising Model on a Closed Cayley Tree",

22962: 22908: 22695: 22281: 21937: 16702:, and the cutoff defines the shortest wavelength. Fluctuations of 14195:

The scale of the field can be redefined to absorb the coefficient

12737: 12512:

Writing this out in terms of creation and annihilation operators:

10336:{\textstyle A=2{\sqrt {\cosh(2K)}},K'={\frac {1}{2}}\ln \cosh(2K)} 5247:

Open-source packages implementing these algorithms are available.

19698:. In order to have ferromagnetism, the atoms must have permanent 13023:

Istrail's NP-completeness result for the general spin glass model

9154: 8713:

If we designate the number of sign changes in a configuration as

6095:

change only depends on the nearest-neighbor interaction strength

4340: 22474:"The phase transition in the one-dimensional Ising model with 1/ 19263:

term, and discard everything else. This will generate a flow in

18806:, which is a polynomial equation. By symmetry, the equation for 17178:

term has a linear part. Only the second term, which varies from

16715:

A quick heuristic way of studying the scaling is to cut off the

14127:

is easy, since the only term that contributes is the first one,

14088:, over all the long wavelength fourier components of the spins. 13297:

On a square lattice, symmetries guarantee that the coefficients

12430: 22571:

Quantum Ising Phases and Transitions in Transverse Ising Models

22331: 21992:

Mathematical Proceedings of the Cambridge Philosophical Society

20113:

suggested in 1982 that a dynamical Ising model would provide a

20102: 19419:{\displaystyle e^{J(N_{+}-N_{-})}+e^{J(N_{-}-N_{+})}=2\cosh(J)} 19196:. The higher order coefficients are also similarly restricted. 13563:{\displaystyle P(H)\propto e^{-\int d^{d}x\left}=e^{-\beta F}.} 13117:, so long as not too much exactness is required for the match. 13051:

the general spin glass Ising model is exactly solvable only in

12955:. The spin and energy correlation functions are described by a 4880: 3394: 1265:{\displaystyle Z_{\beta }=\sum _{\sigma }e^{-\beta H(\sigma )}} 23793:

Ising, E. (1925), "Beitrag zur Theorie des Ferromagnetismus",

22265:"M. Suzen "Effective ergodicity in single-spin-flip dynamics"" 21872: 21719:

Construction of an irreducible Markov chain in the Ising model

20786:

is an arbitrary branching ratio (greater than or equal to 2),

20093:

proteins to the flagellar motor, and the condensation of DNA.

16277:

over time τ. The density of these paths at time τ at position

8877:{\displaystyle \langle S_{i}S_{j}\rangle \propto e^{-p|i-j|}.} 5973:

Flip the value of the spin and calculate the new contribution.

23684:

Brush, Stephen G. (1967). "History of the Lenz-Ising Model".

22644:"The Cartoon Picture of Magnets That Has Transformed Science" 22033: 22031: 22029: 19007:

But there it was not a surprise, because it was predicted by

18310:

that the spin is −. This ratio is the Boltzmann factor:

17280:, it is changed by an amount proportional to the value of λ. 17263:{\displaystyle \delta t=\left(2-{B\lambda \over 2}\right)bt} 16738:

Under rescaling, when the system is shrunk by a factor of (1+

16475:{\displaystyle G(r)\approx {e^{-{\sqrt {t}}r} \over r^{d-2}}} 13314:

s in different directions are different, the fluctuations in

12982:. This construction has been carried on the lattice, and the 409:, although there exist different approaches, more related to 93: 16:

Mathematical model of ferromagnetism in statistical mechanics

24439: 23742:, Texts and Monographs in Physics, Berlin: Springer-Verlag, 22568:

Suzuki, Sei; Inoue, Jun-ichi; Chakrabarti, Bikas K. (2012).

19757:. The probability cost of one more atom is a factor of exp(− 19683:

The original motivation for the model was the phenomenon of

12951:

At the critical point, the two-dimensional Ising model is a

12742:

Onsager famously announced the following expression for the

8717:, the difference in energy from the lowest energy state is 2 6064: 4049:

is increasing with respect to any set of coupling constants

1911:

in Λ. In this case the Hamiltonian is further simplified to

471:

of lattice sites, each with a set of adjacent sites (e.g. a

24460:

The Cartoon Picture of Magnets That Has Transformed Science

23834: 22946: 22893: 21915: 20432:{\textstyle H=-{\frac {1}{2}}\,\sum J_{i,k}\,S_{i}\,S_{k},} 18621:= 1 + ε, just below the critical temperature, the value of 16524: 16269:

The interpretation of the integral representation over the

12986:, conjecturally describing the critical point, is unknown. 8793:. This essentially completes the mathematical description. 5946: 4502:{\displaystyle M={\frac {1}{N}}\sum _{i=1}^{N}\sigma _{i}.} 25015:

Autoregressive conditional heteroskedasticity (ARCH) model

23301:

Proceedings of the National Academy of Sciences of the USA

22680: 22026: 18186:

The magnetization exponent is altered proportionately to:

14346:

is a parameter which goes through zero at the transition.

14096:. It is similar to the Lagrangian in of a scalar field in 12857:{\displaystyle M=\left(1-\left^{-2}\right)^{\frac {1}{8}}} 12152:

is a spin configuration of a one-dimensional Ising model.

10694:{\displaystyle \beta 'J={\frac {1}{2}}\ln \cosh(2\beta J)} 9384:{\displaystyle \sigma _{n}^{z}=\prod _{i=1}^{n}T_{i}^{x},} 7541:{\displaystyle f(\beta ,0)=-{\frac {1}{\beta }}\ln \left.} 21530:{\displaystyle T_{C}={\frac {2\epsilon }{k_{\text{B}}}}.} 19020:

dimensions, the perturbative expansion parameter is 2/3.

17907:

is the combinatorial factors from joining and splitting.

16828:{\displaystyle \nabla ^{2}H+tH=-{\lambda \over 6}H^{3}.} 4715:, which is less than the total number of paths of length 3011:

where the total sum in the first term does not depend on

650:{\displaystyle {\sigma }=\{\sigma _{k}\}_{k\in \Lambda }} 425:

relevant to large neural networks as one of its possible

24543:

Independent and identically distributed random variables

23231: 23171: 22832: 21371:

utilized above and in earlier sections; it is valued at

16505:

is just proportional to the density and only shifts the

15503:

from electrostatics. Define an electric field analog by

15050:) is the analytic continuation to imaginary time of the 11967:{\displaystyle U^{N}=(e^{iH\Delta t})^{N}=\int DXe^{iL}} 9616:

are switched, the Hamiltonian undergoes a transition at

7055:

Then the model factorizes under the change of variables

5184:, meaning all of the sites point in the same direction. 4378: 3700:

gave the first published proof of this formula, using a

3371:

is 1, what is the probability that the spin at position

3240:, which is related to the Ising Hamiltonian as follows, 1466:

The minus sign on each term of the Hamiltonian function

20454:

and ferromagnetic bonds with probability 1 −

19745:

The density of the atoms can be controlled by adding a

6777:

and the spin-spin correlation (i.e. the covariance) is

4848:, separated domains not linked by finite fluctuations. 3431:

In his 1924 PhD thesis, Ising solved the model for the

3426: 2021:

size is at least the size of any other cut, varying S.

25020:

Autoregressive integrated moving average (ARIMA) model

24469:

Simulation of the 2-dimensional Ising model in Julia:

22567: 20364: 20215: 19702:

which are not due to the motion of classical charges.

19234: 19224:

even, and even sites are only connected to odd sites.

18194: 15665:{\displaystyle \int d^{d-1}SE_{r}=\mathrm {constant} } 13372:{\displaystyle Z_{ij}\,\partial _{i}H\,\partial _{j}H} 12486:{\displaystyle \sum C\psi _{i}^{\dagger }\psi _{i}.\,} 10253: 8261: 5979:

If the new energy is more, only keep with probability

5187:

When implementing the algorithm, one must ensure that

4405:

The sum of analytic functions is an analytic function.

1892:⟩ have the same interaction strength. Then we can set 374:

as a simplified model of reality. The two-dimensional

23986:. Harvard University Press, Cambridge Massachusetts, 21645: 21625: 21605: 21585: 21565: 21545: 21427: 21400: 21377: 21350: 21323: 21296: 21247: 21218: 21191: 21142: 21115: 20940: 20907: 20886: 20857: 20824: 20792: 20772: 20564: 20523: 20496: 20321: 20204: 20147: 19941: 19882: 19770: 19599: 19526: 19441: 19284: 19259:

The simplest approximation is to keep only the usual

19045: 18978: 18940: 18824: 18750: 18631: 18453: 18375: 18316: 18192: 18143: 18037: 17926: 17760: 17624: 17487: 17435: 17295: 17213: 17040: 16876: 16773: 16553: 16416: 16366: 16303: 16150: 16017: 15988: 15826: 15740: 15682: 15600: 15544: 15511: 15321: 15078: 14844: 14661: 14427: 14240: 14135: 13955: 13584: 13419: 13327: 13151: 12897: 12870: 12755: 12518: 12447: 12331: 12289: 12243: 12169: 12130: 12064: 11993: 11893: 11847: 11668: 11628: 11601: 11574: 11475: 11446: 11400: 11279: 11250: 10945: 10916: 10889: 10863: 10800: 10771: 10730: 10710: 10639: 10526: 10349: 10148: 9981: 9935: 9906: 9750: 9730: 9688: 9638: 9476: 9459:{\displaystyle \sigma _{n}^{x}=T_{n}^{z}T_{n+1}^{z}.} 9398: 9323: 9175: 9022: 8898: 8809: 8733: 8564: 8384: 8249: 8129: 8091: 7717: 7558: 7452: 7135: 7061: 6955: 6783: 6563: 6416: 6325: 6260: 6191: 6121: 5985: 5812: 5669: 5290: 5070: 4957: 4893: = 10, starting from a random configuration 4773: 4725: 4635: 4570: 4540:

So the average magnetization (over all time) is zero.

4448: 4192: 4170: 4150: 4130: 4110: 4090: 4055: 4022: 3980: 3954: 3883: 3800: 3771: 3744: 3598: 3574:

Phase transition and exact solution in two dimensions

3465: 3248: 3200: 3151: 3086: 3066: 3037: 3017: 2707: 2656: 2633: 2603: 2491: 2449: 2403: 2373: 2346: 2326: 2290: 2263: 2236: 2216: 2187: 2167: 2131: 2032: 1919: 1792: 1739: 1700: 1661: 1596: 1554: 1512: 1472: 1444: 1408: 1385: 1306: 1284: 1213: 1161: 1078: 1050: 1019: 995: 975: 946: 818: 791: 760: 731: 701: 666: 657:

is an assignment of spin value to each lattice site.

609: 554: 527: 501: 481: 457: 22786: 22170:

Beffara, Vincent; Duminil-Copin, Hugo (2012-08-01).

22169: 22105:

Duminil-Copin, Hugo; Tassion, Vincent (2016-04-01).

22104: 16698:

The reason is that there is a cutoff used to define

16691:

Although dimensional analysis shows that both λ and

15798: 9315:. This can be shown by a mapping of Pauli matrices 4918:∈ {−1, +1}: an individual spin site on the lattice, 378:

is one of the simplest statistical models to show a

23064:"Deep Understanding Achieved on the 3d Ising Model" 22733: 18246: 17895:changes, but the same constant appears both in the 16289:The integral representation immediately shows that 14080:

is a statistical path integral. It integrates exp(β

10142:Now, since the cosh function is even, we can solve 9682:satisfies using renormalization. Specifically, let 24275: 23714: 22603:Maris, Humphrey J.; Kadanoff, Leo P. (June 1978). 21651: 21631: 21611: 21591: 21571: 21551: 21529: 21413: 21386: 21363: 21336: 21309: 21282: 21233: 21204: 21177: 21128: 21101: 20923: 20913: 20892: 20872: 20843: 20810: 20778: 20748: 20536: 20509: 20431: 20337: 20307: 20189: 20059: 19928: 19866: 19649: 19571: 19509: 19418: 19244: 19171: 19034: 18997: 18959: 18908: 18818:on the right hand side is 1. Near the transition: 18769: 18726: 18565: 18421: 18359: 18233: 18167: 18123: 18019: 17881: 17740: 17604: 17469: 17417: 17262: 17156: 17019: 16827: 16673: 16491: 16474: 16389: 16348: 16250:This is the inverse of the operator ∇ − 16240: 16120: 15998: 15968: 15781: 15714: 15664: 15562: 15526: 15470: 15304: 15283: 15020: 14755: 14604: 14332: 14177: 14123:In the generic case, choosing the scaling law for 14068: 13935: 13562: 13371: 13287: 12968:to Onsager's solution in the two-dimensional case. 12918:are horizontal and vertical interaction energies. 12910: 12883: 12856: 12731:matrix can be diagonalized by Fourier transforms. 12717: 12485: 12404: 12310: 12264: 12205: 12144: 12115:{\displaystyle |A\rangle =\sum _{S}A(S)|S\rangle } 12114: 12026: 11966: 11875: 11796: 11640: 11614: 11587: 11558: 11459: 11432: 11384: 11263: 11233: 10929: 10902: 10875: 10806: 10786: 10757: 10716: 10693: 10625: 10512: 10335: 10239: 10134: 9967: 9921: 9892: 9736: 9716: 9674: 9598: 9458: 9383: 9297: 9133: 9000: 8876: 8779: 8678: 8538: 8370: 8235: 8115: 8077: 7686: 7540: 7434: 7119: 7047: 6942:The proof of this result is a simple computation. 6906: 6769: 6529: 6373: 6308: 6239: 6169: 6033: 5976:If the new energy is less, keep the flipped value. 5952: 5764: 5653: 5139: 5052: 4832:{\displaystyle \sum _{L}L^{2}4^{2L}e^{-4\beta L},} 4831: 4759: 4681:{\displaystyle \sum _{L}(2d)^{L}\varepsilon ^{L},} 4680: 4610: 4501: 4295: 4176: 4156: 4136: 4116: 4096: 4084:The Simon-Lieb inequality states that for any set 4068: 4041: 4005: 3966: 3937: 3867: 3784: 3757: 3655: 3560: 3345: 3232: 3186: 3138:{\displaystyle \sum _{ij\in \delta (V^{+})}J_{ij}} 3137: 3072: 3052: 3023: 3001: 2688: 2642: 2619: 2587: 2483:the weighted undirected graph G can be defined as 2471: 2435: 2386: 2359: 2332: 2312: 2276: 2249: 2222: 2202: 2173: 2153: 2114: 1998: 1988: 1871: 1766:, there is no external influence on the spin site. 1758: 1719: 1680: 1618: 1576: 1534: 1487: 1450: 1430: 1391: 1369: 1290: 1264: 1197: 1145: 1062: 1025: 1001: 981: 961: 930: 799: 773: 743: 717: 684: 649: 591: 540: 513: 487: 463: 23996: 22037: 13945:The denominator in this expression is called the 11786: 11693: 10835:gives analyticity of the thermodynamic functions. 6545:can be any number, since in this simplified case 4006:{\displaystyle \langle \sigma _{A}\rangle \geq 0} 3439:. Namely, for any positive β, the correlations ⟨σ 25760: 24902:Stochastic chains with memory of variable length 23436: 20478:Cayley tree topologies and large neural networks 20358:can also be described, by the usual Hamiltonian 18602:Above the critical temperature, fluctuations in 18302:This energy cost gives the ratio of probability 18168:{\displaystyle \lambda ={\varepsilon \over 3B}} 17470:{\displaystyle \delta \lambda =-3B\lambda ^{2}b} 14069:{\displaystyle Z=\int DH\,e^{-\int d^{d}x\left}} 12921:A complete derivation was only given in 1951 by 7704:= 0, i.e. in the limit β → ∞ there is no decay. 6589: 5282:tells us that the following equation must hold: 4909:= |Λ|: the total number of sites on the lattice, 3792:on the lattice, the following inequality holds, 24299: 23851:Itzykson, Claude; Drouffe, Jean-Michel (1989), 23850: 23829:Itzykson, Claude; Drouffe, Jean-Michel (1989), 23828: 23238:Proceedings of the National Academy of Sciences 22471: 20927: 20443:-variables describe the Ising spins, while the 19014: 12738:Onsager's formula for spontaneous magnetization 12226: 9968:{\displaystyle \sigma _{2},\sigma _{4},\cdots } 8780:{\displaystyle {\frac {p}{1-p}}=e^{-2\beta J}.} 6034:{\displaystyle e^{-\beta (H_{\nu }-H_{\mu })}.} 5962:The basic form of the algorithm is as follows: 4429:represented by black and white, they look like 3389: 24403:A larger/more complicated 2D Ising java applet 23868: 23638:Exactly solved models in statistical mechanics 23298: 23000: 22526:Exactly solved models in statistical mechanics 16845:)Λ, it will be a slightly different identity. 15795:fixes the overall normalization of the field. 14373:To find the magnetization, fix the scaling of 9155:One-dimensional solution with transverse field 8801:, so the correlations fall off exponentially. 4518:So for every configuration with magnetization 4319:This inequality is proven first for a type of 385:The Ising model was invented by the physicist 24491: 24471:https://github.com/cossio/SquareIsingModel.jl 23393: 22602: 22327:"For example, SquareIsingModel.jl (in Julia)" 22237: 19663:, this quickly flows to zero coupling. Large 18998:{\displaystyle H\propto \varepsilon ^{0.125}} 18960:{\displaystyle H\propto \varepsilon ^{0.308}} 17032:The integral should be split into two parts: 14076:and the integral over all possible values of 12508:destroys two spin-flips on neighboring sites. 12431:Spin flip creation and annihilation operators 8789:The problem is reduced to independent biased 8694:. This gives the formula of the free energy. 5966:Pick a spin site using selection probability 4936:Since every spin site has ±1 spin, there are 4522:there is a configuration with magnetization − 1731:desires to line up in the negative direction, 1692:desires to line up in the positive direction, 495:-dimensional lattice. For each lattice site 311: 23889:Ross Kindermann and J. Laurie Snell (1980), 23623:, Burlington: University of Vermont: 1–118, 23576:: CS1 maint: multiple names: authors list ( 21848:Contribution to the Theory of Ferromagnetism 21599:) and/or neighbor-to-neighbor interactions ( 21259: 21253: 20964: 20941: 19978: 19966: 19807: 19795: 19271:at the fixed point of λ in the ε expansion. 18422:{\displaystyle p={1 \over 1+e^{-2\beta JH}}} 17910: 16356:. The decay factor appropriate for position 15728:can be found by integrating with respect to 15389: 15362: 15297:is the proportionality constant. So knowing 15145: 15079: 14905: 14878: 14872: 14845: 14805:term is 2, while the scale dimension of the 14588: 14566: 14513: 14491: 14476: 14453: 13648: 13585: 13082:and not the spins. Once the correlations in 13059: 13035:proved that the spin glass Ising model on a 12505:produces two spin-flips on neighboring sites 12139: 12109: 12073: 11680: 11674: 8833: 8810: 7588: 7559: 6930:→ 0, though, the inverse correlation length 6842: 6829: 6826: 6813: 6807: 6784: 6385:is analytic in the thermodynamic parameters. 6099:, the Ising model and its variants such the 6069:It is possible to view the Ising model as a 5109: 5097: 4996: 4984: 4881:Monte Carlo methods for numerical simulation 4436:A quantitative measure of the excess is the 4287: 4264: 4261: 4238: 4216: 4193: 4036: 4023: 3994: 3981: 3932: 3903: 3897: 3884: 3859: 3846: 3843: 3830: 3824: 3801: 3623: 3599: 3490: 3466: 3406:-dimensional lattice, namely, Λ = 1958: 1946: 1881:When the external field is zero everywhere, 1828: 1816: 1314: 1307: 1298:of the spins ("observable"), one denotes by 956: 947: 851: 842: 632: 618: 586: 568: 23892:Markov Random Fields and Their Applications 23372: 21989: 19027:and predated the perturbative ε expansion. 18972:This is also true in two dimensions, where 18360:{\displaystyle {p \over 1-p}=e^{2\beta JH}} 18295:, the energy cost for flipping a spin is ±2 16864:with one dangling line, so proportional to 16719:wavenumbers at a point λ. Fourier modes of 14622:by dimensional analysis, it is the shifted 10843: 10794:by iterating the functional equation until 6410:sites with periodic boundary conditions is 6374:{\displaystyle J_{ij}\sim |i-j|^{-\alpha }} 6240:{\displaystyle J_{ij}\sim |i-j|^{-\alpha }} 6170:{\displaystyle J_{ij}\sim |i-j|^{-\alpha }} 4703: − 2)/2 (the straight line) and ( 4042:{\displaystyle \langle \sigma _{A}\rangle } 3715: 3233:{\displaystyle \left|\delta (V^{+})\right|} 2436:{\displaystyle \left|\delta (V^{+})\right|} 25030:Autoregressive–moving-average (ARMA) model 24498: 24484: 23734: 22240:Monte Carlo Methods in Statistical Physics 21833: 16852:term. In a Feynman diagram expansion, the 14092:is a "Euclidean" Lagrangian for the field 13572:The statistical average of any product of 13093:For any value of the slowly varying field 4851: 4440:, which is the average value of the spin: 4330: 318: 304: 29: 24366:Ising model at The Net Advance of Physics 24341: 24288:, North-Holland Publishing Co.: 230–242, 23997:Montroll, Elliott W.; Potts, Renfrey B.; 23875:. Cambridge: Cambridge University Press. 23721:, Cambridge: Cambridge University Press, 23712: 23662: 23596: 23515: 23489: 23378: 23366: 23347: 23330: 23320: 23275: 23257: 23132: 23068:Journal Club for Condensed Matter Physics 23014: 22961: 22907: 22846: 22801: 22747: 22694: 22393: 22391: 22371: 22280: 22233: 22231: 22229: 22227: 22225: 22223: 22221: 22219: 22217: 22215: 22187: 22122: 21936: 21860: 21283:{\displaystyle \sum _{\{s\}}e^{-\beta H}} 20415: 20404: 20384: 18770:{\displaystyle H={\sqrt {3\varepsilon }}} 18579:< 1 there is only the one solution at 16750:) by dimensional analysis. The change in 16710: 16670: 14158: 13971: 13832: 13666: 13355: 13341: 13288:{\displaystyle \beta F=\int d^{d}x\left.} 13128:there and in the neighboring regions. So 12482: 11826:, it is the sum over all spin histories. 10343:. Now we have a self-similarity relation: 7696:hence it decays exponentially as soon as 6401: 6065:Viewing the Ising model as a Markov chain 439:(Max-Cut) problem that can be solved via 364:magnetic dipole moments of atomic "spins" 24505: 24449:image denoising example with Ising Model 24393:A dynamical 2D Ising java applet by UCSC 24388:Science World article on the Ising Model 23831:Théorie statistique des champs, Volume 1 23061: 23055: 21559:and the site-to-site interaction energy 20757: 20481: 20117:to a neural network which is capable of 18306:that the spin is + to the probability 1− 14638:is very small. The fractional change in 4884: 4079: 3393: 3382:is changed, is there a phase transition? 1198:{\displaystyle \beta =1/(k_{\text{B}}T)} 592:{\displaystyle \sigma _{k}\in \{-1,+1\}} 24247: 24219: 24131: 24100: 24061: 23982:Barry M. McCoy and Tai Tsun Wu (1973), 23118: 22401:Statistical Mechanics: Rigorous Results 20873:{\displaystyle J\equiv \beta \epsilon } 20554: 20550: 17186:, contributes to the critical scaling. 16349:{\displaystyle e^{-t\tau }=e^{-tr^{2}}} 16258:-space, acting on the unit function in 14107: 12499:moves a spin-flip one unit to the right 12206:{\displaystyle Z=\mathrm {tr} (T^{N}).} 10854: 5150: 4398:The exponential function is everywhere 4321:positively-correlated percolation model 3727: 3681: 402: 25761: 25336:Doob's martingale convergence theorems 23634: 22561: 22522: 22397: 22388: 22238:Newman, M.E.J.; Barkema, G.T. (1999). 22212: 22111:Communications in Mathematical Physics 22056:Communications in Mathematical Physics 20190:{\displaystyle E=-\sum _{i}h_{i}S_{i}} 19876:Rewriting the bits in terms of spins, 17166:The first part is not proportional to 15782:{\displaystyle G(r)={C \over r^{d-2}}} 14178:{\displaystyle F=\int d^{d}x\,AH^{2}.} 12953:two-dimensional conformal field theory 12502:moves a spin-flip one unit to the left 10758:{\displaystyle f(\beta )\approx \ln 2} 5796:(μ, ν). Metropolis sets the larger of 5220:(using single-spin-flip dynamics) is 2 4622:choices for where to go at each step. 3712:in direct response to Onsager's work. 2230:-depended subsets, those with spin up 25088:Constant elasticity of variance (CEV) 25078:Chan–Karolyi–Longstaff–Sanders (CKLS) 24479: 24327: 23792: 23771: 23683: 23615: 23353: 23092: 22424: 22262: 22176:Probability Theory and Related Fields 22049: 21670: 20545: 19764:So the energy of the lattice gas is: 16746:coefficient scales up by a factor (1+ 14407:There is one subtle point. The field 9675:{\displaystyle f(\beta ,0)=f(\beta )} 6309:{\displaystyle J_{ij}\sim |i-j|^{-2}} 4379:No phase transitions in finite volume 421: 394: 24180: 23966: 22994: 22940: 22641: 22605:"Teaching the renormalization group" 22345: 22256: 21799:Two-dimensional critical Ising model 19742:for each occupied neighboring pair. 19572:{\displaystyle \Delta F=\ln(\cosh).} 18740:to discard the unstable solution at 17174:. It is caused by the fact that the 16674:{\displaystyle F=\int d^{4}x\left\,} 13321:On any lattice, the derivative term 12947:Two-dimensional critical Ising model 12922: 8116:{\displaystyle V_{\sigma ,\sigma '}} 5213:and any possible new state's energy 4688:which goes to zero when ε is small. 3697: 3669:in 1936, using what is now called a 3427:No phase transition in one dimension 390: 227:Grand potential / Landau free energy 23062:Kadanoff, Leo P. (April 30, 2014). 22263:Süzen, Mehmet (29 September 2014). 20354:With the Ising model the so-called 18291:→ ∞ is smooth. In terms of the new 15715:{\displaystyle E={C \over r^{d-1}}} 15592: − 1 dimensional sphere, 14773:proportional to the square root of 14646:is fixed the shift looks infinite. 14634:is finite, but near the transition 14333:{\displaystyle F=\int d^{d}x\left,} 13101:and its gradients. The free energy 12989: 12962: 8702:The energy of the lowest state is − 4863:there is a unique critical point). 4420: 4391:The partition function is a sum of 1431:{\displaystyle P_{\beta }(\sigma )} 13: 25575:Skorokhod's representation theorem 25356:Law of large numbers (weak/strong) 22472:Fröhlich, J.; Spencer, T. (1982). 20924:Krizan, Barth & Glasser (1983) 19929:{\displaystyle B_{i}=(S_{i}+1)/2.} 19527: 19216:odd and the even sites those with 18851: 18834: 17891:In other dimensions, the constant 17842: 17828: 17784: 17676: 17639: 17276:is rescaled, but its dimension is 17130: 16941: 16904: 16775: 16596: 16529:dimensions – renormalization group 15828: 15658: 15655: 15652: 15649: 15646: 15643: 15640: 15637: 15545: 15518: 15484:are singular when points collide. 15415: 15337: 14668: 14664: 14642:is very large, and in units where 14306: 14025: 13886: 13720: 13488: 13357: 13343: 13233: 12323:can be written in terms of these: 12180: 12177: 11921: 11865: 11655: 9627: 7550:With the same change of variables 6599: 3961: 1771: 1647:(σ) also explains how a spin site 962:{\displaystyle \langle ij\rangle } 738: 679: 642: 599:, representing the site's spin. A 508: 458: 14: 25800: 25545:Martingale representation theorem 24359: 23943:Critical Properties of φ-Theories 23941:and Schulte-Frohlinde, V (2001), 23895:. American Mathematical Society. 23869:Friedli, S.; Velenik, Y. (2017). 21866: 21136:is equal to the number of bonds, 20844:{\displaystyle \tau \equiv t^{2}} 19715:for each configuration of atoms. 19015:Low dimensions – block spins 16731:links up lines carrying momentum 16390:{\displaystyle e^{-{\sqrt {t}}r}} 12959:, which has been exactly solved. 10817: 10765:, so we can numerically evaluate 4866: 4699:, the area is somewhere between ( 4308: 4144:(e.g. the boundary of a box with 2154:{\displaystyle \sigma _{i}=\pm 1} 1205:, and the normalization constant 426: 25590:Stochastic differential equation 25480:Doob's optional stopping theorem 25475:Doob–Meyer decomposition theorem 24398:A dynamical 2D Ising java applet 23911:Gauge Fields in Condensed Matter 23641:, London: Academic Press Inc. , 23095:"The Ising Model Is NP-Complete" 22529:, London: Academic Press Inc. , 19208:), the odd sites are those with 18247:Infinite dimensions – mean field 15563:{\displaystyle \nabla \cdot E=0} 14377:so that λ is one. Now the field 14368: 12940: 12311:{\displaystyle \sigma _{i}^{z}.} 12265:{\displaystyle \sigma _{i}^{x}.} 11876:{\displaystyle U=e^{iH\Delta t}} 6110: 5250: 4932:∈ {−1, +1}: state of the system. 3367:If a spin at any given position 1402:The configuration probabilities 1379:the expectation (mean) value of 25460:Convergence of random variables 25346:Fisher–Tippett–Gnedenko theorem 24008:Journal of Mathematical Physics 23984:The Two-Dimensional Ising Model 23584: 23532: 23465: 23430: 23387: 23292: 23225: 23165: 23112: 23086: 22887: 22826: 22780: 22727: 22674: 22660: 22635: 22596: 22516: 22465: 22418: 22339: 22319: 22163: 22038:Montroll, Potts & Ward 1963 21724:Geometrically frustrated magnet 20811:{\displaystyle t\equiv \tanh J} 20349: 20096: 19673: 19650:{\displaystyle 3J'=\ln(\cosh).} 19035:Migdal–Kadanoff renormalization 18744:= 0, the stable solutions are: 18279:. It is convenient to redefine 16492:Symanzik polymer interpretation 15305:The critical two-point function 14792: 12027:{\displaystyle T_{C_{1}C_{2}}.} 9724:be the partition function with 9717:{\displaystyle Z_{N}(\beta ,J)} 7107: 6922:(β) are positive functions for 6073:, as the immediate probability 4760:{\displaystyle N(L)<4^{2L}.} 2181:partitions the set of vertices 685:{\displaystyle i,j\in \Lambda } 25058:Binomial options pricing model 24425:Wolfram Demonstrations Project 23855:, Cambridge University Press, 23003:Journal of High Energy Physics 22950:Journal of Statistical Physics 22736:Journal of High Energy Physics 22642:Wood, Charlie (24 June 2020). 22098: 22043: 21983: 21925:Journal of Statistical Physics 21909: 21853: 21839: 21826: 21518: 21515: 21499: 21487: 21471: 21462: 21178:{\displaystyle g_{mn}(l)t^{l}} 21162: 21156: 21086: 21080: 21020: 21007: 21004: 20992: 20743: 20737: 20677: 20665: 20660: 20648: 20636: 20624: 20612: 20600: 20044: 20029: 19915: 19896: 19709: 19641: 19638: 19629: 19620: 19563: 19560: 19551: 19542: 19510:{\displaystyle F=\log(\cosh).} 19501: 19498: 19495: 19469: 19463: 19454: 19413: 19410: 19384: 19378: 19361: 19335: 19319: 19293: 18878: 18863: 18846: 18837: 18779:The spontaneous magnetization 18699: 18686: 18674: 18662: 18656: 18644: 18560: 18548: 18175:where the scale dimensions of 17942: 17930: 17812: 17802: 17793: 17767: 17709: 17699: 17673: 17661: 17654: 17646: 17372: 17352: 17117: 17098: 17011: 16992: 16974: 16964: 16938: 16926: 16919: 16911: 16604: 16592: 16426: 16420: 16400:A heuristic approximation for 16202: 16183: 16160: 16154: 16027: 16021: 15957: 15951: 15855: 15806:) away from the critical point 15750: 15744: 15451: 15445: 15433: 15427: 15406: 15386: 15380: 15374: 15368: 15278: 15246: 15237: 15205: 15199: 15167: 15142: 15126: 15117: 15104: 15098: 15085: 14993: 14981: 14956: 14946: 14926: 14914: 14902: 14896: 14890: 14884: 14869: 14863: 14857: 14851: 14579: 14572: 14554: 14547: 14526: 14519: 14504: 14497: 14466: 14459: 14438: 14431: 14313: 14303: 14033: 14021: 13894: 13882: 13818: 13805: 13796: 13783: 13777: 13764: 13728: 13716: 13645: 13632: 13623: 13610: 13604: 13591: 13552: 13546: 13496: 13484: 13429: 13423: 13246: 13229: 13068: 12197: 12184: 12132: 12102: 12098: 12092: 12066: 11930: 11907: 11568:When the interaction energies 11543: 11534: 11525: 11509: 11467:occurs at the following point 11225: 11222: 11209: 11200: 11181: 11169: 11156: 11147: 11128: 11116: 11097: 11088: 11069: 11060: 10781: 10775: 10740: 10734: 10688: 10676: 10620: 10609: 10583: 10574: 10536: 10530: 10507: 10496: 10429: 10420: 10382: 10376: 10330: 10321: 10280: 10271: 10234: 10231: 10205: 10199: 10126: 10123: 10120: 10094: 10088: 10076: 10070: 10067: 10064: 10038: 10032: 10020: 10004: 9992: 9882: 9856: 9833: 9807: 9773: 9761: 9711: 9699: 9669: 9663: 9654: 9642: 9486: 9480: 9185: 9179: 9050: 9038: 8986: 8973: 8967: 8955: 8865: 8851: 8640: 8624: 8394: 8388: 8352: 8340: 8284: 8272: 7727: 7721: 7468: 7456: 7444:Therefore, the free energy is 7145: 7139: 6965: 6959: 6895: 6881: 6877: 6871: 6857: 6851: 6726: 6710: 6634: 6628: 6596: 6579: 6567: 6426: 6420: 6358: 6343: 6293: 6278: 6224: 6209: 6154: 6139: 6023: 5997: 5879: 5853: 5828: 5816: 5754: 5728: 5708: 5696: 5688: 5676: 5643: 5617: 5595: 5582: 5554: 5541: 5508: 5502: 5487: 5481: 5459: 5447: 5439: 5427: 5412: 5400: 5394: 5382: 5374: 5362: 5356: 5344: 5329: 5317: 5309: 5297: 5080: 5074: 4967: 4961: 4897: 4735: 4729: 4656: 4646: 4611:{\displaystyle N(L)=(2d)^{L},} 4596: 4586: 4580: 4574: 4556:The number of paths of length 4413:This was first established by 3644: 3638: 3546: 3532: 3528: 3522: 3332: 3319: 3287: 3281: 3258: 3252: 3222: 3209: 3187:{\displaystyle W_{ij}=-J_{ij}} 3117: 3104: 3047: 3041: 2974: 2961: 2920: 2914: 2866: 2853: 2815: 2802: 2764: 2751: 2721: 2715: 2564: 2551: 2513: 2500: 2472:{\displaystyle \delta (V^{+})} 2466: 2453: 2425: 2412: 2313:{\displaystyle \delta (V^{+})} 2307: 2294: 2197: 2191: 2074: 2068: 2042: 2036: 1929: 1923: 1802: 1796: 1482: 1476: 1425: 1419: 1364: 1358: 1345: 1339: 1257: 1251: 1192: 1176: 1123: 1117: 1095: 1089: 828: 822: 338:), named after the physicists 1: 25525:Kolmogorov continuity theorem 25361:Law of the iterated logarithm 24430:Phase transitions on lattices 23608: 20928:Glasser & Goldberg (1983) 18810:must only have odd powers of 17618:is determined by the formula 17193:by an amount proportional to 15814:does not equal zero, so that 14216:in the system of units where 11433:{\displaystyle J_{1}=J_{2}=J} 9147:away from β = ∞. A sign of a 8550:is the highest eigenvalue of 5191:(μ, ν) is selected such that 5162:Metropolis–Hastings algorithm 2689:{\displaystyle W_{ij}=W_{ji}} 2627:denotes a weight of the edge 2161:. A given spin configuration 1461: 744:{\displaystyle j\in \Lambda } 521:there is a discrete variable 514:{\displaystyle k\in \Lambda } 446: 405:). It is usually solved by a 25530:Kolmogorov extension theorem 25209:Generalized queueing network 24717:Interacting particle systems 24322:10.1016/0378-4371(83)90138-3 24294:10.1016/0378-4371(83)90157-7 24270:10.1016/0378-4371(79)90134-1 24106:Supplemento al Nuovo Cimento 23947:World Scientific (Singapore) 23915:World Scientific (Singapore) 23565:10.1016/0378-4371(90)90196-Y 21820: 21394:.) The critical temperature 20135:principle of maximum entropy 20085:is the number of neighbors. 19678: 15576:is spherically symmetric in 14419:split in the following way: 14396:has dimension 2 − 14199:, and then it is clear that 13033:Sandia National Laboratories 9014:is the free energy density: 5242:periodic boundary conditions 4549:spins separated by distance 4371:statistical system has many 3967:{\displaystyle B=\emptyset } 3578:The Ising model undergoes a 3390:Basic properties and history 3355: 3080:is equivalent to minimizing 1584:, the interaction is called 1542:, the interaction is called 1063:{\displaystyle \beta \geq 0} 7: 24662:Continuous-time random walk 24456:provide a good introduction 24454:David Tong's Lecture Notes 24222:American Journal of Physics 23673:Encyclopedia of Mathematics 22609:American Journal of Physics 22050:Simon, Barry (1980-10-01). 21691: 21364:{\displaystyle \sigma _{i}} 19267:, analogous to the flow in 19181:By translation invariance, 18794:The mean-field exponent is 16500:interaction is 4. The term 15999:{\displaystyle {\sqrt {t}}} 15588:. Integrating over a large 11440:, the critical temperature 10520:Taking the limit, we obtain 8697: 6319:In the case of interaction 6103:can be seen as a form of a 5238:lattice coordination number 5155: 3785:{\displaystyle \sigma _{B}} 3758:{\displaystyle \sigma _{A}} 3684:). Onsager showed that the 3145:. Defining the edge weight 1577:{\displaystyle J_{ij}<0} 1535:{\displaystyle J_{ij}>0} 1009:are nearest neighbors. The 660:For any two adjacent sites 541:{\displaystyle \sigma _{k}} 69:Indistinguishable particles 10: 25805: 25670:Extreme value theory (EVT) 25470:Doob decomposition theorem 24762:Ornstein–Uhlenbeck process 24533:Chinese restaurant process 24352:10.1103/PhysRevE.90.032141 23837:), EDP Sciences Editions, 23635:Baxter, Rodney J. (1982), 22926:10.1103/PhysRevD.86.025022 22865:10.1103/PhysRevE.93.012144 22812:10.1103/PhysRevX.13.021009 22523:Baxter, Rodney J. (1982), 22346:Teif, Vladimir B. (2007). 22291:10.1103/PhysRevE.90.032141 21784:Square-lattice Ising model 21729:Classical Heisenberg model 20465: 18614:solution to the equation. 18250: 15584:is the radial gradient of 15527:{\displaystyle E=\nabla G} 14769:< 0, the minima are at 12944: 12230:in terms of Pauli matrices 12145:{\displaystyle |S\rangle } 11271:which solves the equation 10850:Square lattice Ising model 10847: 9922:{\displaystyle K:=\beta J} 4870: 4855: 4339:' arguments in support of 4312: 3738:Given any subset of spins 3731: 3451:⟩ decay exponentially in | 3053:{\displaystyle H(\sigma )} 1720:{\displaystyle h_{j}<0} 1681:{\displaystyle h_{j}>0} 1488:{\displaystyle H(\sigma )} 441:combinatorial optimization 376:square-lattice Ising model 25738: 25642: 25550:Optional stopping theorem 25447: 25409: 25351:Large deviation principle 25318: 25232: 25189: 25156: 25103:Heath–Jarrow–Morton (HJM) 25048: 25040:Moving-average (MA) model 25025:Autoregressive (AR) model 25005: 24915: 24850:Hidden Markov model (HMM) 24832: 24784:Schramm–Loewner evolution 24588: 24513: 24466:article about Ising model 24118:Planar Ising Correlations 23972:Physikalische Zeitschrift 23748:10.1007/978-3-662-03952-6 23706:10.1103/RevModPhys.39.883 23686:Reviews of Modern Physics 22980:10.1007/s10955-014-1042-7 22580:10.1007/978-3-642-33039-1 22189:10.1007/s00440-011-0353-8 22141:10.1007/s00220-015-2480-z 22012:10.1017/S0305004100019174 21955:10.1007/s10955-014-1042-7 21652:{\displaystyle \epsilon } 21612:{\displaystyle \epsilon } 21572:{\displaystyle \epsilon } 20893:{\displaystyle \epsilon } 18179:is altered by an amount λ 17911:Wilson–Fisher fixed point 15499:, which can be solved by 15309:To determine the form of 13060:Four dimensions and above 12744:spontaneous magnetization 10787:{\displaystyle f(\beta )} 9010:The logarithm divided by 8558:is the other eigenvalue: 5201:single-spin-flip dynamics 4164:being inside the box and 3694:spontaneous magnetization 3665:This was first proven by 2257:and those with spin down 1035:configuration probability 800:{\displaystyle {\sigma }} 781:interacting with it. The 25465:Doléans-Dade exponential 25295:Progressively measurable 25093:Cox–Ingersoll–Ross (CIR) 23926:(also available online: 23621:Master of Science Thesis 23151:10.1103/PhysRevE.58.6399 23093:Cipra, Barry A. (2000). 21744:Ising critical exponents 21734:Quantum Heisenberg model 20123:recurrent neural network 19245:{\textstyle {\sqrt {2}}} 18439:, so the mean value is 2 14120:needs to be determined. 13017:Ising critical exponents 12124:where each basis vector 10844:Onsager's exact solution 9167:Hamiltonian is given by 6937: 6082:Markov chain Monte Carlo 4395:over all configurations. 3722:correlation inequalities 3716:Correlation inequalities 3031:, imply that minimizing 1619:{\displaystyle J_{ij}=0} 464:{\displaystyle \Lambda } 358:. The model consists of 25774:Exactly solvable models 25685:Mathematical statistics 25675:Large deviations theory 25505:Infinitesimal generator 25366:Maximal ergodic theorem 25285:Piecewise-deterministic 24887:Random dynamical system 24752:Markov additive process 24423:by Enrique Zeleny, the 24143:, ACM, pp. 87–96, 24132:Istrail, Sorin (2000), 23779:(2nd ed.), Wiley, 23459:10.1103/PhysRev.108.171 23416:10.1103/PhysRev.106.620 23259:10.1073/pnas.1521241113 23195:10.1126/science.1182105 23033:10.1007/JHEP06(2015)174 22766:10.1007/JHEP08(2015)022 22713:10.1007/JHEP07(2013)055 22398:Ruelle, David (1999) . 21789:Swendsen–Wang algorithm 21632:{\displaystyle \gamma } 21592:{\displaystyle \gamma } 21552:{\displaystyle \gamma } 21234:{\displaystyle {Z_{N}}} 20914:{\displaystyle \gamma } 20779:{\displaystyle \gamma } 15038:is large. The function 14415:. To see how, consider 13379:is a positive definite 5166:selection probabilities 4858:Kramers–Wannier duality 4852:Kramers–Wannier duality 4560:on a square lattice in 4533:as with magnetization − 4526:with equal probability. 4331:Historical significance 3073:{\displaystyle \sigma } 3024:{\displaystyle \sigma } 2333:{\displaystyle \sigma } 2223:{\displaystyle \sigma } 2174:{\displaystyle \sigma } 1780: = 0 for all 1759:{\displaystyle h_{j}=0} 1643:The sign convention of 1451:{\displaystyle \sigma } 753:external magnetic field 112:Thermodynamic ensembles 64:Spin–statistics theorem 25520:Karhunen–Loève theorem 25455:Cameron–Martin formula 25419:Burkholder–Davis–Gundy 24814:Variance gamma process 24421:Ising Model simulation 24373:, "The Ising model is 24206:10.1103/PhysRev.85.808 24120:. Birkhäuser, Boston, 24104:(1949), "Discussion", 24087:10.1103/PhysRev.65.117 23713:Baierlein, R. (1999), 23322:10.1073/pnas.79.8.2554 21681: 21653: 21633: 21613: 21593: 21573: 21553: 21531: 21415: 21388: 21365: 21338: 21311: 21284: 21235: 21206: 21179: 21130: 21103: 21066: 20915: 20894: 20874: 20845: 20812: 20780: 20763: 20750: 20703: 20538: 20537:{\displaystyle 10^{5}} 20511: 20510:{\displaystyle 10^{4}} 20487: 20433: 20339: 20338:{\displaystyle J_{ij}} 20309: 20191: 20061: 19930: 19868: 19651: 19573: 19511: 19420: 19274:To find the change in 19246: 19173: 18999: 18961: 18910: 18771: 18728: 18567: 18423: 18361: 18235: 18169: 18125: 18021: 17883: 17742: 17606: 17471: 17419: 17264: 17197:. The total change in 17158: 17021: 16829: 16711:Wilson renormalization 16675: 16476: 16391: 16350: 16242: 16122: 16000: 15970: 15783: 15716: 15666: 15564: 15536:Away from the origin: 15528: 15491:At the critical point 15472: 15285: 15022: 14809:term is 4 − 14757: 14606: 14392:is dimensionless, and 14334: 14179: 14070: 13937: 13564: 13373: 13289: 13218: 12912: 12885: 12858: 12719: 12487: 12406: 12312: 12266: 12207: 12146: 12116: 12028: 11968: 11877: 11798: 11642: 11616: 11589: 11560: 11461: 11434: 11386: 11265: 11235: 10931: 10904: 10877: 10808: 10788: 10759: 10718: 10717:{\displaystyle \beta } 10695: 10627: 10514: 10337: 10241: 10136: 9969: 9929:. We sum over each of 9923: 9894: 9738: 9718: 9676: 9600: 9460: 9385: 9362: 9299: 9135: 9002: 8878: 8781: 8680: 8540: 8372: 8237: 8117: 8079: 7688: 7542: 7436: 7321: 7121: 7049: 6908: 6771: 6531: 6402:Ising's exact solution 6375: 6310: 6241: 6171: 6107:for opinion dynamics. 6035: 5954: 5766: 5655: 5141: 5054: 4894: 4846:superselection sectors 4833: 4761: 4682: 4612: 4503: 4485: 4297: 4178: 4158: 4138: 4118: 4098: 4070: 4043: 4007: 3968: 3939: 3869: 3786: 3759: 3657: 3562: 3399: 3347: 3234: 3188: 3139: 3074: 3054: 3025: 3003: 2690: 2644: 2621: 2620:{\displaystyle W_{ij}} 2589: 2473: 2437: 2388: 2361: 2334: 2314: 2278: 2251: 2224: 2204: 2175: 2155: 2116: 1990: 1873: 1760: 1721: 1682: 1620: 1578: 1536: 1489: 1452: 1432: 1393: 1371: 1292: 1266: 1199: 1147: 1064: 1039:Boltzmann distribution 1027: 1003: 983: 963: 932: 801: 775: 745: 719: 718:{\displaystyle J_{ij}} 686: 651: 593: 542: 515: 489: 465: 407:transfer-matrix method 25779:Statistical mechanics 25650:Actuarial mathematics 25612:Uniform integrability 25607:Stratonovich integral 25535:Lévy–Prokhorov metric 25439:Marcinkiewicz–Zygmund 25326:Central limit theorem 24928:Gaussian random field 24757:McKean–Vlasov process 24677:Dyson Brownian motion 24538:Galton–Watson process 24383:online edition (.pdf) 24149:10.1145/335305.335316 23777:Statistical mechanics 23740:Statistical mechanics 22425:Dyson, F. J. (1969). 21887:10.1287/opre.36.3.493 21676: 21654: 21634: 21614: 21594: 21574: 21554: 21532: 21416: 21414:{\displaystyle T_{C}} 21389: 21387:{\displaystyle \pm 1} 21366: 21339: 21337:{\displaystyle S_{i}} 21312: 21310:{\displaystyle s_{i}} 21285: 21236: 21207: 21205:{\displaystyle 2^{N}} 21180: 21131: 21129:{\displaystyle N_{b}} 21104: 21046: 20916: 20895: 20875: 20846: 20813: 20781: 20761: 20751: 20683: 20539: 20512: 20485: 20434: 20340: 20310: 20192: 20062: 19931: 19869: 19652: 19574: 19512: 19421: 19247: 19174: 19000: 18962: 18911: 18772: 18729: 18568: 18424: 18362: 18283:to absorb the factor 18236: 18170: 18126: 18022: 17884: 17743: 17607: 17472: 17420: 17265: 17159: 17022: 16830: 16676: 16477: 16392: 16351: 16243: 16123: 16001: 15971: 15784: 15717: 15667: 15565: 15529: 15473: 15286: 15023: 14758: 14607: 14335: 14180: 14084:) over all values of 14071: 13938: 13565: 13383:, and can be used to 13374: 13290: 13198: 13136:and its derivatives. 12935:Toeplitz determinants 12913: 12911:{\displaystyle J_{2}} 12886: 12884:{\displaystyle J_{1}} 12859: 12720: 12488: 12407: 12313: 12267: 12208: 12147: 12117: 12029: 11969: 11878: 11799: 11650:geometric frustration 11643: 11617: 11615:{\displaystyle J_{2}} 11590: 11588:{\displaystyle J_{1}} 11561: 11462: 11460:{\displaystyle T_{c}} 11435: 11387: 11266: 11264:{\displaystyle T_{c}} 11236: 10932: 10930:{\displaystyle J_{2}} 10905: 10903:{\displaystyle J_{1}} 10878: 10809: 10789: 10760: 10719: 10696: 10628: 10515: 10338: 10242: 10137: 9970: 9924: 9895: 9739: 9719: 9677: 9601: 9461: 9386: 9342: 9300: 9136: 9003: 8879: 8782: 8681: 8541: 8373: 8238: 8118: 8080: 7689: 7543: 7437: 7301: 7122: 7050: 6909: 6772: 6532: 6376: 6311: 6242: 6172: 6047:The change in energy 6036: 5955: 5767: 5656: 5142: 5055: 4922: = 1, ..., 4888: 4834: 4762: 4683: 4613: 4504: 4465: 4298: 4179: 4159: 4139: 4119: 4099: 4080:Simon-Lieb inequality 4071: 4069:{\displaystyle J_{B}} 4044: 4008: 3969: 3940: 3870: 3787: 3760: 3706:Fredholm determinants 3686:correlation functions 3658: 3563: 3397: 3348: 3235: 3189: 3140: 3075: 3055: 3026: 3004: 2691: 2645: 2622: 2590: 2474: 2438: 2389: 2387:{\displaystyle V^{-}} 2362: 2360:{\displaystyle V^{+}} 2335: 2315: 2279: 2277:{\displaystyle V^{-}} 2252: 2250:{\displaystyle V^{+}} 2225: 2205: 2176: 2156: 2117: 1991: 1874: 1761: 1722: 1683: 1621: 1579: 1537: 1490: 1453: 1433: 1394: 1372: 1293: 1267: 1200: 1148: 1065: 1028: 1004: 984: 969:indicates that sites 964: 933: 802: 776: 774:{\displaystyle h_{j}} 746: 720: 687: 652: 594: 543: 516: 490: 466: 356:statistical mechanics 217:Helmholtz free energy 146:Isoenthalpic–isobaric 23:Statistical mechanics 25789:NP-complete problems 25725:Time series analysis 25680:Mathematical finance 25565:Reflection principle 24892:Regenerative process 24692:Fleming–Viot process 24507:Stochastic processes 24116:John Palmer (2007), 22404:. World Scientific. 21643: 21623: 21603: 21583: 21563: 21543: 21425: 21398: 21375: 21348: 21321: 21294: 21245: 21216: 21189: 21140: 21113: 20938: 20905: 20884: 20855: 20822: 20790: 20770: 20562: 20521: 20494: 20362: 20319: 20202: 20145: 19939: 19880: 19768: 19753:atoms is changed by 19597: 19524: 19439: 19282: 19232: 19043: 18976: 18938: 18822: 18748: 18629: 18451: 18373: 18314: 18287:, so that the limit 18190: 18141: 18035: 17924: 17758: 17622: 17485: 17433: 17293: 17211: 17038: 16874: 16771: 16551: 16414: 16364: 16301: 16148: 16015: 15986: 15982:small compared with 15824: 15738: 15680: 15598: 15542: 15509: 15319: 15076: 14842: 14659: 14425: 14238: 14133: 14108:Dimensional analysis 14104:(thus "Euclidean"). 14098:quantum field theory 13953: 13582: 13417: 13325: 13149: 13043:. That is, assuming 12895: 12868: 12753: 12516: 12445: 12329: 12287: 12241: 12167: 12128: 12062: 11991: 11891: 11845: 11666: 11626: 11599: 11572: 11473: 11444: 11398: 11277: 11248: 10943: 10914: 10887: 10861: 10829:Griffiths inequality 10798: 10769: 10728: 10708: 10637: 10524: 10347: 10251: 10146: 9979: 9933: 9904: 9748: 9728: 9686: 9636: 9474: 9396: 9321: 9173: 9020: 8896: 8807: 8731: 8562: 8382: 8378:In matrix formalism 8247: 8127: 8089: 7715: 7556: 7450: 7133: 7059: 6953: 6781: 6561: 6414: 6323: 6258: 6189: 6119: 5983: 5810: 5667: 5288: 5151:Metropolis algorithm 5068: 4955: 4771: 4723: 4633: 4568: 4446: 4417:in the Ising model. 4368:Josiah Willard Gibbs 4357:statistical behavior 4190: 4168: 4148: 4128: 4108: 4088: 4053: 4020: 3978: 3952: 3881: 3798: 3769: 3742: 3734:Griffiths inequality 3728:Griffiths inequality 3708:, proved in 1951 by 3596: 3463: 3246: 3198: 3149: 3084: 3064: 3035: 3015: 2705: 2654: 2631: 2601: 2489: 2447: 2401: 2371: 2344: 2324: 2288: 2261: 2234: 2214: 2203:{\displaystyle V(G)} 2185: 2165: 2129: 2030: 1917: 1790: 1737: 1698: 1659: 1594: 1552: 1510: 1470: 1442: 1406: 1383: 1304: 1282: 1211: 1159: 1076: 1048: 1026:{\displaystyle \mu } 1017: 993: 973: 944: 816: 809:Hamiltonian function 789: 758: 729: 699: 664: 607: 552: 525: 499: 479: 455: 411:quantum field theory 25720:Stochastic analysis 25560:Quadratic variation 25555:Prokhorov's theorem 25490:Feynman–Kac formula 24960:Markov random field 24608:Birth–death process 24314:1983PhyA..117..670G 24262:1979PhyA...99..268J 24234:1970AmJPh..38.1033G 24198:1952PhRv...85..808Y 24079:1944PhRv...65..117O 24021:1963JMP.....4..308M 23833:, Savoirs actuels ( 23807:1925ZPhy...31..253I 23698:1967RvMP...39..883B 23557:1990PhyA..164..221W 23508:10.1038/nature04701 23500:2006Natur.440.1007S 23484:(7087): 1007–1012, 23451:1957PhRv..108..171J 23408:1957PhRv..106..620J 23313:1982PNAS...79.2554H 23250:2016PNAS..113.4982V 23187:2010Sci...327..685B 23143:1998PhRvE..58.6399S 23025:2015JHEP...06..174S 22972:2014JSP...157..869E 22918:2012PhRvD..86b5022E 22857:2016PhRvE..93a2144D 22758:2015JHEP...08..022C 22705:2013JHEP...07..055B 22494:1982CMaPh..84...87F 22478:interaction energy" 22443:1969CMaPh..12...91D 22242:. Clarendon Press. 22133:2016CMaPh.343..725D 22068:1980CMaPh..77..111S 22004:1936PCPS...32..477P 21947:2014JSP...157..869E 21875:Operations Research 21773:Ashkin–Teller model 21714:Conformal bootstrap 20139:Lagrange multiplier 20115:first approximation 15350: 13576:terms is equal to: 13402:accidental symmetry 13013:conformal bootstrap 12533: 12468: 12401: 12380: 12359: 12304: 12258: 11641:{\displaystyle h=0} 11040: 11009: 10876:{\displaystyle h=0} 9744:sites. Now we have: 9608:Since the roles of 9592: 9561: 9540: 9452: 9431: 9413: 9377: 9338: 9291: 9260: 9239: 8477: 8459: 8441: 7365: 7339: 7074: 5197:thermal equilibrium 4942:Monte Carlo methods 4402:as a function of β. 4349:James Clerk Maxwell 3974:, the special case 1043:inverse temperature 785:of a configuration 154:Isothermal–isobaric 57:Particle statistics 25690:Probability theory 25570:Skorokhod integral 25540:Malliavin calculus 25123:Korn-Kreer-Lenssen 25007:Time series models 24970:Pitman–Yor process 24408:2020-11-25 at the 24381:, Vol. 33, No. 6; 24371:Barry Arthur Cipra 23815:10.1007/BF02980577 22502:10.1007/BF01208373 22451:10.1007/BF01645907 22364:10.1093/nar/gkm268 22076:10.1007/BF01982711 22040:, pp. 308–309 21836:, Chapters VI-VII. 21649: 21629: 21609: 21589: 21569: 21549: 21527: 21411: 21384: 21361: 21334: 21307: 21280: 21263: 21231: 21202: 21175: 21126: 21099: 20911: 20890: 20870: 20841: 20808: 20776: 20764: 20746: 20534: 20507: 20488: 20429: 20335: 20305: 20284: 20238: 20224: 20187: 20166: 20057: 20028: 19982: 19926: 19864: 19850: 19811: 19747:chemical potential 19647: 19569: 19507: 19416: 19242: 19169: 19064: 18995: 18957: 18906: 18767: 18724: 18563: 18435:and 1 − 18419: 18357: 18231: 18203: 18165: 18121: 18119: 18017: 17879: 17738: 17602: 17600: 17467: 17415: 17260: 17154: 17017: 16825: 16754:for infinitesimal 16671: 16472: 16387: 16346: 16238: 16118: 15996: 15966: 15779: 15712: 15662: 15560: 15524: 15497:Laplace's equation 15468: 15466: 15336: 15281: 15052:Feynman propagator 15018: 14753: 14602: 14330: 14175: 14066: 13947:partition function 13933: 13560: 13369: 13285: 13078:the statistics of 13006:critical exponents 12976:Alexander Polyakov 12908: 12881: 12854: 12715: 12519: 12483: 12454: 12402: 12381: 12366: 12345: 12341: 12308: 12290: 12262: 12244: 12203: 12142: 12112: 12088: 12024: 11964: 11873: 11794: 11710: 11684: 11638: 11612: 11585: 11556: 11457: 11430: 11382: 11261: 11231: 11023: 10992: 10927: 10900: 10873: 10804: 10784: 10755: 10724:is small, we have 10714: 10691: 10623: 10510: 10333: 10237: 10132: 10019: 9965: 9919: 9890: 9788: 9734: 9714: 9672: 9596: 9578: 9577: 9541: 9526: 9525: 9456: 9432: 9417: 9399: 9381: 9363: 9324: 9295: 9277: 9276: 9240: 9225: 9224: 9131: 8998: 8954: 8929: 8914: 8888:partition function 8874: 8777: 8676: 8536: 8463: 8445: 8427: 8368: 8359: 8233: 8113: 8075: 7978: 7768: 7684: 7538: 7432: 7353: 7341: 7327: 7186: 7117: 7062: 7045: 6904: 6767: 6603: 6527: 6513: 6471: 6371: 6306: 6237: 6167: 6031: 5950: 5945: 5762: 5651: 5137: 5113: 5050: 5036: 5000: 4895: 4829: 4783: 4757: 4678: 4645: 4608: 4499: 4385:partition function 4293: 4237: 4174: 4154: 4134: 4114: 4094: 4066: 4039: 4003: 3964: 3935: 3921: 3865: 3782: 3755: 3653: 3558: 3400: 3343: 3291: 3230: 3184: 3135: 3121: 3070: 3050: 3021: 2999: 2997: 2978: 2924: 2870: 2819: 2768: 2686: 2643:{\displaystyle ij} 2640: 2617: 2585: 2568: 2469: 2433: 2384: 2357: 2330: 2310: 2274: 2247: 2220: 2200: 2171: 2151: 2112: 2078: 2009:A subset S of the 1986: 1962: 1869: 1832: 1756: 1717: 1678: 1616: 1574: 1532: 1485: 1448: 1428: 1389: 1367: 1335: 1288: 1276:partition function 1262: 1236: 1195: 1143: 1060: 1023: 999: 979: 959: 928: 904: 855: 797: 771: 741: 715: 682: 647: 601:spin configuration 589: 538: 511: 485: 461: 360:discrete variables 348:mathematical model 94:Anyonic statistics 25756: 25755: 25710:Signal processing 25429:Doob's upcrossing 25424:Doob's martingale 25388:Engelbert–Schmidt 25331:Donsker's theorem 25265:Feller-continuous 25133:Rendleman–Bartter 24923:Dirichlet process 24840:Branching process 24809:Telegraph process 24702:Geometric process 24682:Empirical process 24672:Diffusion process 24528:Branching process 24523:Bernoulli process 24445:Ising Model code 24330:Physical Review E 24242:10.1119/1.1976530 24126:978-0-8176-4248-8 24029:10.1063/1.1703955 23862:978-0-521-40805-9 23844:978-2-86883-360-0 23786:978-0-471-81518-1 23757:978-3-540-64883-3 23728:978-0-521-59082-2 23648:978-0-12-083180-7 23356:IEEE Transactions 23181:(5966): 685–689. 23121:Physical Review E 22841:(12144): 012144. 22835:Physical Review E 22790:Physical Review X 22589:978-3-642-33038-4 22536:978-0-12-083180-7 22411:978-981-4495-00-4 22352:Nucleic Acids Res 22269:Physical Review E 21834:Gallavotti (1999) 21709:Boltzmann machine 21522: 21507: 21479: 21459: 21248: 20719: 20681: 20616: 20382: 20275: 20226: 20223: 20157: 20141:for each neuron: 20125:was published by 20019: 20017: 19961: 19959: 19841: 19790: 19788: 19240: 19052: 18858: 18765: 18722: 18587:= 0 is unstable. 18537: 18417: 18333: 18259:mean-field theory 18253:Mean-field theory 18224: 18202: 18163: 18096: 18055: 18004: 17977: 17974: 17877: 17852: 17822: 17736: 17719: 17593: 17565: 17538: 17505: 17410: 17382: 17328: 17247: 17203:operator products 17121: 17065: 17015: 16984: 16810: 16686:Feynman expansion 16653: 16625: 16589: 16514:fractal dimension 16470: 16447: 16380: 16236: 16105: 16075: 16063: 15994: 15935: 15894: 15879: 15777: 15710: 15070:) for each pair. 15016: 14966: 14675: 13928: 12980:Vladimir Dotsenko 12851: 12332: 12079: 11698: 11669: 11554: 11529: 11523: 11370: 11322: 10990: 10833:cluster expansion 10807:{\displaystyle K} 10662: 10604: 10586: 10550: 10470: 10450: 10432: 10396: 10358: 10307: 10283: 10010: 9779: 9737:{\displaystyle N} 9568: 9498: 9267: 9197: 9121: 8945: 8920: 8912: 8905: 8750: 8671: 8516: 8221: 8172: 8085:The coefficients 7943: 7733: 7669: 7485: 7322: 7151: 6757: 6651: 6617: 6588: 6504: 6438: 5938: 5892: 5712: 5601: 5569: 5528: 5512: 5463: 5416: 5333: 5105: 5092: 5027: 4992: 4979: 4774: 4636: 4618:since there are 2 4463: 4364:quantum mechanics 4222: 4177:{\displaystyle y} 4157:{\displaystyle x} 4137:{\displaystyle y} 4117:{\displaystyle x} 4097:{\displaystyle S} 3906: 3264: 3087: 2944: 2897: 2836: 2785: 2734: 2534: 2532: 2051: 1954: 1941: 1824: 1811: 1586:antiferromagnetic 1392:{\displaystyle f} 1326: 1291:{\displaystyle f} 1278:. For a function 1227: 1186: 1138: 1037:is given by the 1002:{\displaystyle j} 982:{\displaystyle i} 895: 837: 488:{\displaystyle d} 418:mean-field theory 372:phase transitions 328: 327: 222:Gibbs free energy 74:Maxwell–Boltzmann 25796: 25730:Machine learning 25617:Usual hypotheses 25500:Girsanov theorem 25485:Dynkin's formula 25250:Continuous paths 25158:Actuarial models 25098:Garman–Kohlhagen 25068:Black–Karasinski 25063:Black–Derman–Toy 25050:Financial models 24916:Fields and other 24845:Gaussian process 24794:Sigma-martingale 24598:Additive process 24500: 24493: 24486: 24477: 24476: 24354: 24345: 24324: 24296: 24272: 24244: 24228:(8): 1033–1036, 24216: 24177: 24138: 24113: 24097: 24073:(3–4): 117–149, 24058: 24057: 24056: 24047:, archived from 23979: 23958:(also available 23886: 23865: 23847: 23825: 23789: 23768: 23731: 23720: 23709: 23680: 23659: 23631: 23603: 23602: 23600: 23588: 23582: 23581: 23575: 23567: 23536: 23530: 23529: 23519: 23493: 23469: 23463: 23462: 23434: 23428: 23427: 23391: 23385: 23384: 23382: 23370: 23364: 23363: 23362:(21): 1197–1206. 23351: 23345: 23344: 23334: 23324: 23307:(8): 2554–2558, 23296: 23290: 23289: 23279: 23261: 23229: 23223: 23222: 23169: 23163: 23162: 23136: 23127:(5): 6399–6406. 23116: 23110: 23109: 23099: 23090: 23084: 23083: 23081: 23079: 23074:on July 22, 2015 23070:. Archived from 23059: 23053: 23052: 23018: 22998: 22992: 22991: 22965: 22956:(4–5): 869–914. 22944: 22938: 22937: 22911: 22891: 22885: 22884: 22850: 22830: 22824: 22823: 22805: 22784: 22778: 22777: 22751: 22731: 22725: 22724: 22698: 22678: 22672: 22671: 22664: 22658: 22657: 22655: 22654: 22639: 22633: 22632: 22600: 22594: 22593: 22565: 22559: 22558: 22557: 22556: 22547:, archived from 22520: 22514: 22513: 22482:Comm. Math. Phys 22469: 22463: 22462: 22431:Comm. Math. Phys 22422: 22416: 22415: 22395: 22386: 22385: 22375: 22343: 22337: 22336: 22323: 22317: 22316: 22314: 22313: 22284: 22260: 22254: 22253: 22235: 22210: 22209: 22191: 22167: 22161: 22160: 22126: 22102: 22096: 22095: 22047: 22041: 22035: 22024: 22023: 21987: 21981: 21980: 21978: 21977: 21971: 21965:. Archived from 21940: 21931:(4–5): 869–914. 21922: 21913: 21907: 21906: 21870: 21864: 21861:Baierlein (1999) 21857: 21851: 21843: 21837: 21830: 21759:Maximal evenness 21704:Binder parameter 21658: 21656: 21655: 21650: 21638: 21636: 21635: 21630: 21618: 21616: 21615: 21610: 21598: 21596: 21595: 21590: 21578: 21576: 21575: 21570: 21558: 21556: 21555: 21550: 21536: 21534: 21533: 21528: 21523: 21521: 21508: 21503: 21480: 21475: 21461: 21460: 21457: 21450: 21442: 21437: 21436: 21420: 21418: 21417: 21412: 21410: 21409: 21393: 21391: 21390: 21385: 21370: 21368: 21367: 21362: 21360: 21359: 21343: 21341: 21340: 21335: 21333: 21332: 21316: 21314: 21313: 21308: 21306: 21305: 21289: 21287: 21286: 21281: 21279: 21278: 21262: 21241:is derived from 21240: 21238: 21237: 21232: 21230: 21229: 21228: 21211: 21209: 21208: 21203: 21201: 21200: 21184: 21182: 21181: 21176: 21174: 21173: 21155: 21154: 21135: 21133: 21132: 21127: 21125: 21124: 21108: 21106: 21105: 21100: 21098: 21097: 21079: 21078: 21065: 21060: 21045: 21044: 21035: 21034: 21033: 21032: 20991: 20990: 20982: 20981: 20980: 20963: 20962: 20953: 20952: 20920: 20918: 20917: 20912: 20899: 20897: 20896: 20891: 20879: 20877: 20876: 20871: 20850: 20848: 20847: 20842: 20840: 20839: 20817: 20815: 20814: 20809: 20785: 20783: 20782: 20777: 20755: 20753: 20752: 20747: 20736: 20735: 20720: 20718: 20717: 20705: 20702: 20697: 20682: 20680: 20663: 20643: 20617: 20615: 20598: 20590: 20543: 20541: 20540: 20535: 20533: 20532: 20516: 20514: 20513: 20508: 20506: 20505: 20438: 20436: 20435: 20430: 20425: 20424: 20414: 20413: 20403: 20402: 20383: 20375: 20344: 20342: 20341: 20336: 20334: 20333: 20314: 20312: 20311: 20306: 20304: 20303: 20294: 20293: 20283: 20271: 20270: 20261: 20260: 20251: 20250: 20237: 20225: 20216: 20196: 20194: 20193: 20188: 20186: 20185: 20176: 20175: 20165: 20121:. This learning 20107:action potential 20101:The activity of 20066: 20064: 20063: 20058: 20056: 20055: 20027: 20018: 20010: 20005: 20004: 19995: 19994: 19981: 19960: 19952: 19935: 19933: 19932: 19927: 19922: 19908: 19907: 19892: 19891: 19873: 19871: 19870: 19865: 19863: 19862: 19849: 19837: 19836: 19827: 19826: 19810: 19789: 19781: 19700:magnetic moments 19656: 19654: 19653: 19648: 19610: 19578: 19576: 19575: 19570: 19516: 19514: 19513: 19508: 19494: 19493: 19481: 19480: 19425: 19423: 19422: 19417: 19409: 19408: 19396: 19395: 19365: 19364: 19360: 19359: 19347: 19346: 19323: 19322: 19318: 19317: 19305: 19304: 19251: 19249: 19248: 19243: 19241: 19236: 19178: 19176: 19175: 19170: 19162: 19161: 19152: 19151: 19142: 19141: 19132: 19131: 19122: 19121: 19097: 19096: 19087: 19086: 19077: 19076: 19063: 19004: 19002: 19001: 18996: 18994: 18993: 18966: 18964: 18963: 18958: 18956: 18955: 18915: 18913: 18912: 18907: 18899: 18898: 18859: 18857: 18849: 18832: 18776: 18774: 18773: 18768: 18766: 18758: 18733: 18731: 18730: 18725: 18723: 18718: 18717: 18716: 18707: 18706: 18684: 18572: 18570: 18569: 18564: 18538: 18536: 18535: 18534: 18506: 18505: 18504: 18476: 18428: 18426: 18425: 18420: 18418: 18416: 18415: 18414: 18383: 18366: 18364: 18363: 18358: 18356: 18355: 18334: 18332: 18318: 18240: 18238: 18237: 18232: 18230: 18226: 18225: 18217: 18204: 18195: 18174: 18172: 18171: 18166: 18164: 18162: 18151: 18133:The coefficient 18130: 18128: 18127: 18122: 18120: 18097: 18092: 18084: 18056: 18051: 18043: 18026: 18024: 18023: 18018: 18016: 18015: 18005: 18003: 17995: 17994: 17985: 17978: 17976: 17975: 17967: 17958: 17888: 17886: 17885: 17880: 17878: 17876: 17875: 17874: 17858: 17853: 17851: 17850: 17849: 17833: 17831: 17823: 17821: 17820: 17819: 17797: 17792: 17791: 17782: 17781: 17754:Λ divided by Λ: 17747: 17745: 17744: 17739: 17737: 17735: 17734: 17722: 17720: 17718: 17717: 17716: 17697: 17693: 17692: 17682: 17680: 17679: 17657: 17649: 17614:The coefficient 17611: 17609: 17608: 17603: 17601: 17594: 17589: 17575: 17566: 17561: 17553: 17544: 17540: 17539: 17534: 17526: 17506: 17501: 17493: 17476: 17474: 17473: 17468: 17463: 17462: 17424: 17422: 17421: 17416: 17411: 17406: 17405: 17404: 17391: 17383: 17381: 17380: 17379: 17364: 17363: 17347: 17339: 17338: 17329: 17324: 17323: 17322: 17309: 17269: 17267: 17266: 17261: 17253: 17249: 17248: 17243: 17235: 17163: 17161: 17160: 17155: 17138: 17137: 17122: 17120: 17110: 17109: 17097: 17096: 17083: 17066: 17064: 17063: 17051: 17026: 17024: 17023: 17018: 17016: 17014: 17004: 17003: 16987: 16985: 16983: 16982: 16981: 16962: 16958: 16957: 16947: 16945: 16944: 16922: 16914: 16889: 16888: 16834: 16832: 16831: 16826: 16821: 16820: 16811: 16803: 16783: 16782: 16680: 16678: 16677: 16672: 16669: 16665: 16664: 16663: 16654: 16652: 16641: 16636: 16635: 16626: 16618: 16613: 16612: 16607: 16595: 16590: 16582: 16572: 16571: 16486:Bessel functions 16481: 16479: 16478: 16473: 16471: 16469: 16468: 16453: 16452: 16448: 16443: 16433: 16396: 16394: 16393: 16388: 16386: 16385: 16381: 16376: 16355: 16353: 16352: 16347: 16345: 16344: 16343: 16342: 16319: 16318: 16247: 16245: 16244: 16239: 16237: 16235: 16228: 16227: 16214: 16209: 16208: 16195: 16194: 16127: 16125: 16124: 16119: 16117: 16116: 16106: 16104: 16096: 16095: 16086: 16076: 16074: 16073: 16068: 16064: 16053: 16043: 16005: 16003: 16002: 15997: 15995: 15990: 15975: 15973: 15972: 15967: 15941: 15937: 15936: 15934: 15926: 15918: 15916: 15915: 15895: 15893: 15882: 15880: 15878: 15877: 15859: 15836: 15835: 15788: 15786: 15785: 15780: 15778: 15776: 15775: 15757: 15721: 15719: 15718: 15713: 15711: 15709: 15708: 15690: 15671: 15669: 15668: 15663: 15661: 15632: 15631: 15619: 15618: 15580:dimensions, and 15569: 15567: 15566: 15561: 15533: 15531: 15530: 15525: 15477: 15475: 15474: 15469: 15467: 15423: 15422: 15412: 15410: 15361: 15357: 15349: 15344: 15326: 15325: 15290: 15288: 15287: 15282: 15277: 15276: 15261: 15260: 15236: 15235: 15220: 15219: 15198: 15197: 15182: 15181: 15160: 15159: 15141: 15140: 15116: 15115: 15097: 15096: 15027: 15025: 15024: 15019: 15017: 15015: 15008: 15007: 14997: 14996: 14969: 14967: 14965: 14964: 14963: 14944: 14936: 14762: 14760: 14759: 14754: 14746: 14745: 14715: 14711: 14710: 14709: 14694: 14693: 14676: 14674: 14663: 14611: 14609: 14608: 14603: 14601: 14600: 14595: 14591: 14587: 14586: 14562: 14561: 14534: 14533: 14512: 14511: 14484: 14483: 14474: 14473: 14446: 14445: 14339: 14337: 14336: 14331: 14326: 14322: 14321: 14320: 14296: 14295: 14280: 14279: 14259: 14258: 14220:does not absorb 14184: 14182: 14181: 14176: 14171: 14170: 14154: 14153: 14075: 14073: 14072: 14067: 14065: 14064: 14063: 14059: 14058: 14057: 14042: 14041: 14036: 14024: 14013: 14012: 13992: 13991: 13942: 13940: 13939: 13934: 13929: 13927: 13926: 13925: 13924: 13920: 13919: 13918: 13903: 13902: 13897: 13885: 13874: 13873: 13853: 13852: 13821: 13817: 13816: 13795: 13794: 13776: 13775: 13760: 13759: 13758: 13754: 13753: 13752: 13737: 13736: 13731: 13719: 13708: 13707: 13687: 13686: 13655: 13644: 13643: 13622: 13621: 13603: 13602: 13569: 13567: 13566: 13561: 13556: 13555: 13528: 13527: 13526: 13522: 13521: 13520: 13505: 13504: 13499: 13487: 13476: 13475: 13455: 13454: 13378: 13376: 13375: 13370: 13365: 13364: 13351: 13350: 13340: 13339: 13313: 13294: 13292: 13291: 13286: 13281: 13277: 13270: 13269: 13254: 13253: 13241: 13240: 13228: 13227: 13217: 13212: 13194: 13193: 13173: 13172: 12990:Phase transition 12963:Three dimensions 12917: 12915: 12914: 12909: 12907: 12906: 12890: 12888: 12887: 12882: 12880: 12879: 12863: 12861: 12860: 12855: 12853: 12852: 12844: 12842: 12838: 12837: 12836: 12828: 12824: 12823: 12822: 12801: 12800: 12724: 12722: 12721: 12716: 12711: 12710: 12699: 12698: 12697: 12686: 12685: 12680: 12679: 12678: 12667: 12666: 12654: 12653: 12638: 12637: 12622: 12621: 12606: 12605: 12600: 12599: 12598: 12587: 12586: 12574: 12573: 12558: 12557: 12552: 12551: 12550: 12532: 12527: 12492: 12490: 12489: 12484: 12478: 12477: 12467: 12462: 12411: 12409: 12408: 12403: 12400: 12395: 12379: 12374: 12358: 12353: 12340: 12317: 12315: 12314: 12309: 12303: 12298: 12271: 12269: 12268: 12263: 12257: 12252: 12235:spin at site i: 12212: 12210: 12209: 12204: 12196: 12195: 12183: 12151: 12149: 12148: 12143: 12135: 12121: 12119: 12118: 12113: 12105: 12087: 12069: 12033: 12031: 12030: 12025: 12020: 12019: 12018: 12017: 12008: 12007: 11973: 11971: 11970: 11965: 11963: 11962: 11938: 11937: 11928: 11927: 11903: 11902: 11882: 11880: 11879: 11874: 11872: 11871: 11803: 11801: 11800: 11795: 11790: 11789: 11783: 11779: 11778: 11777: 11753: 11752: 11726: 11725: 11709: 11697: 11696: 11683: 11647: 11645: 11644: 11639: 11621: 11619: 11618: 11613: 11611: 11610: 11594: 11592: 11591: 11586: 11584: 11583: 11565: 11563: 11562: 11557: 11555: 11547: 11530: 11528: 11524: 11519: 11498: 11490: 11485: 11484: 11466: 11464: 11463: 11458: 11456: 11455: 11439: 11437: 11436: 11431: 11423: 11422: 11410: 11409: 11391: 11389: 11388: 11383: 11375: 11371: 11369: 11368: 11367: 11354: 11353: 11352: 11339: 11327: 11323: 11321: 11320: 11319: 11306: 11305: 11304: 11291: 11270: 11268: 11267: 11262: 11260: 11259: 11240: 11238: 11237: 11232: 11221: 11220: 11199: 11198: 11168: 11167: 11146: 11145: 11115: 11114: 11087: 11086: 11053: 11052: 11039: 11031: 11022: 11021: 11008: 11000: 10991: 10989: 10988: 10987: 10971: 10936: 10934: 10933: 10928: 10926: 10925: 10909: 10907: 10906: 10901: 10899: 10898: 10882: 10880: 10879: 10874: 10825:Peierls argument 10813: 10811: 10810: 10805: 10793: 10791: 10790: 10785: 10764: 10762: 10761: 10756: 10723: 10721: 10720: 10715: 10700: 10698: 10697: 10692: 10663: 10655: 10647: 10632: 10630: 10629: 10624: 10619: 10605: 10597: 10592: 10588: 10587: 10567: 10551: 10543: 10519: 10517: 10516: 10511: 10506: 10495: 10494: 10490: 10471: 10469: 10465: 10453: 10451: 10443: 10438: 10434: 10433: 10413: 10397: 10389: 10375: 10374: 10359: 10351: 10342: 10340: 10339: 10334: 10308: 10300: 10295: 10284: 10264: 10246: 10244: 10243: 10238: 10230: 10229: 10217: 10216: 10186: 10185: 10184: 10183: 10174: 10173: 10164: 10141: 10139: 10138: 10133: 10119: 10118: 10106: 10105: 10063: 10062: 10050: 10049: 10018: 9991: 9990: 9974: 9972: 9971: 9966: 9958: 9957: 9945: 9944: 9928: 9926: 9925: 9920: 9899: 9897: 9896: 9891: 9886: 9885: 9881: 9880: 9868: 9867: 9855: 9854: 9837: 9836: 9832: 9831: 9819: 9818: 9806: 9805: 9787: 9760: 9759: 9743: 9741: 9740: 9735: 9723: 9721: 9720: 9715: 9698: 9697: 9681: 9679: 9678: 9673: 9605: 9603: 9602: 9597: 9591: 9586: 9576: 9560: 9555: 9539: 9534: 9524: 9465: 9463: 9462: 9457: 9451: 9446: 9430: 9425: 9412: 9407: 9390: 9388: 9387: 9382: 9376: 9371: 9361: 9356: 9337: 9332: 9304: 9302: 9301: 9296: 9290: 9285: 9275: 9259: 9254: 9238: 9233: 9223: 9165:transverse-field 9149:phase transition 9140: 9138: 9137: 9132: 9127: 9123: 9122: 9120: 9119: 9118: 9093: 9092: 9074: 9007: 9005: 9004: 8999: 8994: 8993: 8953: 8941: 8940: 8939: 8938: 8928: 8913: 8910: 8883: 8881: 8880: 8875: 8870: 8869: 8868: 8854: 8832: 8831: 8822: 8821: 8786: 8784: 8783: 8778: 8773: 8772: 8751: 8749: 8735: 8685: 8683: 8682: 8677: 8672: 8670: 8669: 8648: 8647: 8623: 8622: 8607: 8590: 8589: 8574: 8573: 8545: 8543: 8542: 8537: 8532: 8528: 8527: 8526: 8521: 8517: 8515: 8514: 8505: 8504: 8495: 8476: 8471: 8458: 8453: 8440: 8435: 8423: 8419: 8418: 8377: 8375: 8374: 8369: 8364: 8363: 8356: 8355: 8326: 8325: 8306: 8305: 8288: 8287: 8242: 8240: 8239: 8234: 8232: 8231: 8230: 8222: 8217: 8209: 8202: 8201: 8200: 8178: 8177: 8173: 8168: 8160: 8150: 8149: 8148: 8122: 8120: 8119: 8114: 8112: 8111: 8110: 8084: 8082: 8081: 8076: 8071: 8070: 8069: 8068: 8056: 8055: 8038: 8037: 8036: 8035: 8023: 8022: 8008: 8007: 8006: 8005: 7993: 7992: 7977: 7976: 7975: 7957: 7956: 7939: 7938: 7937: 7936: 7927: 7926: 7906: 7905: 7904: 7903: 7880: 7879: 7878: 7877: 7868: 7867: 7847: 7846: 7845: 7844: 7824: 7823: 7822: 7821: 7812: 7811: 7791: 7790: 7789: 7788: 7767: 7766: 7765: 7747: 7746: 7693: 7691: 7690: 7685: 7680: 7679: 7674: 7670: 7668: 7667: 7666: 7648: 7647: 7634: 7633: 7632: 7614: 7613: 7600: 7587: 7586: 7571: 7570: 7547: 7545: 7544: 7539: 7534: 7530: 7529: 7528: 7510: 7509: 7486: 7478: 7441: 7439: 7438: 7433: 7428: 7427: 7416: 7412: 7411: 7410: 7392: 7391: 7367: 7366: 7361: 7340: 7335: 7320: 7315: 7294: 7293: 7292: 7291: 7282: 7281: 7252: 7251: 7250: 7249: 7240: 7239: 7219: 7218: 7217: 7216: 7207: 7206: 7185: 7184: 7183: 7165: 7164: 7126: 7124: 7123: 7118: 7103: 7102: 7087: 7086: 7070: 7054: 7052: 7051: 7046: 7041: 7037: 7036: 7035: 7026: 7025: 7001: 7000: 6991: 6990: 6913: 6911: 6910: 6905: 6900: 6899: 6898: 6884: 6841: 6840: 6825: 6824: 6806: 6805: 6796: 6795: 6776: 6774: 6773: 6768: 6763: 6759: 6758: 6756: 6755: 6734: 6733: 6709: 6708: 6693: 6676: 6675: 6652: 6644: 6618: 6616: 6605: 6602: 6536: 6534: 6533: 6528: 6523: 6522: 6512: 6497: 6496: 6481: 6480: 6470: 6390:nearest neighbor 6380: 6378: 6377: 6372: 6370: 6369: 6361: 6346: 6338: 6337: 6315: 6313: 6312: 6307: 6305: 6304: 6296: 6281: 6273: 6272: 6246: 6244: 6243: 6238: 6236: 6235: 6227: 6212: 6204: 6203: 6176: 6174: 6173: 6168: 6166: 6165: 6157: 6142: 6134: 6133: 6040: 6038: 6037: 6032: 6027: 6026: 6022: 6021: 6009: 6008: 5959: 5957: 5956: 5951: 5949: 5948: 5939: 5936: 5916: 5915: 5903: 5902: 5893: 5890: 5883: 5882: 5878: 5877: 5865: 5864: 5771: 5769: 5768: 5763: 5758: 5757: 5753: 5752: 5740: 5739: 5713: 5711: 5691: 5671: 5660: 5658: 5657: 5652: 5647: 5646: 5642: 5641: 5629: 5628: 5602: 5600: 5599: 5598: 5594: 5593: 5570: 5562: 5559: 5558: 5557: 5553: 5552: 5529: 5521: 5518: 5513: 5511: 5501: 5500: 5490: 5480: 5479: 5469: 5464: 5462: 5442: 5422: 5417: 5415: 5377: 5339: 5334: 5332: 5312: 5292: 5280:Detailed balance 5258:Since there are 5177:detailed balance 5146: 5144: 5143: 5138: 5133: 5132: 5123: 5122: 5112: 5103: 5059: 5057: 5056: 5051: 5046: 5045: 5035: 5020: 5019: 5010: 5009: 4999: 4990: 4873:Lee–Yang theorem 4838: 4836: 4835: 4830: 4825: 4824: 4806: 4805: 4793: 4792: 4782: 4766: 4764: 4763: 4758: 4753: 4752: 4687: 4685: 4684: 4679: 4674: 4673: 4664: 4663: 4644: 4617: 4615: 4614: 4609: 4604: 4603: 4508: 4506: 4505: 4500: 4495: 4494: 4484: 4479: 4464: 4456: 4421:Peierls droplets 4353:Ludwig Boltzmann 4302: 4300: 4299: 4294: 4286: 4285: 4276: 4275: 4260: 4259: 4250: 4249: 4236: 4215: 4214: 4205: 4204: 4184:being outside), 4183: 4181: 4180: 4175: 4163: 4161: 4160: 4155: 4143: 4141: 4140: 4135: 4123: 4121: 4120: 4115: 4103: 4101: 4100: 4095: 4075: 4073: 4072: 4067: 4065: 4064: 4048: 4046: 4045: 4040: 4035: 4034: 4012: 4010: 4009: 4004: 3993: 3992: 3973: 3971: 3970: 3965: 3944: 3942: 3941: 3936: 3931: 3930: 3920: 3896: 3895: 3874: 3872: 3871: 3866: 3858: 3857: 3842: 3841: 3823: 3822: 3813: 3812: 3791: 3789: 3788: 3783: 3781: 3780: 3764: 3762: 3761: 3756: 3754: 3753: 3678:Lars Onsager 3671:Peierls argument 3662: 3660: 3659: 3654: 3631: 3630: 3621: 3620: 3611: 3610: 3588:disordered phase 3580:phase transition 3567: 3565: 3564: 3559: 3554: 3550: 3549: 3535: 3498: 3497: 3488: 3487: 3478: 3477: 3437:phase transition 3423: = 0. 3419: = 1, 3352: 3350: 3349: 3344: 3339: 3335: 3331: 3330: 3304: 3303: 3290: 3239: 3237: 3236: 3231: 3229: 3225: 3221: 3220: 3193: 3191: 3190: 3185: 3183: 3182: 3164: 3163: 3144: 3142: 3141: 3136: 3134: 3133: 3120: 3116: 3115: 3079: 3077: 3076: 3071: 3059: 3057: 3056: 3051: 3030: 3028: 3027: 3022: 3008: 3006: 3005: 3000: 2998: 2991: 2990: 2977: 2973: 2972: 2937: 2936: 2923: 2887: 2883: 2882: 2869: 2865: 2864: 2832: 2831: 2818: 2814: 2813: 2781: 2780: 2767: 2763: 2762: 2695: 2693: 2692: 2687: 2685: 2684: 2669: 2668: 2649: 2647: 2646: 2641: 2626: 2624: 2623: 2618: 2616: 2615: 2594: 2592: 2591: 2586: 2581: 2580: 2567: 2563: 2562: 2533: 2525: 2520: 2516: 2512: 2511: 2478: 2476: 2475: 2470: 2465: 2464: 2442: 2440: 2439: 2434: 2432: 2428: 2424: 2423: 2393: 2391: 2390: 2385: 2383: 2382: 2366: 2364: 2363: 2358: 2356: 2355: 2339: 2337: 2336: 2331: 2319: 2317: 2316: 2311: 2306: 2305: 2283: 2281: 2280: 2275: 2273: 2272: 2256: 2254: 2253: 2248: 2246: 2245: 2229: 2227: 2226: 2221: 2209: 2207: 2206: 2201: 2180: 2178: 2177: 2172: 2160: 2158: 2157: 2152: 2141: 2140: 2121: 2119: 2118: 2113: 2111: 2110: 2101: 2100: 2091: 2090: 2077: 1995: 1993: 1992: 1987: 1982: 1981: 1972: 1971: 1961: 1952: 1878: 1876: 1875: 1870: 1865: 1864: 1855: 1854: 1845: 1844: 1831: 1822: 1765: 1763: 1762: 1757: 1749: 1748: 1727:, the spin site 1726: 1724: 1723: 1718: 1710: 1709: 1688:, the spin site 1687: 1685: 1684: 1679: 1671: 1670: 1626:, the spins are 1625: 1623: 1622: 1617: 1609: 1608: 1583: 1581: 1580: 1575: 1567: 1566: 1541: 1539: 1538: 1533: 1525: 1524: 1494: 1492: 1491: 1486: 1457: 1455: 1454: 1449: 1437: 1435: 1434: 1429: 1418: 1417: 1398: 1396: 1395: 1390: 1376: 1374: 1373: 1368: 1357: 1356: 1334: 1322: 1321: 1297: 1295: 1294: 1289: 1271: 1269: 1268: 1263: 1261: 1260: 1235: 1223: 1222: 1204: 1202: 1201: 1196: 1188: 1187: 1184: 1175: 1152: 1150: 1149: 1144: 1139: 1137: 1136: 1127: 1126: 1102: 1088: 1087: 1069: 1067: 1066: 1061: 1032: 1030: 1029: 1024: 1008: 1006: 1005: 1000: 988: 986: 985: 980: 968: 966: 965: 960: 937: 935: 934: 929: 924: 923: 914: 913: 903: 888: 887: 878: 877: 868: 867: 854: 807:is given by the 806: 804: 803: 798: 796: 780: 778: 777: 772: 770: 769: 750: 748: 747: 742: 724: 722: 721: 716: 714: 713: 691: 689: 688: 683: 656: 654: 653: 648: 646: 645: 630: 629: 614: 598: 596: 595: 590: 564: 563: 547: 545: 544: 539: 537: 536: 520: 518: 517: 512: 494: 492: 491: 486: 470: 468: 467: 462: 399:Lars Onsager 387:Wilhelm Lenz 380:phase transition 336:Lenz–Ising model 320: 313: 306: 99:Braid statistics 33: 19: 18: 25804: 25803: 25799: 25798: 25797: 25795: 25794: 25793: 25759: 25758: 25757: 25752: 25734: 25695:Queueing theory 25638: 25580:Skorokhod space 25443: 25434:Kunita–Watanabe 25405: 25371:Sanov's theorem 25341:Ergodic theorem 25314: 25310:Time-reversible 25228: 25191:Queueing models 25185: 25181:Sparre–Anderson 25171:Cramér–Lundberg 25152: 25138:SABR volatility 25044: 25001: 24953:Boolean network 24911: 24897:Renewal process 24828: 24777:Non-homogeneous 24767:Poisson process 24657:Contact process 24620:Brownian motion 24590:Continuous time 24584: 24578:Maximal entropy 24509: 24504: 24464:Quanta Magazine 24410:Wayback Machine 24362: 24357: 24276:Krizan, J. E.; 24186:Physical Review 24159: 24136: 24067:Physical Review 24054: 24052: 23883: 23863: 23845: 23787: 23758: 23729: 23717:Thermal Physics 23649: 23611: 23606: 23589: 23585: 23569: 23568: 23537: 23533: 23470: 23466: 23439:Physical Review 23435: 23431: 23396:Physical Review 23392: 23388: 23371: 23367: 23352: 23348: 23297: 23293: 23230: 23226: 23170: 23166: 23134:physics/9901052 23117: 23113: 23097: 23091: 23087: 23077: 23075: 23060: 23056: 22999: 22995: 22945: 22941: 22892: 22888: 22831: 22827: 22785: 22781: 22732: 22728: 22679: 22675: 22666: 22665: 22661: 22652: 22650: 22648:Quanta Magazine 22640: 22636: 22621:10.1119/1.11224 22601: 22597: 22590: 22566: 22562: 22554: 22552: 22537: 22521: 22517: 22470: 22466: 22423: 22419: 22412: 22396: 22389: 22344: 22340: 22335:. 28 June 2022. 22325: 22324: 22320: 22311: 22309: 22261: 22257: 22250: 22236: 22213: 22168: 22164: 22103: 22099: 22048: 22044: 22036: 22027: 21988: 21984: 21975: 21973: 21969: 21920: 21914: 21910: 21871: 21867: 21858: 21854: 21844: 21840: 21831: 21827: 21823: 21818: 21804:Wolff algorithm 21694: 21644: 21641: 21640: 21624: 21621: 21620: 21604: 21601: 21600: 21584: 21581: 21580: 21564: 21561: 21560: 21544: 21541: 21540: 21502: 21474: 21456: 21452: 21451: 21443: 21441: 21432: 21428: 21426: 21423: 21422: 21405: 21401: 21399: 21396: 21395: 21376: 21373: 21372: 21355: 21351: 21349: 21346: 21345: 21328: 21324: 21322: 21319: 21318: 21301: 21297: 21295: 21292: 21291: 21268: 21264: 21252: 21246: 21243: 21242: 21224: 21220: 21219: 21217: 21214: 21213: 21196: 21192: 21190: 21187: 21186: 21169: 21165: 21147: 21143: 21141: 21138: 21137: 21120: 21116: 21114: 21111: 21110: 21093: 21089: 21071: 21067: 21061: 21050: 21040: 21036: 21028: 21024: 21023: 21019: 20983: 20976: 20972: 20971: 20970: 20958: 20954: 20948: 20944: 20939: 20936: 20935: 20906: 20903: 20902: 20885: 20882: 20881: 20856: 20853: 20852: 20835: 20831: 20823: 20820: 20819: 20791: 20788: 20787: 20771: 20768: 20767: 20731: 20727: 20713: 20709: 20704: 20698: 20687: 20664: 20644: 20642: 20599: 20591: 20589: 20563: 20560: 20559: 20528: 20524: 20522: 20519: 20518: 20501: 20497: 20495: 20492: 20491: 20480: 20468: 20448: 20420: 20416: 20409: 20405: 20392: 20388: 20374: 20363: 20360: 20359: 20352: 20326: 20322: 20320: 20317: 20316: 20299: 20295: 20289: 20285: 20279: 20266: 20262: 20256: 20252: 20243: 20239: 20230: 20214: 20203: 20200: 20199: 20181: 20177: 20171: 20167: 20161: 20146: 20143: 20142: 20127:Shun'ichi Amari 20099: 20051: 20047: 20023: 20009: 20000: 19996: 19990: 19986: 19965: 19951: 19940: 19937: 19936: 19918: 19903: 19899: 19887: 19883: 19881: 19878: 19877: 19858: 19854: 19845: 19832: 19828: 19822: 19818: 19794: 19780: 19769: 19766: 19765: 19741: 19733: 19723: 19712: 19681: 19676: 19603: 19598: 19595: 19594: 19525: 19522: 19521: 19489: 19485: 19476: 19472: 19440: 19437: 19436: 19434: 19404: 19400: 19391: 19387: 19355: 19351: 19342: 19338: 19331: 19327: 19313: 19309: 19300: 19296: 19289: 19285: 19283: 19280: 19279: 19235: 19233: 19230: 19229: 19186: 19157: 19153: 19147: 19143: 19137: 19133: 19127: 19123: 19108: 19104: 19092: 19088: 19082: 19078: 19069: 19065: 19056: 19044: 19041: 19040: 19037: 19017: 18989: 18985: 18977: 18974: 18973: 18951: 18947: 18939: 18936: 18935: 18894: 18890: 18850: 18833: 18831: 18823: 18820: 18819: 18757: 18749: 18746: 18745: 18712: 18708: 18702: 18698: 18685: 18683: 18630: 18627: 18626: 18518: 18514: 18507: 18488: 18484: 18477: 18475: 18452: 18449: 18448: 18398: 18394: 18387: 18382: 18374: 18371: 18370: 18342: 18338: 18322: 18317: 18315: 18312: 18311: 18255: 18249: 18216: 18209: 18205: 18193: 18191: 18188: 18187: 18155: 18150: 18142: 18139: 18138: 18118: 18117: 18098: 18085: 18083: 18080: 18079: 18057: 18044: 18042: 18038: 18036: 18033: 18032: 17996: 17990: 17986: 17984: 17983: 17979: 17966: 17962: 17957: 17925: 17922: 17921: 17913: 17870: 17866: 17862: 17857: 17845: 17841: 17837: 17832: 17824: 17815: 17811: 17801: 17796: 17787: 17783: 17777: 17773: 17759: 17756: 17755: 17730: 17726: 17721: 17712: 17708: 17698: 17688: 17684: 17683: 17681: 17653: 17645: 17638: 17634: 17623: 17620: 17619: 17599: 17598: 17576: 17574: 17567: 17554: 17552: 17549: 17548: 17527: 17525: 17518: 17514: 17507: 17494: 17492: 17488: 17486: 17483: 17482: 17458: 17454: 17434: 17431: 17430: 17400: 17396: 17392: 17390: 17375: 17371: 17359: 17355: 17351: 17346: 17334: 17330: 17318: 17314: 17310: 17308: 17294: 17291: 17290: 17236: 17234: 17227: 17223: 17212: 17209: 17208: 17133: 17129: 17105: 17101: 17092: 17088: 17087: 17082: 17059: 17055: 17050: 17039: 17036: 17035: 16999: 16995: 16991: 16986: 16977: 16973: 16963: 16953: 16949: 16948: 16946: 16918: 16910: 16903: 16899: 16884: 16880: 16875: 16872: 16871: 16860:gives a change 16816: 16812: 16802: 16778: 16774: 16772: 16769: 16768: 16713: 16659: 16655: 16645: 16640: 16631: 16627: 16617: 16608: 16603: 16602: 16591: 16581: 16580: 16576: 16567: 16563: 16552: 16549: 16548: 16531: 16494: 16458: 16454: 16442: 16438: 16434: 16432: 16415: 16412: 16411: 16375: 16371: 16367: 16365: 16362: 16361: 16338: 16334: 16327: 16323: 16308: 16304: 16302: 16299: 16298: 16223: 16219: 16218: 16213: 16190: 16186: 16179: 16175: 16149: 16146: 16145: 16097: 16091: 16087: 16085: 16081: 16077: 16069: 16052: 16048: 16047: 16042: 16016: 16013: 16012: 15989: 15987: 15984: 15983: 15927: 15919: 15917: 15905: 15901: 15900: 15896: 15886: 15881: 15867: 15863: 15858: 15831: 15827: 15825: 15822: 15821: 15808: 15765: 15761: 15756: 15739: 15736: 15735: 15698: 15694: 15689: 15681: 15678: 15677: 15636: 15627: 15623: 15608: 15604: 15599: 15596: 15595: 15543: 15540: 15539: 15510: 15507: 15506: 15465: 15464: 15454: 15418: 15414: 15411: 15409: 15403: 15402: 15392: 15345: 15340: 15332: 15328: 15322: 15320: 15317: 15316: 15307: 15269: 15265: 15253: 15249: 15228: 15224: 15212: 15208: 15190: 15186: 15174: 15170: 15155: 15151: 15133: 15129: 15111: 15107: 15092: 15088: 15077: 15074: 15073: 15003: 14999: 14998: 14974: 14970: 14968: 14959: 14955: 14945: 14937: 14935: 14843: 14840: 14839: 14795: 14741: 14737: 14705: 14701: 14689: 14685: 14681: 14677: 14667: 14662: 14660: 14657: 14656: 14596: 14582: 14578: 14557: 14553: 14543: 14539: 14538: 14529: 14525: 14507: 14503: 14479: 14475: 14469: 14465: 14441: 14437: 14426: 14423: 14422: 14381:has dimension − 14371: 14316: 14312: 14291: 14287: 14275: 14271: 14267: 14263: 14254: 14250: 14239: 14236: 14235: 14166: 14162: 14149: 14145: 14134: 14131: 14130: 14110: 14053: 14049: 14037: 14032: 14031: 14020: 14008: 14004: 14000: 13996: 13987: 13983: 13976: 13972: 13954: 13951: 13950: 13914: 13910: 13898: 13893: 13892: 13881: 13869: 13865: 13861: 13857: 13848: 13844: 13837: 13833: 13822: 13812: 13808: 13790: 13786: 13771: 13767: 13748: 13744: 13732: 13727: 13726: 13715: 13703: 13699: 13695: 13691: 13682: 13678: 13671: 13667: 13656: 13654: 13639: 13635: 13617: 13613: 13598: 13594: 13583: 13580: 13579: 13536: 13532: 13516: 13512: 13500: 13495: 13494: 13483: 13471: 13467: 13463: 13459: 13450: 13446: 13439: 13435: 13418: 13415: 13414: 13399: 13392: 13360: 13356: 13346: 13342: 13332: 13328: 13326: 13323: 13322: 13311: 13309: 13302: 13265: 13261: 13249: 13245: 13236: 13232: 13223: 13219: 13213: 13202: 13189: 13185: 13181: 13177: 13168: 13164: 13150: 13147: 13146: 13139:By symmetry in 13071: 13062: 13025: 12992: 12984:continuum limit 12965: 12949: 12943: 12902: 12898: 12896: 12893: 12892: 12875: 12871: 12869: 12866: 12865: 12843: 12829: 12818: 12814: 12796: 12792: 12779: 12775: 12774: 12767: 12763: 12762: 12754: 12751: 12750: 12740: 12700: 12693: 12689: 12688: 12687: 12681: 12674: 12670: 12669: 12668: 12662: 12658: 12643: 12639: 12633: 12629: 12611: 12607: 12601: 12594: 12590: 12589: 12588: 12582: 12578: 12563: 12559: 12553: 12546: 12542: 12541: 12540: 12528: 12523: 12517: 12514: 12513: 12473: 12469: 12463: 12458: 12446: 12443: 12442: 12433: 12396: 12385: 12375: 12370: 12354: 12349: 12336: 12330: 12327: 12326: 12299: 12294: 12288: 12285: 12284: 12253: 12248: 12242: 12239: 12238: 12232: 12191: 12187: 12176: 12168: 12165: 12164: 12131: 12129: 12126: 12125: 12101: 12083: 12065: 12063: 12060: 12059: 12053: 12046: 12013: 12009: 12003: 11999: 11998: 11994: 11992: 11989: 11988: 11985:transfer matrix 11955: 11951: 11933: 11929: 11914: 11910: 11898: 11894: 11892: 11889: 11888: 11858: 11854: 11846: 11843: 11842: 11785: 11784: 11761: 11757: 11736: 11732: 11731: 11727: 11715: 11711: 11702: 11692: 11691: 11673: 11667: 11664: 11663: 11658: 11656:Transfer matrix 11627: 11624: 11623: 11606: 11602: 11600: 11597: 11596: 11579: 11575: 11573: 11570: 11569: 11546: 11518: 11499: 11491: 11489: 11480: 11476: 11474: 11471: 11470: 11451: 11447: 11445: 11442: 11441: 11418: 11414: 11405: 11401: 11399: 11396: 11395: 11363: 11359: 11355: 11348: 11344: 11340: 11338: 11334: 11315: 11311: 11307: 11300: 11296: 11292: 11290: 11286: 11278: 11275: 11274: 11255: 11251: 11249: 11246: 11245: 11216: 11212: 11194: 11190: 11163: 11159: 11141: 11137: 11110: 11106: 11082: 11078: 11048: 11044: 11032: 11027: 11017: 11013: 11001: 10996: 10983: 10979: 10975: 10970: 10944: 10941: 10940: 10937:, respectively 10921: 10917: 10915: 10912: 10911: 10894: 10890: 10888: 10885: 10884: 10862: 10859: 10858: 10852: 10846: 10820: 10799: 10796: 10795: 10770: 10767: 10766: 10729: 10726: 10725: 10709: 10706: 10705: 10654: 10640: 10638: 10635: 10634: 10612: 10596: 10566: 10562: 10558: 10542: 10525: 10522: 10521: 10499: 10486: 10482: 10478: 10461: 10457: 10452: 10442: 10412: 10408: 10404: 10388: 10370: 10366: 10350: 10348: 10345: 10344: 10299: 10288: 10263: 10252: 10249: 10248: 10225: 10221: 10212: 10208: 10179: 10175: 10169: 10165: 10157: 10156: 10152: 10147: 10144: 10143: 10114: 10110: 10101: 10097: 10058: 10054: 10045: 10041: 10014: 9986: 9982: 9980: 9977: 9976: 9953: 9949: 9940: 9936: 9934: 9931: 9930: 9905: 9902: 9901: 9876: 9872: 9863: 9859: 9850: 9846: 9842: 9838: 9827: 9823: 9814: 9810: 9801: 9797: 9793: 9789: 9783: 9755: 9751: 9749: 9746: 9745: 9729: 9726: 9725: 9693: 9689: 9687: 9684: 9683: 9637: 9634: 9633: 9630: 9628:Renormalization 9587: 9582: 9572: 9556: 9545: 9535: 9530: 9502: 9475: 9472: 9471: 9447: 9436: 9426: 9421: 9408: 9403: 9397: 9394: 9393: 9372: 9367: 9357: 9346: 9333: 9328: 9322: 9319: 9318: 9286: 9281: 9271: 9255: 9244: 9234: 9229: 9201: 9174: 9171: 9170: 9157: 9105: 9101: 9094: 9079: 9075: 9073: 9066: 9062: 9021: 9018: 9017: 8989: 8985: 8949: 8934: 8930: 8924: 8919: 8915: 8909: 8897: 8894: 8893: 8864: 8850: 8843: 8839: 8827: 8823: 8817: 8813: 8808: 8805: 8804: 8759: 8755: 8739: 8734: 8732: 8729: 8728: 8700: 8693: 8689: 8656: 8652: 8643: 8639: 8612: 8608: 8606: 8582: 8578: 8569: 8565: 8563: 8560: 8559: 8557: 8549: 8522: 8510: 8506: 8500: 8496: 8494: 8490: 8489: 8482: 8478: 8472: 8467: 8454: 8449: 8436: 8431: 8414: 8410: 8406: 8383: 8380: 8379: 8358: 8357: 8333: 8329: 8327: 8315: 8311: 8308: 8307: 8295: 8291: 8289: 8268: 8264: 8257: 8256: 8248: 8245: 8244: 8223: 8210: 8208: 8207: 8203: 8193: 8183: 8179: 8161: 8159: 8158: 8154: 8141: 8134: 8130: 8128: 8125: 8124: 8103: 8096: 8092: 8090: 8087: 8086: 8064: 8060: 8051: 8047: 8046: 8042: 8031: 8027: 8018: 8014: 8013: 8009: 8001: 7997: 7988: 7984: 7983: 7979: 7971: 7967: 7952: 7948: 7947: 7932: 7928: 7922: 7918: 7911: 7907: 7899: 7895: 7888: 7884: 7873: 7869: 7863: 7859: 7852: 7848: 7840: 7836: 7829: 7825: 7817: 7813: 7807: 7803: 7796: 7792: 7784: 7780: 7773: 7769: 7761: 7757: 7742: 7738: 7737: 7716: 7713: 7712: 7675: 7656: 7652: 7640: 7636: 7635: 7622: 7618: 7606: 7602: 7601: 7599: 7595: 7594: 7576: 7572: 7566: 7562: 7557: 7554: 7553: 7518: 7514: 7502: 7498: 7497: 7493: 7477: 7451: 7448: 7447: 7417: 7400: 7396: 7384: 7380: 7379: 7375: 7374: 7357: 7346: 7342: 7331: 7326: 7316: 7305: 7287: 7283: 7271: 7267: 7260: 7256: 7245: 7241: 7235: 7231: 7224: 7220: 7212: 7208: 7202: 7198: 7191: 7187: 7179: 7175: 7160: 7156: 7155: 7134: 7131: 7130: 7092: 7088: 7082: 7078: 7066: 7060: 7057: 7056: 7031: 7027: 7015: 7011: 6996: 6992: 6986: 6982: 6981: 6977: 6954: 6951: 6950: 6940: 6894: 6880: 6864: 6860: 6836: 6832: 6820: 6816: 6801: 6797: 6791: 6787: 6782: 6779: 6778: 6742: 6738: 6729: 6725: 6698: 6694: 6692: 6668: 6664: 6663: 6659: 6643: 6609: 6604: 6592: 6562: 6559: 6558: 6518: 6514: 6508: 6486: 6482: 6476: 6472: 6442: 6415: 6412: 6411: 6404: 6388:In the case of 6362: 6357: 6356: 6342: 6330: 6326: 6324: 6321: 6320: 6297: 6292: 6291: 6277: 6265: 6261: 6259: 6256: 6255: 6250:In the case of 6228: 6223: 6222: 6208: 6196: 6192: 6190: 6187: 6186: 6181:In the case of 6177:with α > 1. 6158: 6153: 6152: 6138: 6126: 6122: 6120: 6117: 6116: 6113: 6094: 6079: 6067: 6060: 6053: 6017: 6013: 6004: 6000: 5990: 5986: 5984: 5981: 5980: 5944: 5943: 5935: 5933: 5927: 5926: 5911: 5907: 5898: 5894: 5889: 5887: 5873: 5869: 5860: 5856: 5846: 5842: 5835: 5834: 5811: 5808: 5807: 5800:(μ, ν) or 5787: 5780: 5748: 5744: 5735: 5731: 5721: 5717: 5692: 5672: 5670: 5668: 5665: 5664: 5637: 5633: 5624: 5620: 5610: 5606: 5589: 5585: 5575: 5571: 5561: 5560: 5548: 5544: 5534: 5530: 5520: 5519: 5517: 5496: 5492: 5491: 5475: 5471: 5470: 5468: 5443: 5423: 5421: 5378: 5340: 5338: 5313: 5293: 5291: 5289: 5286: 5285: 5253: 5219: 5212: 5175:(μ, ν) so that 5158: 5153: 5128: 5124: 5118: 5114: 5096: 5069: 5066: 5065: 5041: 5037: 5031: 5015: 5011: 5005: 5001: 4983: 4956: 4953: 4952: 4917: 4900: 4883: 4875: 4869: 4860: 4854: 4811: 4807: 4798: 4794: 4788: 4784: 4778: 4772: 4769: 4768: 4745: 4741: 4724: 4721: 4720: 4669: 4665: 4659: 4655: 4640: 4634: 4631: 4630: 4599: 4595: 4569: 4566: 4565: 4490: 4486: 4480: 4469: 4455: 4447: 4444: 4443: 4431:television snow 4423: 4381: 4333: 4317: 4311: 4281: 4277: 4271: 4267: 4255: 4251: 4245: 4241: 4226: 4210: 4206: 4200: 4196: 4191: 4188: 4187: 4169: 4166: 4165: 4149: 4146: 4145: 4129: 4126: 4125: 4109: 4106: 4105: 4089: 4086: 4085: 4082: 4060: 4056: 4054: 4051: 4050: 4030: 4026: 4021: 4018: 4017: 3988: 3984: 3979: 3976: 3975: 3953: 3950: 3949: 3926: 3922: 3910: 3891: 3887: 3882: 3879: 3878: 3853: 3849: 3837: 3833: 3818: 3814: 3808: 3804: 3799: 3796: 3795: 3776: 3772: 3770: 3767: 3766: 3749: 3745: 3743: 3740: 3739: 3736: 3730: 3718: 3626: 3622: 3616: 3612: 3606: 3602: 3597: 3594: 3593: 3576: 3545: 3531: 3515: 3511: 3493: 3489: 3483: 3479: 3473: 3469: 3464: 3461: 3460: 3450: 3444: 3429: 3418: 3392: 3358: 3326: 3322: 3315: 3311: 3296: 3292: 3268: 3247: 3244: 3243: 3216: 3212: 3205: 3201: 3199: 3196: 3195: 3175: 3171: 3156: 3152: 3150: 3147: 3146: 3126: 3122: 3111: 3107: 3091: 3085: 3082: 3081: 3065: 3062: 3061: 3036: 3033: 3032: 3016: 3013: 3012: 2996: 2995: 2983: 2979: 2968: 2964: 2948: 2929: 2925: 2901: 2885: 2884: 2875: 2871: 2860: 2856: 2840: 2824: 2820: 2809: 2805: 2789: 2773: 2769: 2758: 2754: 2738: 2724: 2708: 2706: 2703: 2702: 2699:The identities 2677: 2673: 2661: 2657: 2655: 2652: 2651: 2632: 2629: 2628: 2608: 2604: 2602: 2599: 2598: 2573: 2569: 2558: 2554: 2538: 2524: 2507: 2503: 2496: 2492: 2490: 2487: 2486: 2460: 2456: 2448: 2445: 2444: 2419: 2415: 2408: 2404: 2402: 2399: 2398: 2378: 2374: 2372: 2369: 2368: 2351: 2347: 2345: 2342: 2341: 2325: 2322: 2321: 2301: 2297: 2289: 2286: 2285: 2284:. We denote by 2268: 2264: 2262: 2259: 2258: 2241: 2237: 2235: 2232: 2231: 2215: 2212: 2211: 2186: 2183: 2182: 2166: 2163: 2162: 2136: 2132: 2130: 2127: 2126: 2106: 2102: 2096: 2092: 2083: 2079: 2055: 2031: 2028: 2027: 2007: 1977: 1973: 1967: 1963: 1945: 1918: 1915: 1914: 1897: 1860: 1856: 1850: 1846: 1837: 1833: 1815: 1791: 1788: 1787: 1774: 1772:Simplifications 1769: 1744: 1740: 1738: 1735: 1734: 1705: 1701: 1699: 1696: 1695: 1666: 1662: 1660: 1657: 1656: 1633: 1601: 1597: 1595: 1592: 1591: 1559: 1555: 1553: 1550: 1549: 1517: 1513: 1511: 1508: 1507: 1471: 1468: 1467: 1464: 1443: 1440: 1439: 1413: 1409: 1407: 1404: 1403: 1384: 1381: 1380: 1352: 1348: 1330: 1317: 1313: 1305: 1302: 1301: 1283: 1280: 1279: 1241: 1237: 1231: 1218: 1214: 1212: 1209: 1208: 1183: 1179: 1171: 1160: 1157: 1156: 1132: 1128: 1107: 1103: 1101: 1083: 1079: 1077: 1074: 1073: 1049: 1046: 1045: 1018: 1015: 1014: 1011:magnetic moment 994: 991: 990: 974: 971: 970: 945: 942: 941: 919: 915: 909: 905: 899: 883: 879: 873: 869: 860: 856: 841: 817: 814: 813: 792: 790: 787: 786: 765: 761: 759: 756: 755: 730: 727: 726: 706: 702: 700: 697: 696: 665: 662: 661: 635: 631: 625: 621: 610: 608: 605: 604: 559: 555: 553: 550: 549: 532: 528: 526: 523: 522: 500: 497: 496: 480: 477: 476: 456: 453: 452: 451:Consider a set 449: 362:that represent 324: 295: 294: 240: 232: 231: 207:Internal energy 202: 192: 191: 167: 159: 158: 138:Grand canonical 114: 104: 103: 59: 17: 12: 11: 5: 25802: 25792: 25791: 25786: 25784:Lattice models 25781: 25776: 25771: 25754: 25753: 25751: 25750: 25745: 25743:List of topics 25739: 25736: 25735: 25733: 25732: 25727: 25722: 25717: 25712: 25707: 25702: 25700:Renewal theory 25697: 25692: 25687: 25682: 25677: 25672: 25667: 25665:Ergodic theory 25662: 25657: 25655:Control theory 25652: 25646: 25644: 25640: 25639: 25637: 25636: 25635: 25634: 25629: 25619: 25614: 25609: 25604: 25599: 25598: 25597: 25587: 25585:Snell envelope 25582: 25577: 25572: 25567: 25562: 25557: 25552: 25547: 25542: 25537: 25532: 25527: 25522: 25517: 25512: 25507: 25502: 25497: 25492: 25487: 25482: 25477: 25472: 25467: 25462: 25457: 25451: 25449: 25445: 25444: 25442: 25441: 25436: 25431: 25426: 25421: 25415: 25413: 25407: 25406: 25404: 25403: 25384:Borel–Cantelli 25373: 25368: 25363: 25358: 25353: 25348: 25343: 25338: 25333: 25328: 25322: 25320: 25319:Limit theorems 25316: 25315: 25313: 25312: 25307: 25302: 25297: 25292: 25287: 25282: 25277: 25272: 25267: 25262: 25257: 25252: 25247: 25242: 25236: 25234: 25230: 25229: 25227: 25226: 25221: 25216: 25211: 25206: 25201: 25195: 25193: 25187: 25186: 25184: 25183: 25178: 25173: 25168: 25162: 25160: 25154: 25153: 25151: 25150: 25145: 25140: 25135: 25130: 25125: 25120: 25115: 25110: 25105: 25100: 25095: 25090: 25085: 25080: 25075: 25070: 25065: 25060: 25054: 25052: 25046: 25045: 25043: 25042: 25037: 25032: 25027: 25022: 25017: 25011: 25009: 25003: 25002: 25000: 24999: 24994: 24989: 24988: 24987: 24982: 24972: 24967: 24962: 24957: 24956: 24955: 24950: 24940: 24938:Hopfield model 24935: 24930: 24925: 24919: 24917: 24913: 24912: 24910: 24909: 24904: 24899: 24894: 24889: 24884: 24883: 24882: 24877: 24872: 24867: 24857: 24855:Markov process 24852: 24847: 24842: 24836: 24834: 24830: 24829: 24827: 24826: 24824:Wiener sausage 24821: 24819:Wiener process 24816: 24811: 24806: 24801: 24799:Stable process 24796: 24791: 24789:Semimartingale 24786: 24781: 24780: 24779: 24774: 24764: 24759: 24754: 24749: 24744: 24739: 24734: 24732:Jump diffusion 24729: 24724: 24719: 24714: 24709: 24707:Hawkes process 24704: 24699: 24694: 24689: 24687:Feller process 24684: 24679: 24674: 24669: 24664: 24659: 24654: 24652:Cauchy process 24649: 24648: 24647: 24642: 24637: 24632: 24627: 24617: 24616: 24615: 24605: 24603:Bessel process 24600: 24594: 24592: 24586: 24585: 24583: 24582: 24581: 24580: 24575: 24570: 24565: 24555: 24550: 24545: 24540: 24535: 24530: 24525: 24519: 24517: 24511: 24510: 24503: 24502: 24495: 24488: 24480: 24474: 24473: 24467: 24457: 24451: 24442: 24437: 24432: 24427: 24418: 24412: 24400: 24395: 24390: 24385: 24368: 24361: 24360:External links 24358: 24356: 24355: 24325: 24308:(2): 670–672, 24297: 24273: 24256:(1): 268–280, 24245: 24217: 24192:(5): 808–816, 24178: 24158:978-1581131840 24157: 24129: 24114: 24098: 24059: 24015:(2): 308–322, 23994: 23980: 23964: 23936: 23904: 23887: 23881: 23866: 23861: 23848: 23843: 23826: 23801:(1): 253–258, 23790: 23785: 23769: 23756: 23736:Gallavotti, G. 23732: 23727: 23710: 23692:(4): 883–893. 23681: 23660: 23647: 23632: 23612: 23610: 23607: 23605: 23604: 23583: 23551:(2): 221–239, 23531: 23464: 23445:(2): 171–190, 23429: 23402:(4): 620–630, 23386: 23365: 23346: 23291: 23244:(18): 4982–7. 23224: 23164: 23111: 23085: 23054: 22993: 22939: 22886: 22825: 22779: 22726: 22673: 22659: 22634: 22615:(6): 652–657. 22595: 22588: 22560: 22535: 22515: 22464: 22417: 22410: 22387: 22338: 22318: 22255: 22248: 22211: 22182:(3): 511–542. 22162: 22117:(2): 725–745. 22097: 22062:(2): 111–126. 22042: 22025: 21982: 21908: 21881:(3): 493–513. 21865: 21852: 21838: 21824: 21822: 21819: 21817: 21816: 21811: 21806: 21801: 21796: 21791: 21786: 21781: 21776: 21766: 21764:Order operator 21761: 21756: 21754:Kuramoto model 21751: 21746: 21741: 21736: 21731: 21726: 21721: 21716: 21711: 21706: 21701: 21695: 21693: 21690: 21648: 21639:or increasing 21628: 21608: 21588: 21568: 21548: 21526: 21520: 21517: 21514: 21511: 21506: 21501: 21498: 21495: 21492: 21489: 21486: 21483: 21478: 21473: 21470: 21467: 21464: 21455: 21449: 21446: 21440: 21435: 21431: 21408: 21404: 21383: 21380: 21358: 21354: 21331: 21327: 21304: 21300: 21277: 21274: 21271: 21267: 21261: 21258: 21255: 21251: 21227: 21223: 21199: 21195: 21172: 21168: 21164: 21161: 21158: 21153: 21150: 21146: 21123: 21119: 21096: 21092: 21088: 21085: 21082: 21077: 21074: 21070: 21064: 21059: 21056: 21053: 21049: 21043: 21039: 21031: 21027: 21022: 21018: 21015: 21012: 21009: 21006: 21003: 21000: 20997: 20994: 20989: 20986: 20979: 20975: 20969: 20966: 20961: 20957: 20951: 20947: 20943: 20910: 20889: 20869: 20866: 20863: 20860: 20838: 20834: 20830: 20827: 20807: 20804: 20801: 20798: 20795: 20775: 20745: 20742: 20739: 20734: 20730: 20726: 20723: 20716: 20712: 20708: 20701: 20696: 20693: 20690: 20686: 20679: 20676: 20673: 20670: 20667: 20662: 20659: 20656: 20653: 20650: 20647: 20641: 20638: 20635: 20632: 20629: 20626: 20623: 20620: 20614: 20611: 20608: 20605: 20602: 20597: 20594: 20588: 20585: 20582: 20579: 20576: 20573: 20570: 20567: 20555:Jellito (1979) 20551:Glasser (1970) 20531: 20527: 20504: 20500: 20479: 20476: 20467: 20464: 20446: 20428: 20423: 20419: 20412: 20408: 20401: 20398: 20395: 20391: 20387: 20381: 20378: 20373: 20370: 20367: 20351: 20348: 20332: 20329: 20325: 20302: 20298: 20292: 20288: 20282: 20278: 20274: 20269: 20265: 20259: 20255: 20249: 20246: 20242: 20236: 20233: 20229: 20222: 20219: 20213: 20210: 20207: 20184: 20180: 20174: 20170: 20164: 20160: 20156: 20153: 20150: 20098: 20095: 20054: 20050: 20046: 20043: 20040: 20037: 20034: 20031: 20026: 20022: 20016: 20013: 20008: 20003: 19999: 19993: 19989: 19985: 19980: 19977: 19974: 19971: 19968: 19964: 19958: 19955: 19950: 19947: 19944: 19925: 19921: 19917: 19914: 19911: 19906: 19902: 19898: 19895: 19890: 19886: 19861: 19857: 19853: 19848: 19844: 19840: 19835: 19831: 19825: 19821: 19817: 19814: 19809: 19806: 19803: 19800: 19797: 19793: 19787: 19784: 19779: 19776: 19773: 19737: 19729: 19721: 19711: 19708: 19685:ferromagnetism 19680: 19677: 19675: 19672: 19646: 19643: 19640: 19637: 19634: 19631: 19628: 19625: 19622: 19619: 19616: 19613: 19609: 19606: 19602: 19568: 19565: 19562: 19559: 19556: 19553: 19550: 19547: 19544: 19541: 19538: 19535: 19532: 19529: 19506: 19503: 19500: 19497: 19492: 19488: 19484: 19479: 19475: 19471: 19468: 19465: 19462: 19459: 19456: 19453: 19450: 19447: 19444: 19432: 19415: 19412: 19407: 19403: 19399: 19394: 19390: 19386: 19383: 19380: 19377: 19374: 19371: 19368: 19363: 19358: 19354: 19350: 19345: 19341: 19337: 19334: 19330: 19326: 19321: 19316: 19312: 19308: 19303: 19299: 19295: 19292: 19288: 19239: 19184: 19168: 19165: 19160: 19156: 19150: 19146: 19140: 19136: 19130: 19126: 19120: 19117: 19114: 19111: 19107: 19103: 19100: 19095: 19091: 19085: 19081: 19075: 19072: 19068: 19062: 19059: 19055: 19051: 19048: 19036: 19033: 19016: 19013: 18992: 18988: 18984: 18981: 18954: 18950: 18946: 18943: 18905: 18902: 18897: 18893: 18889: 18886: 18883: 18880: 18877: 18874: 18871: 18868: 18865: 18862: 18856: 18853: 18848: 18845: 18842: 18839: 18836: 18830: 18827: 18764: 18761: 18756: 18753: 18721: 18715: 18711: 18705: 18701: 18697: 18694: 18691: 18688: 18682: 18679: 18676: 18673: 18670: 18667: 18664: 18661: 18658: 18655: 18652: 18649: 18646: 18643: 18640: 18637: 18634: 18562: 18559: 18556: 18553: 18550: 18547: 18544: 18541: 18533: 18530: 18527: 18524: 18521: 18517: 18513: 18510: 18503: 18500: 18497: 18494: 18491: 18487: 18483: 18480: 18474: 18471: 18468: 18465: 18462: 18459: 18456: 18413: 18410: 18407: 18404: 18401: 18397: 18393: 18390: 18386: 18381: 18378: 18354: 18351: 18348: 18345: 18341: 18337: 18331: 18328: 18325: 18321: 18251:Main article: 18248: 18245: 18229: 18223: 18220: 18215: 18212: 18208: 18201: 18198: 18161: 18158: 18154: 18149: 18146: 18116: 18113: 18110: 18107: 18104: 18101: 18099: 18095: 18091: 18088: 18082: 18081: 18078: 18075: 18072: 18069: 18066: 18063: 18060: 18058: 18054: 18050: 18047: 18041: 18040: 18014: 18011: 18008: 18002: 17999: 17993: 17989: 17982: 17973: 17970: 17965: 17961: 17956: 17953: 17950: 17947: 17944: 17941: 17938: 17935: 17932: 17929: 17912: 17909: 17873: 17869: 17865: 17861: 17856: 17848: 17844: 17840: 17836: 17830: 17827: 17818: 17814: 17810: 17807: 17804: 17800: 17795: 17790: 17786: 17780: 17776: 17772: 17769: 17766: 17763: 17733: 17729: 17725: 17715: 17711: 17707: 17704: 17701: 17696: 17691: 17687: 17678: 17675: 17672: 17669: 17666: 17663: 17660: 17656: 17652: 17648: 17644: 17641: 17637: 17633: 17630: 17627: 17597: 17592: 17588: 17585: 17582: 17579: 17573: 17570: 17568: 17564: 17560: 17557: 17551: 17550: 17547: 17543: 17537: 17533: 17530: 17524: 17521: 17517: 17513: 17510: 17508: 17504: 17500: 17497: 17491: 17490: 17466: 17461: 17457: 17453: 17450: 17447: 17444: 17441: 17438: 17414: 17409: 17403: 17399: 17395: 17389: 17386: 17378: 17374: 17370: 17367: 17362: 17358: 17354: 17350: 17345: 17342: 17337: 17333: 17327: 17321: 17317: 17313: 17307: 17304: 17301: 17298: 17259: 17256: 17252: 17246: 17242: 17239: 17233: 17230: 17226: 17222: 17219: 17216: 17153: 17150: 17147: 17144: 17141: 17136: 17132: 17128: 17125: 17119: 17116: 17113: 17108: 17104: 17100: 17095: 17091: 17086: 17081: 17078: 17075: 17072: 17069: 17062: 17058: 17054: 17049: 17046: 17043: 17013: 17010: 17007: 17002: 16998: 16994: 16990: 16980: 16976: 16972: 16969: 16966: 16961: 16956: 16952: 16943: 16940: 16937: 16934: 16931: 16928: 16925: 16921: 16917: 16913: 16909: 16906: 16902: 16898: 16895: 16892: 16887: 16883: 16879: 16824: 16819: 16815: 16809: 16806: 16801: 16798: 16795: 16792: 16789: 16786: 16781: 16777: 16712: 16709: 16668: 16662: 16658: 16651: 16648: 16644: 16639: 16634: 16630: 16624: 16621: 16616: 16611: 16606: 16601: 16598: 16594: 16588: 16585: 16579: 16575: 16570: 16566: 16562: 16559: 16556: 16530: 16525:4 − 16523: 16493: 16490: 16467: 16464: 16461: 16457: 16451: 16446: 16441: 16437: 16431: 16428: 16425: 16422: 16419: 16384: 16379: 16374: 16370: 16341: 16337: 16333: 16330: 16326: 16322: 16317: 16314: 16311: 16307: 16234: 16231: 16226: 16222: 16217: 16212: 16207: 16204: 16201: 16198: 16193: 16189: 16185: 16182: 16178: 16174: 16171: 16168: 16165: 16162: 16159: 16156: 16153: 16115: 16112: 16109: 16103: 16100: 16094: 16090: 16084: 16080: 16072: 16067: 16062: 16059: 16056: 16051: 16046: 16041: 16038: 16035: 16032: 16029: 16026: 16023: 16020: 15993: 15965: 15962: 15959: 15956: 15953: 15950: 15947: 15944: 15940: 15933: 15930: 15925: 15922: 15914: 15911: 15908: 15904: 15899: 15892: 15889: 15885: 15876: 15873: 15870: 15866: 15862: 15857: 15854: 15851: 15848: 15845: 15842: 15839: 15834: 15830: 15807: 15797: 15774: 15771: 15768: 15764: 15760: 15755: 15752: 15749: 15746: 15743: 15707: 15704: 15701: 15697: 15693: 15688: 15685: 15660: 15657: 15654: 15651: 15648: 15645: 15642: 15639: 15635: 15630: 15626: 15622: 15617: 15614: 15611: 15607: 15603: 15559: 15556: 15553: 15550: 15547: 15523: 15520: 15517: 15514: 15501:Gauss's method 15463: 15460: 15457: 15455: 15453: 15450: 15447: 15444: 15441: 15438: 15435: 15432: 15429: 15426: 15421: 15417: 15413: 15408: 15405: 15404: 15401: 15398: 15395: 15393: 15391: 15388: 15385: 15382: 15379: 15376: 15373: 15370: 15367: 15364: 15360: 15356: 15353: 15348: 15343: 15339: 15335: 15331: 15327: 15324: 15306: 15303: 15280: 15275: 15272: 15268: 15264: 15259: 15256: 15252: 15248: 15245: 15242: 15239: 15234: 15231: 15227: 15223: 15218: 15215: 15211: 15207: 15204: 15201: 15196: 15193: 15189: 15185: 15180: 15177: 15173: 15169: 15166: 15163: 15158: 15154: 15150: 15147: 15144: 15139: 15136: 15132: 15128: 15125: 15122: 15119: 15114: 15110: 15106: 15103: 15100: 15095: 15091: 15087: 15084: 15081: 15056:Wick's theorem 15014: 15011: 15006: 15002: 14995: 14992: 14989: 14986: 14983: 14980: 14977: 14973: 14962: 14958: 14954: 14951: 14948: 14943: 14940: 14934: 14931: 14928: 14925: 14922: 14919: 14916: 14913: 14910: 14907: 14904: 14901: 14898: 14895: 14892: 14889: 14886: 14883: 14880: 14877: 14874: 14871: 14868: 14865: 14862: 14859: 14856: 14853: 14850: 14847: 14794: 14791: 14777:. So Landau's 14752: 14749: 14744: 14740: 14736: 14733: 14730: 14727: 14724: 14721: 14718: 14714: 14708: 14704: 14700: 14697: 14692: 14688: 14684: 14680: 14673: 14670: 14666: 14630:at any finite 14599: 14594: 14590: 14585: 14581: 14577: 14574: 14571: 14568: 14565: 14560: 14556: 14552: 14549: 14546: 14542: 14537: 14532: 14528: 14524: 14521: 14518: 14515: 14510: 14506: 14502: 14499: 14496: 14493: 14490: 14487: 14482: 14478: 14472: 14468: 14464: 14461: 14458: 14455: 14452: 14449: 14444: 14440: 14436: 14433: 14430: 14370: 14367: 14361:term or the (∇ 14329: 14325: 14319: 14315: 14311: 14308: 14305: 14302: 14299: 14294: 14290: 14286: 14283: 14278: 14274: 14270: 14266: 14262: 14257: 14253: 14249: 14246: 14243: 14174: 14169: 14165: 14161: 14157: 14152: 14148: 14144: 14141: 14138: 14109: 14106: 14062: 14056: 14052: 14048: 14045: 14040: 14035: 14030: 14027: 14023: 14019: 14016: 14011: 14007: 14003: 13999: 13995: 13990: 13986: 13982: 13979: 13975: 13970: 13967: 13964: 13961: 13958: 13932: 13923: 13917: 13913: 13909: 13906: 13901: 13896: 13891: 13888: 13884: 13880: 13877: 13872: 13868: 13864: 13860: 13856: 13851: 13847: 13843: 13840: 13836: 13831: 13828: 13825: 13820: 13815: 13811: 13807: 13804: 13801: 13798: 13793: 13789: 13785: 13782: 13779: 13774: 13770: 13766: 13763: 13757: 13751: 13747: 13743: 13740: 13735: 13730: 13725: 13722: 13718: 13714: 13711: 13706: 13702: 13698: 13694: 13690: 13685: 13681: 13677: 13674: 13670: 13665: 13662: 13659: 13653: 13650: 13647: 13642: 13638: 13634: 13631: 13628: 13625: 13620: 13616: 13612: 13609: 13606: 13601: 13597: 13593: 13590: 13587: 13559: 13554: 13551: 13548: 13545: 13542: 13539: 13535: 13531: 13525: 13519: 13515: 13511: 13508: 13503: 13498: 13493: 13490: 13486: 13482: 13479: 13474: 13470: 13466: 13462: 13458: 13453: 13449: 13445: 13442: 13438: 13434: 13431: 13428: 13425: 13422: 13395: 13390: 13381:quadratic form 13368: 13363: 13359: 13354: 13349: 13345: 13338: 13335: 13331: 13307: 13300: 13284: 13280: 13276: 13273: 13268: 13264: 13260: 13257: 13252: 13248: 13244: 13239: 13235: 13231: 13226: 13222: 13216: 13211: 13208: 13205: 13201: 13197: 13192: 13188: 13184: 13180: 13176: 13171: 13167: 13163: 13160: 13157: 13154: 13070: 13067: 13061: 13058: 13024: 13021: 12997:critical point 12991: 12988: 12964: 12961: 12945:Main article: 12942: 12939: 12905: 12901: 12878: 12874: 12850: 12847: 12841: 12835: 12832: 12827: 12821: 12817: 12813: 12810: 12807: 12804: 12799: 12795: 12791: 12788: 12785: 12782: 12778: 12773: 12770: 12766: 12761: 12758: 12739: 12736: 12714: 12709: 12706: 12703: 12696: 12692: 12684: 12677: 12673: 12665: 12661: 12657: 12652: 12649: 12646: 12642: 12636: 12632: 12628: 12625: 12620: 12617: 12614: 12610: 12604: 12597: 12593: 12585: 12581: 12577: 12572: 12569: 12566: 12562: 12556: 12549: 12545: 12539: 12536: 12531: 12526: 12522: 12510: 12509: 12506: 12503: 12500: 12481: 12476: 12472: 12466: 12461: 12457: 12453: 12450: 12432: 12429: 12399: 12394: 12391: 12388: 12384: 12378: 12373: 12369: 12365: 12362: 12357: 12352: 12348: 12344: 12339: 12335: 12307: 12302: 12297: 12293: 12261: 12256: 12251: 12247: 12231: 12225: 12219:, we can find 12202: 12199: 12194: 12190: 12186: 12182: 12179: 12175: 12172: 12141: 12138: 12134: 12111: 12108: 12104: 12100: 12097: 12094: 12091: 12086: 12082: 12078: 12075: 12072: 12068: 12051: 12044: 12023: 12016: 12012: 12006: 12002: 11997: 11961: 11958: 11954: 11950: 11947: 11944: 11941: 11936: 11932: 11926: 11923: 11920: 11917: 11913: 11909: 11906: 11901: 11897: 11870: 11867: 11864: 11861: 11857: 11853: 11850: 11793: 11788: 11782: 11776: 11773: 11770: 11767: 11764: 11760: 11756: 11751: 11748: 11745: 11742: 11739: 11735: 11730: 11724: 11721: 11718: 11714: 11708: 11705: 11701: 11695: 11690: 11687: 11682: 11679: 11676: 11672: 11657: 11654: 11637: 11634: 11631: 11609: 11605: 11582: 11578: 11553: 11550: 11545: 11542: 11539: 11536: 11533: 11527: 11522: 11517: 11514: 11511: 11508: 11505: 11502: 11497: 11494: 11488: 11483: 11479: 11454: 11450: 11429: 11426: 11421: 11417: 11413: 11408: 11404: 11381: 11378: 11374: 11366: 11362: 11358: 11351: 11347: 11343: 11337: 11333: 11330: 11326: 11318: 11314: 11310: 11303: 11299: 11295: 11289: 11285: 11282: 11258: 11254: 11230: 11227: 11224: 11219: 11215: 11211: 11208: 11205: 11202: 11197: 11193: 11189: 11186: 11183: 11180: 11177: 11174: 11171: 11166: 11162: 11158: 11155: 11152: 11149: 11144: 11140: 11136: 11133: 11130: 11127: 11124: 11121: 11118: 11113: 11109: 11105: 11102: 11099: 11096: 11093: 11090: 11085: 11081: 11077: 11074: 11071: 11068: 11065: 11062: 11059: 11056: 11051: 11047: 11043: 11038: 11035: 11030: 11026: 11020: 11016: 11012: 11007: 11004: 10999: 10995: 10986: 10982: 10978: 10974: 10969: 10966: 10963: 10960: 10957: 10954: 10951: 10948: 10924: 10920: 10897: 10893: 10872: 10869: 10866: 10855:Onsager (1944) 10848:Main article: 10845: 10842: 10841: 10840: 10836: 10819: 10818:Two dimensions 10816: 10803: 10783: 10780: 10777: 10774: 10754: 10751: 10748: 10745: 10742: 10739: 10736: 10733: 10713: 10690: 10687: 10684: 10681: 10678: 10675: 10672: 10669: 10666: 10661: 10658: 10653: 10650: 10646: 10643: 10622: 10618: 10615: 10611: 10608: 10603: 10600: 10595: 10591: 10585: 10582: 10579: 10576: 10573: 10570: 10565: 10561: 10557: 10554: 10549: 10546: 10541: 10538: 10535: 10532: 10529: 10509: 10505: 10502: 10498: 10493: 10489: 10485: 10481: 10477: 10474: 10468: 10464: 10460: 10456: 10449: 10446: 10441: 10437: 10431: 10428: 10425: 10422: 10419: 10416: 10411: 10407: 10403: 10400: 10395: 10392: 10387: 10384: 10381: 10378: 10373: 10369: 10365: 10362: 10357: 10354: 10332: 10329: 10326: 10323: 10320: 10317: 10314: 10311: 10306: 10303: 10298: 10294: 10291: 10287: 10282: 10279: 10276: 10273: 10270: 10267: 10262: 10259: 10256: 10236: 10233: 10228: 10224: 10220: 10215: 10211: 10207: 10204: 10201: 10198: 10195: 10192: 10189: 10182: 10178: 10172: 10168: 10163: 10160: 10155: 10151: 10131: 10128: 10125: 10122: 10117: 10113: 10109: 10104: 10100: 10096: 10093: 10090: 10087: 10084: 10081: 10078: 10075: 10072: 10069: 10066: 10061: 10057: 10053: 10048: 10044: 10040: 10037: 10034: 10031: 10028: 10025: 10022: 10017: 10013: 10009: 10006: 10003: 10000: 9997: 9994: 9989: 9985: 9964: 9961: 9956: 9952: 9948: 9943: 9939: 9918: 9915: 9912: 9909: 9889: 9884: 9879: 9875: 9871: 9866: 9862: 9858: 9853: 9849: 9845: 9841: 9835: 9830: 9826: 9822: 9817: 9813: 9809: 9804: 9800: 9796: 9792: 9786: 9782: 9778: 9775: 9772: 9769: 9766: 9763: 9758: 9754: 9733: 9713: 9710: 9707: 9704: 9701: 9696: 9692: 9671: 9668: 9665: 9662: 9659: 9656: 9653: 9650: 9647: 9644: 9641: 9629: 9626: 9595: 9590: 9585: 9581: 9575: 9571: 9567: 9564: 9559: 9554: 9551: 9548: 9544: 9538: 9533: 9529: 9523: 9520: 9517: 9514: 9511: 9508: 9505: 9501: 9497: 9494: 9491: 9488: 9485: 9482: 9479: 9455: 9450: 9445: 9442: 9439: 9435: 9429: 9424: 9420: 9416: 9411: 9406: 9402: 9380: 9375: 9370: 9366: 9360: 9355: 9352: 9349: 9345: 9341: 9336: 9331: 9327: 9294: 9289: 9284: 9280: 9274: 9270: 9266: 9263: 9258: 9253: 9250: 9247: 9243: 9237: 9232: 9228: 9222: 9219: 9216: 9213: 9210: 9207: 9204: 9200: 9196: 9193: 9190: 9187: 9184: 9181: 9178: 9161:Pauli matrices 9156: 9153: 9130: 9126: 9117: 9114: 9111: 9108: 9104: 9100: 9097: 9091: 9088: 9085: 9082: 9078: 9072: 9069: 9065: 9061: 9058: 9055: 9052: 9049: 9046: 9043: 9040: 9037: 9034: 9031: 9028: 9025: 8997: 8992: 8988: 8984: 8981: 8978: 8975: 8972: 8969: 8966: 8963: 8960: 8957: 8952: 8948: 8944: 8937: 8933: 8927: 8923: 8918: 8908: 8904: 8901: 8873: 8867: 8863: 8860: 8857: 8853: 8849: 8846: 8842: 8838: 8835: 8830: 8826: 8820: 8816: 8812: 8776: 8771: 8768: 8765: 8762: 8758: 8754: 8748: 8745: 8742: 8738: 8699: 8696: 8691: 8687: 8675: 8668: 8665: 8662: 8659: 8655: 8651: 8646: 8642: 8638: 8635: 8632: 8629: 8626: 8621: 8618: 8615: 8611: 8605: 8602: 8599: 8596: 8593: 8588: 8585: 8581: 8577: 8572: 8568: 8555: 8547: 8535: 8531: 8525: 8520: 8513: 8509: 8503: 8499: 8493: 8488: 8485: 8481: 8475: 8470: 8466: 8462: 8457: 8452: 8448: 8444: 8439: 8434: 8430: 8426: 8422: 8417: 8413: 8409: 8405: 8402: 8399: 8396: 8393: 8390: 8387: 8367: 8362: 8354: 8351: 8348: 8345: 8342: 8339: 8336: 8332: 8328: 8324: 8321: 8318: 8314: 8310: 8309: 8304: 8301: 8298: 8294: 8290: 8286: 8283: 8280: 8277: 8274: 8271: 8267: 8263: 8262: 8260: 8255: 8252: 8229: 8226: 8220: 8216: 8213: 8206: 8199: 8196: 8192: 8189: 8186: 8182: 8176: 8171: 8167: 8164: 8157: 8153: 8147: 8144: 8140: 8137: 8133: 8109: 8106: 8102: 8099: 8095: 8074: 8067: 8063: 8059: 8054: 8050: 8045: 8041: 8034: 8030: 8026: 8021: 8017: 8012: 8004: 8000: 7996: 7991: 7987: 7982: 7974: 7970: 7966: 7963: 7960: 7955: 7951: 7946: 7942: 7935: 7931: 7925: 7921: 7917: 7914: 7910: 7902: 7898: 7894: 7891: 7887: 7883: 7876: 7872: 7866: 7862: 7858: 7855: 7851: 7843: 7839: 7835: 7832: 7828: 7820: 7816: 7810: 7806: 7802: 7799: 7795: 7787: 7783: 7779: 7776: 7772: 7764: 7760: 7756: 7753: 7750: 7745: 7741: 7736: 7732: 7729: 7726: 7723: 7720: 7683: 7678: 7673: 7665: 7662: 7659: 7655: 7651: 7646: 7643: 7639: 7631: 7628: 7625: 7621: 7617: 7612: 7609: 7605: 7598: 7593: 7590: 7585: 7582: 7579: 7575: 7569: 7565: 7561: 7537: 7533: 7527: 7524: 7521: 7517: 7513: 7508: 7505: 7501: 7496: 7492: 7489: 7484: 7481: 7476: 7473: 7470: 7467: 7464: 7461: 7458: 7455: 7431: 7426: 7423: 7420: 7415: 7409: 7406: 7403: 7399: 7395: 7390: 7387: 7383: 7378: 7373: 7370: 7364: 7360: 7356: 7352: 7349: 7345: 7338: 7334: 7330: 7325: 7319: 7314: 7311: 7308: 7304: 7300: 7297: 7290: 7286: 7280: 7277: 7274: 7270: 7266: 7263: 7259: 7255: 7248: 7244: 7238: 7234: 7230: 7227: 7223: 7215: 7211: 7205: 7201: 7197: 7194: 7190: 7182: 7178: 7174: 7171: 7168: 7163: 7159: 7154: 7150: 7147: 7144: 7141: 7138: 7116: 7113: 7110: 7106: 7101: 7098: 7095: 7091: 7085: 7081: 7077: 7073: 7069: 7065: 7044: 7040: 7034: 7030: 7024: 7021: 7018: 7014: 7010: 7007: 7004: 6999: 6995: 6989: 6985: 6980: 6976: 6973: 6970: 6967: 6964: 6961: 6958: 6939: 6936: 6934:(β) vanishes. 6903: 6897: 6893: 6890: 6887: 6883: 6879: 6876: 6873: 6870: 6867: 6863: 6859: 6856: 6853: 6850: 6847: 6844: 6839: 6835: 6831: 6828: 6823: 6819: 6815: 6812: 6809: 6804: 6800: 6794: 6790: 6786: 6766: 6762: 6754: 6751: 6748: 6745: 6741: 6737: 6732: 6728: 6724: 6721: 6718: 6715: 6712: 6707: 6704: 6701: 6697: 6691: 6688: 6685: 6682: 6679: 6674: 6671: 6667: 6662: 6658: 6655: 6650: 6647: 6642: 6639: 6636: 6633: 6630: 6627: 6624: 6621: 6615: 6612: 6608: 6601: 6598: 6595: 6591: 6587: 6584: 6581: 6578: 6575: 6572: 6569: 6566: 6526: 6521: 6517: 6511: 6507: 6503: 6500: 6495: 6492: 6489: 6485: 6479: 6475: 6469: 6466: 6463: 6460: 6457: 6454: 6451: 6448: 6445: 6441: 6437: 6434: 6431: 6428: 6425: 6422: 6419: 6403: 6400: 6399: 6398: 6386: 6368: 6365: 6360: 6355: 6352: 6349: 6345: 6341: 6336: 6333: 6329: 6317: 6303: 6300: 6295: 6290: 6287: 6284: 6280: 6276: 6271: 6268: 6264: 6248: 6234: 6231: 6226: 6221: 6218: 6215: 6211: 6207: 6202: 6199: 6195: 6164: 6161: 6156: 6151: 6148: 6145: 6141: 6137: 6132: 6129: 6125: 6112: 6109: 6092: 6077: 6066: 6063: 6058: 6051: 6045: 6044: 6041: 6030: 6025: 6020: 6016: 6012: 6007: 6003: 5999: 5996: 5993: 5989: 5977: 5974: 5971: 5947: 5942: 5934: 5932: 5929: 5928: 5925: 5922: 5919: 5914: 5910: 5906: 5901: 5897: 5888: 5886: 5881: 5876: 5872: 5868: 5863: 5859: 5855: 5852: 5849: 5845: 5841: 5840: 5838: 5833: 5830: 5827: 5824: 5821: 5818: 5815: 5785: 5778: 5761: 5756: 5751: 5747: 5743: 5738: 5734: 5730: 5727: 5724: 5720: 5716: 5710: 5707: 5704: 5701: 5698: 5695: 5690: 5687: 5684: 5681: 5678: 5675: 5650: 5645: 5640: 5636: 5632: 5627: 5623: 5619: 5616: 5613: 5609: 5605: 5597: 5592: 5588: 5584: 5581: 5578: 5574: 5568: 5565: 5556: 5551: 5547: 5543: 5540: 5537: 5533: 5527: 5524: 5516: 5510: 5507: 5504: 5499: 5495: 5489: 5486: 5483: 5478: 5474: 5467: 5461: 5458: 5455: 5452: 5449: 5446: 5441: 5438: 5435: 5432: 5429: 5426: 5420: 5414: 5411: 5408: 5405: 5402: 5399: 5396: 5393: 5390: 5387: 5384: 5381: 5376: 5373: 5370: 5367: 5364: 5361: 5358: 5355: 5352: 5349: 5346: 5343: 5337: 5331: 5328: 5325: 5322: 5319: 5316: 5311: 5308: 5305: 5302: 5299: 5296: 5252: 5249: 5236:represent the 5217: 5210: 5157: 5154: 5152: 5149: 5136: 5131: 5127: 5121: 5117: 5111: 5108: 5102: 5099: 5095: 5091: 5088: 5085: 5082: 5079: 5076: 5073: 5049: 5044: 5040: 5034: 5030: 5026: 5023: 5018: 5014: 5008: 5004: 4998: 4995: 4989: 4986: 4982: 4978: 4975: 4972: 4969: 4966: 4963: 4960: 4934: 4933: 4927: 4913: 4910: 4899: 4896: 4882: 4879: 4871:Main article: 4868: 4867:Yang–Lee zeros 4865: 4856:Main article: 4853: 4850: 4828: 4823: 4820: 4817: 4814: 4810: 4804: 4801: 4797: 4791: 4787: 4781: 4777: 4756: 4751: 4748: 4744: 4740: 4737: 4734: 4731: 4728: 4677: 4672: 4668: 4662: 4658: 4654: 4651: 4648: 4643: 4639: 4607: 4602: 4598: 4594: 4591: 4588: 4585: 4582: 4579: 4576: 4573: 4564:dimensions is 4542: 4541: 4538: 4527: 4516: 4498: 4493: 4489: 4483: 4478: 4475: 4472: 4468: 4462: 4459: 4454: 4451: 4422: 4419: 4415:Rudolf Peierls 4407: 4406: 4403: 4396: 4380: 4377: 4332: 4329: 4315:FKG inequality 4313:Main article: 4310: 4309:FKG inequality 4307: 4292: 4289: 4284: 4280: 4274: 4270: 4266: 4263: 4258: 4254: 4248: 4244: 4240: 4235: 4232: 4229: 4225: 4221: 4218: 4213: 4209: 4203: 4199: 4195: 4173: 4153: 4133: 4113: 4104:disconnecting 4093: 4081: 4078: 4063: 4059: 4038: 4033: 4029: 4025: 4002: 3999: 3996: 3991: 3987: 3983: 3963: 3960: 3957: 3934: 3929: 3925: 3919: 3916: 3913: 3909: 3905: 3902: 3899: 3894: 3890: 3886: 3864: 3861: 3856: 3852: 3848: 3845: 3840: 3836: 3832: 3829: 3826: 3821: 3817: 3811: 3807: 3803: 3779: 3775: 3752: 3748: 3732:Main article: 3729: 3726: 3717: 3714: 3667:Rudolf Peierls 3652: 3649: 3646: 3643: 3640: 3637: 3634: 3629: 3625: 3619: 3615: 3609: 3605: 3601: 3575: 3572: 3557: 3553: 3548: 3544: 3541: 3538: 3534: 3530: 3527: 3524: 3521: 3518: 3514: 3510: 3507: 3504: 3501: 3496: 3492: 3486: 3482: 3476: 3472: 3468: 3446: 3440: 3428: 3425: 3414: 3391: 3388: 3387: 3386: 3383: 3376: 3365: 3357: 3354: 3342: 3338: 3334: 3329: 3325: 3321: 3318: 3314: 3310: 3307: 3302: 3299: 3295: 3289: 3286: 3283: 3280: 3277: 3274: 3271: 3267: 3263: 3260: 3257: 3254: 3251: 3228: 3224: 3219: 3215: 3211: 3208: 3204: 3181: 3178: 3174: 3170: 3167: 3162: 3159: 3155: 3132: 3129: 3125: 3119: 3114: 3110: 3106: 3103: 3100: 3097: 3094: 3090: 3069: 3049: 3046: 3043: 3040: 3020: 2994: 2989: 2986: 2982: 2976: 2971: 2967: 2963: 2960: 2957: 2954: 2951: 2947: 2943: 2940: 2935: 2932: 2928: 2922: 2919: 2916: 2913: 2910: 2907: 2904: 2900: 2896: 2893: 2890: 2888: 2886: 2881: 2878: 2874: 2868: 2863: 2859: 2855: 2852: 2849: 2846: 2843: 2839: 2835: 2830: 2827: 2823: 2817: 2812: 2808: 2804: 2801: 2798: 2795: 2792: 2788: 2784: 2779: 2776: 2772: 2766: 2761: 2757: 2753: 2750: 2747: 2744: 2741: 2737: 2733: 2730: 2727: 2725: 2723: 2720: 2717: 2714: 2711: 2710: 2683: 2680: 2676: 2672: 2667: 2664: 2660: 2639: 2636: 2614: 2611: 2607: 2584: 2579: 2576: 2572: 2566: 2561: 2557: 2553: 2550: 2547: 2544: 2541: 2537: 2531: 2528: 2523: 2519: 2515: 2510: 2506: 2502: 2499: 2495: 2468: 2463: 2459: 2455: 2452: 2431: 2427: 2422: 2418: 2414: 2411: 2407: 2381: 2377: 2354: 2350: 2329: 2309: 2304: 2300: 2296: 2293: 2271: 2267: 2244: 2240: 2219: 2199: 2196: 2193: 2190: 2170: 2150: 2147: 2144: 2139: 2135: 2109: 2105: 2099: 2095: 2089: 2086: 2082: 2076: 2073: 2070: 2067: 2064: 2061: 2058: 2054: 2050: 2047: 2044: 2041: 2038: 2035: 2006: 1999:Connection to 1997: 1985: 1980: 1976: 1970: 1966: 1960: 1957: 1951: 1948: 1944: 1940: 1937: 1934: 1931: 1928: 1925: 1922: 1903:for all pairs 1895: 1868: 1863: 1859: 1853: 1849: 1843: 1840: 1836: 1830: 1827: 1821: 1818: 1814: 1810: 1807: 1804: 1801: 1798: 1795: 1773: 1770: 1768: 1767: 1755: 1752: 1747: 1743: 1732: 1716: 1713: 1708: 1704: 1693: 1677: 1674: 1669: 1665: 1653: 1632: 1631: 1628:noninteracting 1615: 1612: 1607: 1604: 1600: 1589: 1573: 1570: 1565: 1562: 1558: 1547: 1531: 1528: 1523: 1520: 1516: 1504: 1484: 1481: 1478: 1475: 1463: 1460: 1447: 1427: 1424: 1421: 1416: 1412: 1388: 1366: 1363: 1360: 1355: 1351: 1347: 1344: 1341: 1338: 1333: 1329: 1325: 1320: 1316: 1312: 1309: 1287: 1259: 1256: 1253: 1250: 1247: 1244: 1240: 1234: 1230: 1226: 1221: 1217: 1194: 1191: 1182: 1178: 1174: 1170: 1167: 1164: 1142: 1135: 1131: 1125: 1122: 1119: 1116: 1113: 1110: 1106: 1100: 1097: 1094: 1091: 1086: 1082: 1059: 1056: 1053: 1022: 998: 978: 958: 955: 952: 949: 927: 922: 918: 912: 908: 902: 898: 894: 891: 886: 882: 876: 872: 866: 863: 859: 853: 850: 847: 844: 840: 836: 833: 830: 827: 824: 821: 795: 768: 764: 740: 737: 734: 725:. Also a site 712: 709: 705: 681: 678: 675: 672: 669: 644: 641: 638: 634: 628: 624: 620: 617: 613: 588: 585: 582: 579: 576: 573: 570: 567: 562: 558: 535: 531: 510: 507: 504: 484: 460: 448: 445: 352:ferromagnetism 326: 325: 323: 322: 315: 308: 300: 297: 296: 293: 292: 287: 282: 277: 272: 267: 262: 257: 252: 247: 241: 238: 237: 234: 233: 230: 229: 224: 219: 214: 209: 203: 198: 197: 194: 193: 190: 189: 184: 179: 174: 168: 165: 164: 161: 160: 157: 156: 148: 140: 132: 124: 122:Microcanonical 115: 110: 109: 106: 105: 102: 101: 96: 91: 89:Parastatistics 86: 81: 76: 71: 66: 60: 55: 54: 51: 50: 49: 48: 46:Kinetic theory 43: 41:Thermodynamics 35: 34: 26: 25: 15: 9: 6: 4: 3: 2: 25801: 25790: 25787: 25785: 25782: 25780: 25777: 25775: 25772: 25770: 25767: 25766: 25764: 25749: 25746: 25744: 25741: 25740: 25737: 25731: 25728: 25726: 25723: 25721: 25718: 25716: 25713: 25711: 25708: 25706: 25703: 25701: 25698: 25696: 25693: 25691: 25688: 25686: 25683: 25681: 25678: 25676: 25673: 25671: 25668: 25666: 25663: 25661: 25658: 25656: 25653: 25651: 25648: 25647: 25645: 25641: 25633: 25630: 25628: 25625: 25624: 25623: 25620: 25618: 25615: 25613: 25610: 25608: 25605: 25603: 25602:Stopping time 25600: 25596: 25593: 25592: 25591: 25588: 25586: 25583: 25581: 25578: 25576: 25573: 25571: 25568: 25566: 25563: 25561: 25558: 25556: 25553: 25551: 25548: 25546: 25543: 25541: 25538: 25536: 25533: 25531: 25528: 25526: 25523: 25521: 25518: 25516: 25513: 25511: 25508: 25506: 25503: 25501: 25498: 25496: 25493: 25491: 25488: 25486: 25483: 25481: 25478: 25476: 25473: 25471: 25468: 25466: 25463: 25461: 25458: 25456: 25453: 25452: 25450: 25446: 25440: 25437: 25435: 25432: 25430: 25427: 25425: 25422: 25420: 25417: 25416: 25414: 25412: 25408: 25401: 25397: 25393: 25392:Hewitt–Savage 25389: 25385: 25381: 25377: 25376:Zero–one laws 25374: 25372: 25369: 25367: 25364: 25362: 25359: 25357: 25354: 25352: 25349: 25347: 25344: 25342: 25339: 25337: 25334: 25332: 25329: 25327: 25324: 25323: 25321: 25317: 25311: 25308: 25306: 25303: 25301: 25298: 25296: 25293: 25291: 25288: 25286: 25283: 25281: 25278: 25276: 25273: 25271: 25268: 25266: 25263: 25261: 25258: 25256: 25253: 25251: 25248: 25246: 25243: 25241: 25238: 25237: 25235: 25231: 25225: 25222: 25220: 25217: 25215: 25212: 25210: 25207: 25205: 25202: 25200: 25197: 25196: 25194: 25192: 25188: 25182: 25179: 25177: 25174: 25172: 25169: 25167: 25164: 25163: 25161: 25159: 25155: 25149: 25146: 25144: 25141: 25139: 25136: 25134: 25131: 25129: 25126: 25124: 25121: 25119: 25116: 25114: 25111: 25109: 25106: 25104: 25101: 25099: 25096: 25094: 25091: 25089: 25086: 25084: 25081: 25079: 25076: 25074: 25073:Black–Scholes 25071: 25069: 25066: 25064: 25061: 25059: 25056: 25055: 25053: 25051: 25047: 25041: 25038: 25036: 25033: 25031: 25028: 25026: 25023: 25021: 25018: 25016: 25013: 25012: 25010: 25008: 25004: 24998: 24995: 24993: 24990: 24986: 24983: 24981: 24978: 24977: 24976: 24975:Point process 24973: 24971: 24968: 24966: 24963: 24961: 24958: 24954: 24951: 24949: 24946: 24945: 24944: 24941: 24939: 24936: 24934: 24933:Gibbs measure 24931: 24929: 24926: 24924: 24921: 24920: 24918: 24914: 24908: 24905: 24903: 24900: 24898: 24895: 24893: 24890: 24888: 24885: 24881: 24878: 24876: 24873: 24871: 24868: 24866: 24863: 24862: 24861: 24858: 24856: 24853: 24851: 24848: 24846: 24843: 24841: 24838: 24837: 24835: 24831: 24825: 24822: 24820: 24817: 24815: 24812: 24810: 24807: 24805: 24802: 24800: 24797: 24795: 24792: 24790: 24787: 24785: 24782: 24778: 24775: 24773: 24770: 24769: 24768: 24765: 24763: 24760: 24758: 24755: 24753: 24750: 24748: 24745: 24743: 24740: 24738: 24735: 24733: 24730: 24728: 24725: 24723: 24722:Itô diffusion 24720: 24718: 24715: 24713: 24710: 24708: 24705: 24703: 24700: 24698: 24697:Gamma process 24695: 24693: 24690: 24688: 24685: 24683: 24680: 24678: 24675: 24673: 24670: 24668: 24665: 24663: 24660: 24658: 24655: 24653: 24650: 24646: 24643: 24641: 24638: 24636: 24633: 24631: 24628: 24626: 24623: 24622: 24621: 24618: 24614: 24611: 24610: 24609: 24606: 24604: 24601: 24599: 24596: 24595: 24593: 24591: 24587: 24579: 24576: 24574: 24571: 24569: 24568:Self-avoiding 24566: 24564: 24561: 24560: 24559: 24556: 24554: 24553:Moran process 24551: 24549: 24546: 24544: 24541: 24539: 24536: 24534: 24531: 24529: 24526: 24524: 24521: 24520: 24518: 24516: 24515:Discrete time 24512: 24508: 24501: 24496: 24494: 24489: 24487: 24482: 24481: 24478: 24472: 24468: 24465: 24461: 24458: 24455: 24452: 24450: 24446: 24443: 24441: 24438: 24436: 24433: 24431: 24428: 24426: 24422: 24419: 24416: 24413: 24411: 24407: 24404: 24401: 24399: 24396: 24394: 24391: 24389: 24386: 24384: 24380: 24376: 24372: 24369: 24367: 24364: 24363: 24353: 24349: 24344: 24339: 24336:(3): 032141, 24335: 24331: 24326: 24323: 24319: 24315: 24311: 24307: 24303: 24298: 24295: 24291: 24287: 24283: 24279: 24274: 24271: 24267: 24263: 24259: 24255: 24251: 24246: 24243: 24239: 24235: 24231: 24227: 24223: 24218: 24215: 24211: 24207: 24203: 24199: 24195: 24191: 24188:, Series II, 24187: 24183: 24179: 24176: 24172: 24168: 24164: 24160: 24154: 24150: 24146: 24142: 24135: 24130: 24127: 24123: 24119: 24115: 24111: 24107: 24103: 24102:Onsager, Lars 24099: 24096: 24092: 24088: 24084: 24080: 24076: 24072: 24069:, Series II, 24068: 24064: 24063:Onsager, Lars 24060: 24051:on 2013-01-12 24050: 24046: 24042: 24038: 24034: 24030: 24026: 24022: 24018: 24014: 24010: 24009: 24004: 24000: 23999:Ward, John C. 23995: 23993: 23992:0-674-91440-6 23989: 23985: 23981: 23977: 23973: 23969: 23965: 23963: 23961: 23956: 23955:981-02-4658-7 23952: 23949:; Paperback 23948: 23944: 23940: 23937: 23935: 23933: 23929: 23924: 23923:9971-5-0210-0 23920: 23917:; Paperback 23916: 23912: 23908: 23905: 23902: 23901:0-8218-3381-2 23898: 23894: 23893: 23888: 23884: 23882:9781107184824 23878: 23874: 23873: 23867: 23864: 23858: 23854: 23849: 23846: 23840: 23836: 23832: 23827: 23824: 23820: 23816: 23812: 23808: 23804: 23800: 23796: 23791: 23788: 23782: 23778: 23774: 23773:Huang, Kerson 23770: 23767: 23763: 23759: 23753: 23749: 23745: 23741: 23737: 23733: 23730: 23724: 23719: 23718: 23711: 23707: 23703: 23699: 23695: 23691: 23687: 23682: 23679: 23675: 23674: 23669: 23668:"Ising model" 23665: 23661: 23658: 23654: 23650: 23644: 23640: 23639: 23633: 23630: 23626: 23622: 23618: 23614: 23613: 23599: 23594: 23587: 23579: 23573: 23566: 23562: 23558: 23554: 23550: 23546: 23542: 23535: 23527: 23523: 23518: 23513: 23509: 23505: 23501: 23497: 23492: 23491:q-bio/0512013 23487: 23483: 23479: 23475: 23468: 23460: 23456: 23452: 23448: 23444: 23440: 23433: 23425: 23421: 23417: 23413: 23409: 23405: 23401: 23397: 23390: 23381: 23376: 23369: 23361: 23357: 23350: 23342: 23338: 23333: 23328: 23323: 23318: 23314: 23310: 23306: 23302: 23295: 23287: 23283: 23278: 23273: 23269: 23265: 23260: 23255: 23251: 23247: 23243: 23239: 23235: 23228: 23220: 23216: 23212: 23208: 23204: 23200: 23196: 23192: 23188: 23184: 23180: 23176: 23168: 23160: 23156: 23152: 23148: 23144: 23140: 23135: 23130: 23126: 23122: 23115: 23107: 23103: 23096: 23089: 23073: 23069: 23065: 23058: 23050: 23046: 23042: 23038: 23034: 23030: 23026: 23022: 23017: 23012: 23008: 23004: 22997: 22989: 22985: 22981: 22977: 22973: 22969: 22964: 22959: 22955: 22951: 22943: 22935: 22931: 22927: 22923: 22919: 22915: 22910: 22905: 22902:(2): 025022. 22901: 22897: 22890: 22882: 22878: 22874: 22870: 22866: 22862: 22858: 22854: 22849: 22844: 22840: 22836: 22829: 22821: 22817: 22813: 22809: 22804: 22799: 22796:(2): 021009. 22795: 22791: 22783: 22775: 22771: 22767: 22763: 22759: 22755: 22750: 22745: 22741: 22737: 22730: 22722: 22718: 22714: 22710: 22706: 22702: 22697: 22692: 22688: 22684: 22677: 22669: 22663: 22649: 22645: 22638: 22630: 22626: 22622: 22618: 22614: 22610: 22606: 22599: 22591: 22585: 22581: 22577: 22573: 22572: 22564: 22551:on 2012-03-20 22550: 22546: 22542: 22538: 22532: 22528: 22527: 22519: 22511: 22507: 22503: 22499: 22495: 22491: 22488:(1): 87–101. 22487: 22483: 22479: 22477: 22468: 22460: 22456: 22452: 22448: 22444: 22440: 22437:(2): 91–107. 22436: 22432: 22428: 22421: 22413: 22407: 22403: 22402: 22394: 22392: 22383: 22379: 22374: 22369: 22365: 22361: 22357: 22353: 22349: 22342: 22334: 22333: 22328: 22322: 22308: 22304: 22300: 22296: 22292: 22288: 22283: 22278: 22275:(3): 032141. 22274: 22270: 22266: 22259: 22251: 22249:9780198517979 22245: 22241: 22234: 22232: 22230: 22228: 22226: 22224: 22222: 22220: 22218: 22216: 22207: 22203: 22199: 22195: 22190: 22185: 22181: 22177: 22173: 22166: 22158: 22154: 22150: 22146: 22142: 22138: 22134: 22130: 22125: 22120: 22116: 22112: 22108: 22101: 22093: 22089: 22085: 22081: 22077: 22073: 22069: 22065: 22061: 22057: 22053: 22046: 22039: 22034: 22032: 22030: 22021: 22017: 22013: 22009: 22005: 22001: 21997: 21993: 21986: 21972:on 2014-04-07 21968: 21964: 21960: 21956: 21952: 21948: 21944: 21939: 21934: 21930: 21926: 21919: 21912: 21904: 21900: 21896: 21892: 21888: 21884: 21880: 21876: 21869: 21863:, Chapter 16. 21862: 21856: 21850: 21849: 21846:Ernst Ising, 21842: 21835: 21829: 21825: 21815: 21812: 21810: 21807: 21805: 21802: 21800: 21797: 21795: 21792: 21790: 21787: 21785: 21782: 21780: 21777: 21774: 21771:(common with 21770: 21767: 21765: 21762: 21760: 21757: 21755: 21752: 21750: 21747: 21745: 21742: 21740: 21737: 21735: 21732: 21730: 21727: 21725: 21722: 21720: 21717: 21715: 21712: 21710: 21707: 21705: 21702: 21700: 21697: 21696: 21689: 21685: 21680: 21675: 21672: 21668: 21664: 21660: 21646: 21626: 21606: 21586: 21566: 21546: 21537: 21524: 21512: 21509: 21504: 21496: 21493: 21490: 21484: 21481: 21476: 21468: 21465: 21453: 21447: 21444: 21438: 21433: 21429: 21406: 21402: 21381: 21378: 21356: 21352: 21329: 21325: 21302: 21298: 21275: 21272: 21269: 21265: 21256: 21249: 21225: 21221: 21197: 21193: 21170: 21166: 21159: 21151: 21148: 21144: 21121: 21117: 21094: 21090: 21083: 21075: 21072: 21068: 21062: 21057: 21054: 21051: 21047: 21041: 21037: 21029: 21025: 21016: 21013: 21010: 21001: 20998: 20995: 20987: 20984: 20977: 20973: 20967: 20959: 20955: 20949: 20945: 20931: 20929: 20925: 20908: 20887: 20867: 20864: 20861: 20858: 20836: 20832: 20828: 20825: 20805: 20802: 20799: 20796: 20793: 20773: 20760: 20756: 20740: 20732: 20728: 20724: 20721: 20714: 20710: 20706: 20699: 20694: 20691: 20688: 20684: 20674: 20671: 20668: 20657: 20654: 20651: 20645: 20639: 20633: 20630: 20627: 20621: 20618: 20609: 20606: 20603: 20595: 20592: 20586: 20583: 20580: 20577: 20574: 20571: 20568: 20565: 20557: 20556: 20552: 20547: 20529: 20525: 20502: 20498: 20484: 20475: 20473: 20463: 20461: 20457: 20453: 20449: 20442: 20426: 20421: 20417: 20410: 20406: 20399: 20396: 20393: 20389: 20385: 20379: 20376: 20371: 20368: 20365: 20357: 20347: 20330: 20327: 20323: 20300: 20296: 20290: 20286: 20280: 20276: 20272: 20267: 20263: 20257: 20253: 20247: 20244: 20240: 20234: 20231: 20227: 20220: 20217: 20211: 20208: 20205: 20182: 20178: 20172: 20168: 20162: 20158: 20154: 20151: 20148: 20140: 20136: 20130: 20128: 20124: 20120: 20116: 20112: 20108: 20104: 20094: 20092: 20086: 20084: 20080: 20077: − 20076: 20072: 20067: 20052: 20048: 20041: 20038: 20035: 20032: 20024: 20020: 20014: 20011: 20006: 20001: 19997: 19991: 19987: 19983: 19975: 19972: 19969: 19962: 19956: 19953: 19948: 19945: 19942: 19923: 19919: 19912: 19909: 19904: 19900: 19893: 19888: 19884: 19874: 19859: 19855: 19851: 19846: 19842: 19838: 19833: 19829: 19823: 19819: 19815: 19812: 19804: 19801: 19798: 19791: 19785: 19782: 19777: 19774: 19771: 19762: 19760: 19756: 19752: 19748: 19743: 19740: 19736: 19732: 19728: 19724: 19716: 19707: 19703: 19701: 19697: 19693: 19688: 19686: 19671: 19668: 19666: 19662: 19657: 19644: 19635: 19632: 19626: 19623: 19617: 19614: 19611: 19607: 19604: 19600: 19592: 19588: 19584: 19579: 19566: 19557: 19554: 19548: 19545: 19539: 19536: 19533: 19530: 19517: 19504: 19490: 19486: 19482: 19477: 19473: 19466: 19460: 19457: 19451: 19448: 19445: 19442: 19431: 19426: 19405: 19401: 19397: 19392: 19388: 19381: 19375: 19372: 19369: 19366: 19356: 19352: 19348: 19343: 19339: 19332: 19328: 19324: 19314: 19310: 19306: 19301: 19297: 19290: 19286: 19277: 19272: 19270: 19266: 19262: 19257: 19253: 19237: 19225: 19223: 19220: + 19219: 19215: 19212: + 19211: 19207: 19203: 19197: 19195: 19192: − 19191: 19187: 19179: 19166: 19163: 19158: 19154: 19148: 19144: 19138: 19134: 19128: 19124: 19118: 19115: 19112: 19109: 19105: 19101: 19098: 19093: 19089: 19083: 19079: 19073: 19070: 19066: 19060: 19057: 19053: 19049: 19046: 19032: 19028: 19026: 19021: 19012: 19010: 19005: 18990: 18986: 18982: 18979: 18970: 18967: 18952: 18948: 18944: 18941: 18931: 18929: 18925: 18921: 18916: 18903: 18900: 18895: 18891: 18887: 18884: 18881: 18875: 18872: 18869: 18866: 18860: 18854: 18843: 18840: 18828: 18825: 18817: 18813: 18809: 18805: 18804:Taylor series 18801: 18797: 18792: 18790: 18786: 18782: 18777: 18762: 18759: 18754: 18751: 18743: 18739: 18734: 18719: 18713: 18709: 18703: 18695: 18692: 18689: 18680: 18677: 18671: 18668: 18665: 18659: 18653: 18650: 18647: 18641: 18638: 18635: 18632: 18624: 18620: 18615: 18613: 18609: 18605: 18600: 18598: 18594: 18588: 18586: 18582: 18578: 18573: 18557: 18554: 18551: 18545: 18542: 18539: 18531: 18528: 18525: 18522: 18519: 18515: 18511: 18508: 18501: 18498: 18495: 18492: 18489: 18485: 18481: 18478: 18472: 18469: 18466: 18463: 18460: 18457: 18454: 18446: 18442: 18438: 18434: 18429: 18411: 18408: 18405: 18402: 18399: 18395: 18391: 18388: 18384: 18379: 18376: 18367: 18352: 18349: 18346: 18343: 18339: 18335: 18329: 18326: 18323: 18319: 18309: 18305: 18300: 18298: 18294: 18290: 18286: 18282: 18278: 18274: 18270: 18267: 18262: 18260: 18254: 18244: 18241: 18227: 18221: 18218: 18213: 18210: 18206: 18199: 18196: 18184: 18182: 18178: 18159: 18156: 18152: 18147: 18144: 18136: 18131: 18114: 18111: 18108: 18105: 18102: 18100: 18093: 18089: 18086: 18076: 18073: 18070: 18067: 18064: 18061: 18059: 18052: 18048: 18045: 18030: 18027: 18012: 18009: 18006: 18000: 17997: 17991: 17987: 17980: 17971: 17968: 17963: 17959: 17954: 17951: 17948: 17945: 17939: 17936: 17933: 17927: 17917: 17908: 17906: 17902: 17898: 17894: 17889: 17871: 17867: 17863: 17859: 17854: 17846: 17838: 17834: 17825: 17816: 17808: 17805: 17798: 17788: 17778: 17774: 17770: 17764: 17761: 17753: 17748: 17731: 17727: 17723: 17713: 17705: 17702: 17694: 17689: 17685: 17670: 17667: 17664: 17658: 17650: 17642: 17635: 17631: 17628: 17625: 17617: 17612: 17595: 17590: 17586: 17583: 17580: 17577: 17571: 17569: 17562: 17558: 17555: 17545: 17541: 17535: 17531: 17528: 17522: 17519: 17515: 17511: 17509: 17502: 17498: 17495: 17480: 17477: 17464: 17459: 17455: 17451: 17448: 17445: 17442: 17439: 17436: 17428: 17425: 17412: 17407: 17401: 17397: 17393: 17387: 17384: 17376: 17368: 17365: 17360: 17356: 17348: 17343: 17340: 17335: 17331: 17325: 17319: 17315: 17311: 17305: 17302: 17299: 17296: 17288: 17286: 17281: 17279: 17275: 17270: 17257: 17254: 17250: 17244: 17240: 17237: 17231: 17228: 17224: 17220: 17217: 17214: 17206: 17204: 17200: 17196: 17192: 17187: 17185: 17181: 17177: 17173: 17169: 17164: 17151: 17148: 17145: 17142: 17139: 17134: 17126: 17123: 17114: 17111: 17106: 17102: 17093: 17089: 17084: 17079: 17076: 17073: 17070: 17067: 17060: 17056: 17052: 17047: 17044: 17041: 17033: 17030: 17027: 17008: 17005: 17000: 16996: 16988: 16978: 16970: 16967: 16959: 16954: 16950: 16935: 16932: 16929: 16923: 16915: 16907: 16900: 16896: 16893: 16890: 16885: 16881: 16877: 16869: 16867: 16863: 16859: 16855: 16851: 16846: 16844: 16840: 16835: 16822: 16817: 16813: 16807: 16804: 16799: 16796: 16793: 16790: 16787: 16784: 16779: 16766: 16763: 16761: 16757: 16753: 16749: 16745: 16741: 16736: 16734: 16730: 16724: 16722: 16718: 16708: 16705: 16701: 16696: 16694: 16689: 16687: 16681: 16666: 16660: 16656: 16649: 16646: 16642: 16637: 16632: 16628: 16622: 16619: 16614: 16609: 16599: 16586: 16583: 16577: 16573: 16568: 16564: 16560: 16557: 16554: 16546: 16544: 16540: 16535: 16528: 16522: 16520: 16519:Kurt Symanzik 16515: 16510: 16508: 16503: 16499: 16489: 16487: 16482: 16465: 16462: 16459: 16455: 16449: 16444: 16439: 16435: 16429: 16423: 16417: 16409: 16407: 16403: 16398: 16382: 16377: 16372: 16368: 16360:is therefore 16359: 16339: 16335: 16331: 16328: 16324: 16320: 16315: 16312: 16309: 16305: 16296: 16292: 16287: 16284: 16280: 16276: 16272: 16267: 16265: 16261: 16257: 16253: 16248: 16232: 16229: 16224: 16220: 16215: 16210: 16205: 16199: 16196: 16191: 16187: 16180: 16176: 16172: 16169: 16166: 16163: 16157: 16151: 16143: 16141: 16137: 16133: 16128: 16113: 16110: 16107: 16101: 16098: 16092: 16088: 16082: 16078: 16070: 16065: 16060: 16057: 16054: 16049: 16044: 16039: 16036: 16033: 16030: 16024: 16018: 16010: 16007: 15991: 15981: 15976: 15963: 15960: 15954: 15948: 15945: 15942: 15938: 15931: 15928: 15923: 15920: 15912: 15909: 15906: 15902: 15897: 15890: 15887: 15883: 15874: 15871: 15868: 15864: 15860: 15852: 15849: 15846: 15843: 15840: 15837: 15832: 15819: 15817: 15813: 15805: 15801: 15796: 15794: 15791:The constant 15789: 15772: 15769: 15766: 15762: 15758: 15753: 15747: 15741: 15733: 15731: 15727: 15722: 15705: 15702: 15699: 15695: 15691: 15686: 15683: 15675: 15672: 15633: 15628: 15624: 15620: 15615: 15612: 15609: 15605: 15601: 15593: 15591: 15587: 15583: 15579: 15575: 15570: 15557: 15554: 15551: 15548: 15537: 15534: 15521: 15515: 15512: 15504: 15502: 15498: 15495:= 0, this is 15494: 15489: 15487: 15483: 15478: 15461: 15458: 15456: 15448: 15442: 15439: 15436: 15430: 15424: 15419: 15399: 15396: 15394: 15383: 15377: 15371: 15365: 15358: 15354: 15351: 15346: 15341: 15333: 15329: 15314: 15312: 15302: 15300: 15296: 15291: 15273: 15270: 15266: 15262: 15257: 15254: 15250: 15243: 15240: 15232: 15229: 15225: 15221: 15216: 15213: 15209: 15202: 15194: 15191: 15187: 15183: 15178: 15175: 15171: 15164: 15161: 15156: 15152: 15148: 15137: 15134: 15130: 15123: 15120: 15112: 15108: 15101: 15093: 15089: 15082: 15071: 15069: 15066: − 15065: 15061: 15057: 15053: 15049: 15046: − 15045: 15041: 15037: 15034: − 15033: 15028: 15012: 15009: 15004: 15000: 14990: 14987: 14984: 14978: 14975: 14971: 14960: 14952: 14949: 14941: 14938: 14932: 14929: 14923: 14920: 14917: 14911: 14908: 14899: 14893: 14887: 14881: 14875: 14866: 14860: 14854: 14848: 14837: 14835: 14830: 14827: 14822: 14820: 14816: 14812: 14808: 14804: 14800: 14790: 14787: 14782: 14780: 14776: 14772: 14768: 14763: 14750: 14747: 14742: 14738: 14734: 14731: 14728: 14725: 14722: 14719: 14716: 14712: 14706: 14702: 14698: 14695: 14690: 14686: 14682: 14678: 14671: 14654: 14652: 14647: 14645: 14641: 14637: 14633: 14629: 14625: 14621: 14617: 14612: 14597: 14592: 14583: 14575: 14569: 14563: 14558: 14550: 14544: 14540: 14535: 14530: 14522: 14516: 14508: 14500: 14494: 14488: 14485: 14480: 14470: 14462: 14456: 14450: 14447: 14442: 14434: 14428: 14420: 14418: 14414: 14410: 14405: 14403: 14399: 14395: 14391: 14388: 14384: 14380: 14376: 14369:Magnetization 14366: 14365:) term to 1. 14364: 14360: 14356: 14352: 14347: 14345: 14340: 14327: 14323: 14317: 14309: 14300: 14297: 14292: 14288: 14284: 14281: 14276: 14272: 14268: 14264: 14260: 14255: 14251: 14247: 14244: 14241: 14233: 14231: 14227: 14223: 14219: 14215: 14210: 14205: 14202: 14198: 14193: 14191: 14185: 14172: 14167: 14163: 14159: 14155: 14150: 14146: 14142: 14139: 14136: 14128: 14126: 14121: 14119: 14115: 14105: 14103: 14099: 14095: 14091: 14087: 14083: 14079: 14060: 14054: 14050: 14046: 14043: 14038: 14028: 14017: 14014: 14009: 14005: 14001: 13997: 13993: 13988: 13984: 13980: 13977: 13973: 13968: 13965: 13962: 13959: 13956: 13948: 13943: 13930: 13921: 13915: 13911: 13907: 13904: 13899: 13889: 13878: 13875: 13870: 13866: 13862: 13858: 13854: 13849: 13845: 13841: 13838: 13834: 13829: 13826: 13823: 13813: 13809: 13802: 13799: 13791: 13787: 13780: 13772: 13768: 13761: 13755: 13749: 13745: 13741: 13738: 13733: 13723: 13712: 13709: 13704: 13700: 13696: 13692: 13688: 13683: 13679: 13675: 13672: 13668: 13663: 13660: 13657: 13651: 13640: 13636: 13629: 13626: 13618: 13614: 13607: 13599: 13595: 13588: 13577: 13575: 13570: 13557: 13549: 13543: 13540: 13537: 13533: 13529: 13523: 13517: 13513: 13509: 13506: 13501: 13491: 13480: 13477: 13472: 13468: 13464: 13460: 13456: 13451: 13447: 13443: 13440: 13436: 13432: 13426: 13420: 13412: 13410: 13405: 13403: 13398: 13393: 13386: 13382: 13366: 13361: 13352: 13347: 13336: 13333: 13329: 13319: 13317: 13310: 13303: 13295: 13282: 13278: 13274: 13271: 13266: 13262: 13258: 13255: 13250: 13242: 13237: 13224: 13220: 13214: 13209: 13206: 13203: 13199: 13195: 13190: 13186: 13182: 13178: 13174: 13169: 13165: 13161: 13158: 13155: 13152: 13144: 13142: 13137: 13135: 13131: 13127: 13123: 13118: 13116: 13112: 13108: 13104: 13100: 13096: 13091: 13089: 13085: 13081: 13076: 13066: 13057: 13054: 13050: 13046: 13042: 13038: 13034: 13030: 13029:Sorin Istrail 13020: 13018: 13014: 13009: 13007: 13002: 12998: 12987: 12985: 12981: 12977: 12972: 12970: 12960: 12958: 12957:minimal model 12954: 12948: 12941:Minimal model 12938: 12936: 12932: 12931:limit formula 12928: 12924: 12919: 12903: 12899: 12876: 12872: 12848: 12845: 12839: 12833: 12830: 12825: 12819: 12815: 12811: 12808: 12805: 12802: 12797: 12793: 12789: 12786: 12783: 12780: 12776: 12771: 12768: 12764: 12759: 12756: 12748: 12745: 12735: 12732: 12730: 12725: 12712: 12707: 12704: 12701: 12694: 12690: 12682: 12675: 12671: 12663: 12659: 12655: 12650: 12647: 12644: 12640: 12634: 12630: 12626: 12623: 12618: 12615: 12612: 12608: 12602: 12595: 12591: 12583: 12579: 12575: 12570: 12567: 12564: 12560: 12554: 12547: 12543: 12537: 12534: 12529: 12524: 12520: 12507: 12504: 12501: 12498: 12497: 12496: 12493: 12479: 12474: 12470: 12464: 12459: 12455: 12451: 12448: 12440: 12438: 12428: 12427:has flipped. 12426: 12421: 12417: 12412: 12397: 12392: 12389: 12386: 12382: 12376: 12371: 12367: 12363: 12360: 12355: 12350: 12346: 12342: 12337: 12333: 12324: 12322: 12318: 12305: 12300: 12295: 12291: 12282: 12280: 12275: 12272: 12259: 12254: 12249: 12245: 12236: 12229: 12224: 12222: 12218: 12213: 12200: 12192: 12188: 12173: 12170: 12162: 12158: 12153: 12136: 12122: 12106: 12095: 12089: 12084: 12080: 12076: 12070: 12057: 12050: 12043: 12039: 12034: 12021: 12014: 12010: 12004: 12000: 11995: 11986: 11981: 11979: 11974: 11959: 11956: 11952: 11948: 11945: 11942: 11939: 11934: 11924: 11918: 11915: 11911: 11904: 11899: 11895: 11886: 11883: 11868: 11862: 11859: 11855: 11851: 11848: 11840: 11836: 11832: 11827: 11825: 11824:path integral 11821: 11818:direction as 11817: 11813: 11810:direction as 11809: 11806:Think of the 11804: 11791: 11780: 11774: 11771: 11768: 11765: 11762: 11758: 11754: 11749: 11746: 11743: 11740: 11737: 11733: 11728: 11722: 11719: 11716: 11712: 11706: 11703: 11699: 11688: 11685: 11677: 11670: 11661: 11653: 11651: 11635: 11632: 11629: 11607: 11603: 11580: 11576: 11566: 11551: 11548: 11540: 11537: 11531: 11520: 11515: 11512: 11506: 11503: 11500: 11495: 11492: 11486: 11481: 11477: 11468: 11452: 11448: 11427: 11424: 11419: 11415: 11411: 11406: 11402: 11392: 11379: 11376: 11372: 11364: 11360: 11356: 11349: 11345: 11341: 11335: 11331: 11328: 11324: 11316: 11312: 11308: 11301: 11297: 11293: 11287: 11283: 11280: 11272: 11256: 11252: 11241: 11228: 11217: 11213: 11206: 11203: 11195: 11191: 11187: 11184: 11178: 11175: 11172: 11164: 11160: 11153: 11150: 11142: 11138: 11134: 11131: 11125: 11122: 11119: 11111: 11107: 11103: 11100: 11094: 11091: 11083: 11079: 11075: 11072: 11066: 11063: 11057: 11054: 11049: 11045: 11041: 11036: 11033: 11028: 11024: 11018: 11014: 11010: 11005: 11002: 10997: 10993: 10984: 10980: 10976: 10972: 10967: 10964: 10961: 10958: 10955: 10952: 10949: 10946: 10938: 10922: 10918: 10895: 10891: 10870: 10867: 10864: 10856: 10851: 10837: 10834: 10830: 10826: 10822: 10821: 10815: 10801: 10778: 10772: 10752: 10749: 10746: 10743: 10737: 10731: 10711: 10702: 10685: 10682: 10679: 10673: 10670: 10667: 10664: 10659: 10656: 10651: 10648: 10644: 10641: 10616: 10613: 10606: 10601: 10598: 10593: 10589: 10580: 10577: 10571: 10568: 10563: 10559: 10555: 10552: 10547: 10544: 10539: 10533: 10527: 10503: 10500: 10491: 10487: 10483: 10479: 10475: 10472: 10466: 10462: 10458: 10454: 10447: 10444: 10439: 10435: 10426: 10423: 10417: 10414: 10409: 10405: 10401: 10398: 10393: 10390: 10385: 10379: 10371: 10367: 10363: 10360: 10355: 10352: 10327: 10324: 10318: 10315: 10312: 10309: 10304: 10301: 10296: 10292: 10289: 10285: 10277: 10274: 10268: 10265: 10260: 10257: 10254: 10226: 10222: 10218: 10213: 10209: 10202: 10196: 10193: 10190: 10187: 10180: 10176: 10170: 10166: 10161: 10158: 10153: 10149: 10129: 10115: 10111: 10107: 10102: 10098: 10091: 10085: 10082: 10079: 10073: 10059: 10055: 10051: 10046: 10042: 10035: 10029: 10026: 10023: 10015: 10011: 10007: 10001: 9998: 9995: 9987: 9983: 9962: 9959: 9954: 9950: 9946: 9941: 9937: 9916: 9913: 9910: 9907: 9887: 9877: 9873: 9869: 9864: 9860: 9851: 9847: 9843: 9839: 9828: 9824: 9820: 9815: 9811: 9802: 9798: 9794: 9790: 9784: 9780: 9776: 9770: 9767: 9764: 9756: 9752: 9731: 9708: 9705: 9702: 9694: 9690: 9666: 9660: 9657: 9651: 9648: 9645: 9639: 9625: 9623: 9619: 9615: 9611: 9606: 9593: 9588: 9583: 9579: 9573: 9569: 9565: 9562: 9557: 9552: 9549: 9546: 9542: 9536: 9531: 9527: 9521: 9518: 9515: 9512: 9509: 9506: 9503: 9499: 9495: 9492: 9489: 9483: 9477: 9469: 9466: 9453: 9448: 9443: 9440: 9437: 9433: 9427: 9422: 9418: 9414: 9409: 9404: 9400: 9391: 9378: 9373: 9368: 9364: 9358: 9353: 9350: 9347: 9343: 9339: 9334: 9329: 9325: 9316: 9314: 9311: ~ 9310: 9305: 9292: 9287: 9282: 9278: 9272: 9268: 9264: 9261: 9256: 9251: 9248: 9245: 9241: 9235: 9230: 9226: 9220: 9217: 9214: 9211: 9208: 9205: 9202: 9198: 9194: 9191: 9188: 9182: 9176: 9168: 9166: 9162: 9152: 9150: 9146: 9141: 9128: 9124: 9115: 9112: 9109: 9106: 9102: 9098: 9095: 9089: 9086: 9083: 9080: 9076: 9070: 9067: 9063: 9059: 9056: 9053: 9047: 9044: 9041: 9035: 9032: 9029: 9026: 9023: 9015: 9013: 9008: 8995: 8990: 8982: 8979: 8976: 8970: 8964: 8961: 8958: 8950: 8946: 8942: 8935: 8931: 8925: 8921: 8916: 8906: 8902: 8899: 8891: 8889: 8884: 8871: 8861: 8858: 8855: 8847: 8844: 8840: 8836: 8828: 8824: 8818: 8814: 8802: 8800: 8794: 8792: 8787: 8774: 8769: 8766: 8763: 8760: 8756: 8752: 8746: 8743: 8740: 8736: 8726: 8724: 8720: 8716: 8711: 8709: 8705: 8695: 8673: 8666: 8663: 8660: 8657: 8653: 8649: 8644: 8636: 8633: 8630: 8627: 8619: 8616: 8613: 8609: 8603: 8600: 8597: 8594: 8591: 8586: 8583: 8579: 8575: 8570: 8566: 8553: 8533: 8529: 8523: 8518: 8511: 8507: 8501: 8497: 8491: 8486: 8483: 8479: 8473: 8468: 8464: 8460: 8455: 8450: 8446: 8442: 8437: 8432: 8428: 8424: 8420: 8415: 8411: 8407: 8403: 8400: 8397: 8391: 8385: 8365: 8360: 8349: 8346: 8343: 8337: 8334: 8330: 8322: 8319: 8316: 8312: 8302: 8299: 8296: 8292: 8281: 8278: 8275: 8269: 8265: 8258: 8253: 8250: 8227: 8224: 8218: 8214: 8211: 8204: 8197: 8194: 8190: 8187: 8184: 8180: 8174: 8169: 8165: 8162: 8155: 8151: 8145: 8142: 8138: 8135: 8131: 8107: 8104: 8100: 8097: 8093: 8072: 8065: 8061: 8057: 8052: 8048: 8043: 8039: 8032: 8028: 8024: 8019: 8015: 8010: 8002: 7998: 7994: 7989: 7985: 7980: 7972: 7968: 7964: 7961: 7958: 7953: 7949: 7944: 7940: 7933: 7929: 7923: 7919: 7915: 7912: 7908: 7900: 7896: 7892: 7889: 7885: 7881: 7874: 7870: 7864: 7860: 7856: 7853: 7849: 7841: 7837: 7833: 7830: 7826: 7818: 7814: 7808: 7804: 7800: 7797: 7793: 7785: 7781: 7777: 7774: 7770: 7762: 7758: 7754: 7751: 7748: 7743: 7739: 7734: 7730: 7724: 7718: 7710: 7705: 7703: 7700:≠ 0; but for 7699: 7694: 7681: 7676: 7671: 7663: 7660: 7657: 7653: 7649: 7644: 7641: 7637: 7629: 7626: 7623: 7619: 7615: 7610: 7607: 7603: 7596: 7591: 7583: 7580: 7577: 7573: 7567: 7563: 7551: 7548: 7535: 7531: 7525: 7522: 7519: 7515: 7511: 7506: 7503: 7499: 7494: 7490: 7487: 7482: 7479: 7474: 7471: 7465: 7462: 7459: 7453: 7445: 7442: 7429: 7424: 7421: 7418: 7413: 7407: 7404: 7401: 7397: 7393: 7388: 7385: 7381: 7376: 7371: 7368: 7362: 7358: 7354: 7350: 7347: 7343: 7336: 7332: 7328: 7323: 7317: 7312: 7309: 7306: 7302: 7298: 7295: 7288: 7284: 7278: 7275: 7272: 7268: 7264: 7261: 7257: 7253: 7246: 7242: 7236: 7232: 7228: 7225: 7221: 7213: 7209: 7203: 7199: 7195: 7192: 7188: 7180: 7176: 7172: 7169: 7166: 7161: 7157: 7152: 7148: 7142: 7136: 7127: 7114: 7111: 7108: 7104: 7099: 7096: 7093: 7089: 7083: 7079: 7075: 7071: 7067: 7063: 7042: 7038: 7032: 7028: 7022: 7019: 7016: 7012: 7008: 7005: 7002: 6997: 6993: 6987: 6983: 6978: 6974: 6971: 6968: 6962: 6956: 6948: 6943: 6935: 6933: 6929: 6925: 6921: 6917: 6901: 6891: 6888: 6885: 6874: 6868: 6865: 6861: 6854: 6848: 6845: 6837: 6833: 6821: 6817: 6810: 6802: 6798: 6792: 6788: 6764: 6760: 6752: 6749: 6746: 6743: 6739: 6735: 6730: 6722: 6719: 6716: 6713: 6705: 6702: 6699: 6695: 6689: 6686: 6683: 6680: 6677: 6672: 6669: 6665: 6660: 6656: 6653: 6648: 6645: 6640: 6637: 6631: 6625: 6622: 6619: 6613: 6610: 6606: 6593: 6585: 6582: 6576: 6573: 6570: 6564: 6556: 6552: 6548: 6544: 6540: 6524: 6519: 6515: 6509: 6505: 6501: 6498: 6493: 6490: 6487: 6483: 6477: 6473: 6467: 6464: 6461: 6458: 6455: 6452: 6449: 6446: 6443: 6439: 6435: 6432: 6429: 6423: 6417: 6409: 6396: 6391: 6387: 6384: 6366: 6363: 6353: 6350: 6347: 6339: 6334: 6331: 6327: 6318: 6301: 6298: 6288: 6285: 6282: 6274: 6269: 6266: 6262: 6253: 6252:ferromagnetic 6249: 6232: 6229: 6219: 6216: 6213: 6205: 6200: 6197: 6193: 6184: 6183:ferromagnetic 6180: 6179: 6178: 6162: 6159: 6149: 6146: 6143: 6135: 6130: 6127: 6123: 6111:One dimension 6108: 6106: 6102: 6098: 6091: 6087: 6083: 6076: 6072: 6062: 6057: 6054: − 6050: 6042: 6028: 6018: 6014: 6010: 6005: 6001: 5994: 5991: 5987: 5978: 5975: 5972: 5969: 5965: 5964: 5963: 5960: 5940: 5930: 5923: 5920: 5917: 5912: 5908: 5904: 5899: 5895: 5884: 5874: 5870: 5866: 5861: 5857: 5850: 5847: 5843: 5836: 5831: 5825: 5822: 5819: 5813: 5805: 5803: 5799: 5795: 5791: 5784: 5777: 5772: 5759: 5749: 5745: 5741: 5736: 5732: 5725: 5722: 5718: 5714: 5705: 5702: 5699: 5693: 5685: 5682: 5679: 5673: 5661: 5648: 5638: 5634: 5630: 5625: 5621: 5614: 5611: 5607: 5603: 5590: 5586: 5579: 5576: 5572: 5566: 5563: 5549: 5545: 5538: 5535: 5531: 5525: 5522: 5514: 5505: 5497: 5493: 5484: 5476: 5472: 5465: 5456: 5453: 5450: 5444: 5436: 5433: 5430: 5424: 5418: 5409: 5406: 5403: 5397: 5391: 5388: 5385: 5379: 5371: 5368: 5365: 5359: 5353: 5350: 5347: 5341: 5335: 5326: 5323: 5320: 5314: 5306: 5303: 5300: 5294: 5283: 5281: 5277: 5273: 5269: 5265: 5261: 5256: 5251:Specification 5248: 5245: 5243: 5239: 5235: 5230: 5228: 5223: 5216: 5209: 5204: 5202: 5198: 5194: 5190: 5185: 5183: 5182:ferromagnetic 5178: 5174: 5170: 5167: 5163: 5148: 5134: 5129: 5125: 5119: 5115: 5106: 5100: 5093: 5089: 5086: 5083: 5077: 5071: 5063: 5047: 5042: 5038: 5032: 5028: 5024: 5021: 5016: 5012: 5006: 5002: 4993: 4987: 4980: 4976: 4973: 4970: 4964: 4958: 4950: 4945: 4943: 4939: 4931: 4928: 4925: 4921: 4916: 4911: 4908: 4905: 4904: 4903: 4892: 4887: 4878: 4874: 4864: 4859: 4849: 4847: 4842: 4839: 4826: 4821: 4818: 4815: 4812: 4808: 4802: 4799: 4795: 4789: 4785: 4779: 4775: 4754: 4749: 4746: 4742: 4738: 4732: 4726: 4718: 4714: 4710: 4706: 4702: 4698: 4693: 4689: 4675: 4670: 4666: 4660: 4652: 4649: 4641: 4637: 4628: 4623: 4621: 4605: 4600: 4592: 4589: 4583: 4577: 4571: 4563: 4559: 4554: 4552: 4546: 4539: 4536: 4532: 4528: 4525: 4521: 4517: 4514: 4513: 4512: 4509: 4496: 4491: 4487: 4481: 4476: 4473: 4470: 4466: 4460: 4457: 4452: 4449: 4441: 4439: 4438:magnetization 4434: 4432: 4426: 4418: 4416: 4411: 4404: 4401: 4397: 4394: 4390: 4389: 4388: 4386: 4376: 4374: 4373:zero-one laws 4369: 4365: 4360: 4358: 4354: 4350: 4345: 4342: 4338: 4328: 4326: 4322: 4316: 4306: 4303: 4290: 4282: 4278: 4272: 4268: 4256: 4252: 4246: 4242: 4233: 4230: 4227: 4223: 4219: 4211: 4207: 4201: 4197: 4185: 4171: 4151: 4131: 4111: 4091: 4077: 4061: 4057: 4031: 4027: 4014: 4000: 3997: 3989: 3985: 3958: 3955: 3946: 3927: 3923: 3917: 3914: 3911: 3907: 3900: 3892: 3888: 3875: 3862: 3854: 3850: 3838: 3834: 3827: 3819: 3815: 3809: 3805: 3793: 3777: 3773: 3750: 3746: 3735: 3725: 3723: 3713: 3711: 3707: 3703: 3702:limit formula 3699: 3695: 3691: 3687: 3683: 3679: 3674: 3672: 3668: 3663: 3650: 3647: 3641: 3635: 3632: 3627: 3617: 3613: 3607: 3603: 3591: 3589: 3585: 3581: 3571: 3568: 3555: 3551: 3542: 3539: 3536: 3525: 3519: 3516: 3512: 3508: 3505: 3502: 3499: 3494: 3484: 3480: 3474: 3470: 3458: 3455: − 3454: 3449: 3443: 3438: 3434: 3424: 3422: 3417: 3413: 3409: 3405: 3396: 3384: 3381: 3377: 3374: 3370: 3366: 3363: 3362: 3361: 3353: 3340: 3336: 3327: 3323: 3316: 3312: 3308: 3305: 3300: 3297: 3293: 3284: 3278: 3275: 3272: 3269: 3265: 3261: 3255: 3249: 3241: 3226: 3217: 3213: 3206: 3202: 3179: 3176: 3172: 3168: 3165: 3160: 3157: 3153: 3130: 3127: 3123: 3112: 3108: 3101: 3098: 3095: 3092: 3088: 3067: 3044: 3038: 3018: 3009: 2992: 2987: 2984: 2980: 2969: 2965: 2958: 2955: 2952: 2949: 2945: 2941: 2938: 2933: 2930: 2926: 2917: 2911: 2908: 2905: 2902: 2898: 2894: 2891: 2889: 2879: 2876: 2872: 2861: 2857: 2850: 2847: 2844: 2841: 2837: 2833: 2828: 2825: 2821: 2810: 2806: 2799: 2796: 2793: 2790: 2786: 2782: 2777: 2774: 2770: 2759: 2755: 2748: 2745: 2742: 2739: 2735: 2731: 2728: 2726: 2718: 2712: 2700: 2697: 2681: 2678: 2674: 2670: 2665: 2662: 2658: 2637: 2634: 2612: 2609: 2605: 2595: 2582: 2577: 2574: 2570: 2559: 2555: 2548: 2545: 2542: 2539: 2535: 2529: 2526: 2521: 2517: 2508: 2504: 2497: 2493: 2484: 2482: 2461: 2457: 2450: 2429: 2420: 2416: 2409: 2405: 2397: 2379: 2375: 2352: 2348: 2327: 2302: 2298: 2291: 2269: 2265: 2242: 2238: 2217: 2194: 2188: 2168: 2148: 2145: 2142: 2137: 2133: 2123: 2107: 2103: 2097: 2093: 2087: 2084: 2080: 2071: 2065: 2062: 2059: 2056: 2052: 2048: 2045: 2039: 2033: 2025: 2022: 2020: 2016: 2015:complementary 2012: 2005: 2002: 1996: 1983: 1978: 1974: 1968: 1964: 1955: 1949: 1942: 1938: 1935: 1932: 1926: 1920: 1912: 1910: 1906: 1902: 1898: 1891: 1886: 1884: 1879: 1866: 1861: 1857: 1851: 1847: 1841: 1838: 1834: 1825: 1819: 1812: 1808: 1805: 1799: 1793: 1785: 1783: 1779: 1753: 1750: 1745: 1741: 1733: 1730: 1714: 1711: 1706: 1702: 1694: 1691: 1675: 1672: 1667: 1663: 1655: 1654: 1652: 1650: 1646: 1641: 1637: 1629: 1613: 1610: 1605: 1602: 1598: 1590: 1587: 1571: 1568: 1563: 1560: 1556: 1548: 1545: 1544:ferromagnetic 1529: 1526: 1521: 1518: 1514: 1506: 1505: 1503: 1502: 1498: 1479: 1473: 1459: 1445: 1422: 1414: 1410: 1400: 1386: 1377: 1361: 1353: 1349: 1342: 1336: 1331: 1327: 1323: 1318: 1310: 1299: 1285: 1277: 1272: 1254: 1248: 1245: 1242: 1238: 1232: 1228: 1224: 1219: 1215: 1206: 1189: 1180: 1172: 1168: 1165: 1162: 1153: 1140: 1133: 1129: 1120: 1114: 1111: 1108: 1104: 1098: 1092: 1084: 1080: 1071: 1057: 1054: 1051: 1044: 1040: 1036: 1020: 1012: 996: 976: 953: 950: 938: 925: 920: 916: 910: 906: 900: 896: 892: 889: 884: 880: 874: 870: 864: 861: 857: 848: 845: 838: 834: 831: 825: 819: 811: 810: 793: 784: 766: 762: 754: 735: 732: 710: 707: 703: 695: 676: 673: 670: 667: 658: 639: 636: 626: 622: 615: 611: 602: 583: 580: 577: 574: 571: 565: 560: 556: 533: 529: 505: 502: 482: 474: 444: 442: 438: 435: 430: 428: 423: 419: 414: 412: 408: 404: 400: 396: 392: 388: 383: 381: 377: 373: 369: 365: 361: 357: 353: 349: 345: 341: 337: 333: 321: 316: 314: 309: 307: 302: 301: 299: 298: 291: 288: 286: 283: 281: 278: 276: 273: 271: 268: 266: 263: 261: 258: 256: 253: 251: 248: 246: 243: 242: 236: 235: 228: 225: 223: 220: 218: 215: 213: 210: 208: 205: 204: 201: 196: 195: 188: 185: 183: 180: 178: 175: 173: 170: 169: 163: 162: 155: 152: 149: 147: 144: 141: 139: 136: 133: 131: 128: 125: 123: 120: 117: 116: 113: 108: 107: 100: 97: 95: 92: 90: 87: 85: 82: 80: 79:Bose–Einstein 77: 75: 72: 70: 67: 65: 62: 61: 58: 53: 52: 47: 44: 42: 39: 38: 37: 36: 32: 28: 27: 24: 21: 20: 25660:Econometrics 25622:Wiener space 25510:Itô integral 25411:Inequalities 25300:Self-similar 25270:Gauss–Markov 25260:Exchangeable 25240:Càdlàg paths 25176:Risk process 25128:LIBOR market 24997:Random graph 24992:Random field 24942: 24804:Superprocess 24742:Lévy process 24737:Jump process 24712:Hunt process 24548:Markov chain 24333: 24329: 24305: 24301: 24285: 24281: 24278:Barth, P. F. 24253: 24249: 24225: 24221: 24189: 24185: 24140: 24117: 24109: 24105: 24070: 24066: 24053:, retrieved 24049:the original 24012: 24006: 23983: 23975: 23971: 23957: 23942: 23925: 23910: 23890: 23871: 23852: 23830: 23798: 23794: 23776: 23739: 23716: 23689: 23685: 23671: 23637: 23620: 23617:Barth, P. F. 23586: 23548: 23544: 23534: 23481: 23477: 23467: 23442: 23438: 23432: 23399: 23395: 23389: 23368: 23359: 23355: 23349: 23304: 23300: 23294: 23241: 23237: 23227: 23178: 23174: 23167: 23124: 23120: 23114: 23105: 23101: 23088: 23076:. Retrieved 23072:the original 23067: 23057: 23006: 23002: 22996: 22953: 22949: 22942: 22899: 22895: 22889: 22838: 22834: 22828: 22793: 22789: 22782: 22739: 22735: 22729: 22686: 22682: 22676: 22662: 22651:. Retrieved 22647: 22637: 22612: 22608: 22598: 22574:. Springer. 22570: 22563: 22553:, retrieved 22549:the original 22525: 22518: 22485: 22481: 22475: 22467: 22434: 22430: 22420: 22400: 22355: 22351: 22341: 22330: 22321: 22310:. Retrieved 22272: 22268: 22258: 22239: 22179: 22175: 22165: 22114: 22110: 22100: 22059: 22055: 22045: 21995: 21991: 21985: 21974:. Retrieved 21967:the original 21928: 21924: 21911: 21878: 21874: 21868: 21855: 21847: 21841: 21828: 21739:Hopfield net 21686: 21682: 21677: 21671:Barth (1981) 21669: 21665: 21661: 21538: 21421:is given by 20932: 20765: 20558: 20546:Barth (1981) 20489: 20469: 20459: 20455: 20451: 20444: 20440: 20356:spin glasses 20353: 20350:Spin glasses 20131: 20100: 20097:Neuroscience 20087: 20082: 20078: 20074: 20070: 20068: 19875: 19763: 19758: 19754: 19750: 19744: 19738: 19734: 19730: 19726: 19719: 19717: 19713: 19704: 19689: 19682: 19674:Applications 19669: 19664: 19660: 19658: 19590: 19586: 19582: 19580: 19518: 19429: 19427: 19275: 19273: 19268: 19264: 19260: 19258: 19254: 19226: 19221: 19217: 19213: 19209: 19205: 19201: 19198: 19193: 19189: 19182: 19180: 19038: 19029: 19025:Leo Kadanoff 19022: 19018: 19006: 18971: 18968: 18932: 18927: 18923: 18919: 18917: 18815: 18811: 18807: 18800:catastrophes 18793: 18784: 18780: 18778: 18741: 18737: 18736:Dividing by 18735: 18622: 18618: 18616: 18611: 18610:or negative 18607: 18603: 18601: 18596: 18592: 18589: 18584: 18580: 18576: 18574: 18444: 18440: 18436: 18432: 18430: 18368: 18307: 18303: 18301: 18296: 18292: 18288: 18284: 18280: 18276: 18272: 18268: 18263: 18256: 18242: 18185: 18180: 18176: 18134: 18132: 18031: 18028: 17918: 17914: 17904: 17900: 17896: 17892: 17890: 17751: 17749: 17615: 17613: 17481: 17478: 17429: 17426: 17289: 17284: 17282: 17273: 17271: 17207: 17198: 17194: 17190: 17188: 17183: 17179: 17175: 17171: 17167: 17165: 17034: 17031: 17028: 16870: 16865: 16861: 16857: 16853: 16849: 16847: 16842: 16838: 16836: 16767: 16764: 16759: 16755: 16751: 16747: 16743: 16739: 16737: 16732: 16728: 16725: 16720: 16716: 16714: 16703: 16699: 16697: 16692: 16690: 16682: 16547: 16538: 16536: 16532: 16526: 16511: 16506: 16501: 16497: 16495: 16483: 16410: 16405: 16401: 16399: 16357: 16294: 16290: 16288: 16282: 16278: 16274: 16270: 16268: 16263: 16259: 16255: 16251: 16249: 16144: 16139: 16135: 16131: 16129: 16011: 16008: 15979: 15977: 15820: 15815: 15811: 15809: 15803: 15799: 15792: 15790: 15734: 15729: 15725: 15723: 15676: 15674:This gives: 15673: 15594: 15589: 15585: 15581: 15577: 15573: 15571: 15538: 15535: 15505: 15492: 15490: 15485: 15481: 15479: 15315: 15310: 15308: 15298: 15294: 15292: 15072: 15067: 15063: 15059: 15047: 15043: 15039: 15035: 15031: 15029: 14838: 14831: 14825: 14823: 14818: 14817:< 4, the 14814: 14810: 14806: 14802: 14798: 14796: 14793:Fluctuations 14785: 14783: 14774: 14770: 14766: 14764: 14655: 14650: 14648: 14643: 14639: 14635: 14631: 14627: 14623: 14619: 14615: 14613: 14421: 14416: 14412: 14408: 14406: 14401: 14397: 14393: 14389: 14386: 14385:/4, so that 14382: 14378: 14374: 14372: 14362: 14358: 14354: 14350: 14348: 14343: 14341: 14234: 14229: 14225: 14221: 14217: 14213: 14208: 14206: 14200: 14196: 14194: 14189: 14186: 14129: 14124: 14122: 14117: 14113: 14112:The form of 14111: 14101: 14093: 14089: 14085: 14081: 14077: 13946: 13944: 13578: 13573: 13571: 13413: 13408: 13406: 13396: 13388: 13384: 13320: 13315: 13305: 13298: 13296: 13145: 13140: 13138: 13133: 13129: 13125: 13121: 13119: 13114: 13110: 13106: 13102: 13098: 13094: 13092: 13087: 13083: 13079: 13074: 13072: 13063: 13048: 13044: 13026: 13010: 12993: 12973: 12969: 12966: 12950: 12920: 12746: 12741: 12733: 12728: 12726: 12511: 12494: 12441: 12436: 12434: 12424: 12419: 12415: 12413: 12325: 12320: 12319: 12283: 12278: 12276: 12273: 12237: 12233: 12227: 12220: 12216: 12214: 12160: 12154: 12123: 12055: 12048: 12041: 12037: 12035: 11982: 11977: 11975: 11887: 11884: 11838: 11834: 11830: 11828: 11819: 11815: 11811: 11807: 11805: 11662: 11659: 11567: 11469: 11393: 11273: 11242: 10939: 10853: 10703: 9631: 9621: 9617: 9613: 9609: 9607: 9470: 9467: 9392: 9317: 9312: 9308: 9306: 9169: 9158: 9142: 9016: 9011: 9009: 8892: 8885: 8803: 8798: 8795: 8788: 8727: 8722: 8718: 8714: 8712: 8707: 8703: 8701: 8551: 7708: 7706: 7701: 7697: 7695: 7552: 7549: 7446: 7443: 7128: 6946: 6944: 6941: 6931: 6927: 6926:> 0. For 6923: 6919: 6915: 6550: 6546: 6542: 6538: 6407: 6405: 6394: 6389: 6254:interaction 6251: 6185:interaction 6182: 6114: 6101:Sznajd model 6096: 6089: 6085: 6074: 6071:Markov chain 6068: 6055: 6048: 6046: 5967: 5961: 5806: 5801: 5797: 5793: 5792:(ν, μ) > 5789: 5782: 5775: 5773: 5662: 5284: 5275: 5271: 5267: 5263: 5259: 5257: 5254: 5246: 5237: 5233: 5231: 5226: 5221: 5214: 5207: 5205: 5200: 5188: 5186: 5172: 5168: 5165: 5159: 5061: 4946: 4937: 4935: 4929: 4923: 4919: 4914: 4906: 4901: 4890: 4876: 4861: 4843: 4840: 4716: 4712: 4708: 4704: 4700: 4696: 4694: 4690: 4626: 4624: 4619: 4561: 4557: 4555: 4550: 4547: 4543: 4534: 4530: 4523: 4519: 4510: 4442: 4437: 4435: 4427: 4424: 4412: 4408: 4392: 4382: 4362:Once modern 4361: 4346: 4334: 4318: 4304: 4186: 4083: 4015: 3947: 3876: 3794: 3737: 3720:A number of 3719: 3675: 3670: 3664: 3592: 3577: 3569: 3456: 3452: 3447: 3441: 3432: 3430: 3420: 3415: 3411: 3407: 3403: 3401: 3379: 3372: 3368: 3359: 3242: 3010: 2701: 2698: 2596: 2485: 2395: 2124: 2026: 2023: 2008: 1913: 1908: 1904: 1900: 1893: 1889: 1887: 1882: 1880: 1786: 1781: 1777: 1775: 1728: 1689: 1648: 1644: 1642: 1638: 1634: 1627: 1500: 1496: 1465: 1401: 1378: 1300: 1273: 1207: 1154: 1072: 1034: 1013:is given by 939: 812: 782: 752: 693: 692:there is an 659: 600: 475:) forming a 450: 431: 427:applications 422:Barth (1981) 415: 395:Ising (1925) 384: 344:Wilhelm Lenz 335: 331: 329: 181: 150: 142: 134: 126: 118: 25769:Spin models 25705:Ruin theory 25643:Disciplines 25515:Itô's lemma 25290:Predictable 24965:Percolation 24948:Potts model 24943:Ising model 24907:White noise 24865:Differences 24727:Itô process 24667:Cox process 24563:Loop-erased 24558:Random walk 24375:NP-complete 24182:Yang, C. N. 23939:Kleinert, H 23907:Kleinert, H 23598:1408.2487v3 22358:(11): e80. 21779:Spin models 21769:Potts model 21699:ANNNI model 20081:)/2, where 19710:Lattice gas 16543:free energy 16271:proper time 15030:valid when 14779:catastrophe 13069:Local field 13041:NP-complete 13039:lattice is 13001:Monte Carlo 12923:Yang (1952) 9975:, to obtain 8791:coin tosses 7129:This gives 6555:free energy 6383:free energy 6105:voter model 5274:(μ, ν) = 1/ 5195:is met. In 4949:Hamiltonian 4898:Definitions 4629:divided by 4325:Potts model 3698:Yang (1952) 3690:free energy 3582:between an 2443:of the cut 2019:maximum cut 2004:maximum cut 694:interaction 437:maximum cut 340:Ernst Ising 332:Ising model 275:von Neumann 84:Fermi–Dirac 25763:Categories 25715:Statistics 25495:Filtration 25396:Kolmogorov 25380:Blumenthal 25305:Stationary 25245:Continuous 25233:Properties 25118:Hull–White 24860:Martingale 24747:Local time 24635:Fractional 24613:pure birth 24055:2009-10-25 23609:References 23539:J-S Wang, 23380:2212.11279 23016:1502.02033 23009:(6): 174. 22848:1501.01776 22803:2210.13482 22749:1503.02011 22689:(7): 055. 22653:2020-06-26 22555:2009-10-25 22312:2022-08-09 22124:1502.03050 21998:(3): 477. 21976:2013-04-21 21749:J. C. Ward 20439:where the 20091:chemotaxis 19659:For small 18266:mean field 14836:averages: 13073:The field 11814:, and the 10814:is small. 5193:ergodicity 4337:Democritus 3375:is also 1? 1462:Discussion 548:such that 447:Definition 239:Scientists 200:Potentials 25627:Classical 24640:Geometric 24630:Excursion 24379:SIAM News 24343:1405.4497 24037:0022-2488 23978:: 613–615 23823:122157319 23678:EMS Press 23666:(2001) , 23664:K. Binder 23545:Physica A 23268:0027-8424 23219:206523521 23203:0036-8075 23102:SIAM News 23041:1029-8479 22988:119627708 22963:1403.4545 22909:1203.6064 22896:Phys. Rev 22820:253107625 22742:(8): 22. 22721:119226610 22696:1304.4110 22629:0002-9505 22510:122722140 22459:122117175 22307:118355454 22282:1405.4497 22198:1432-2064 22157:119330137 22149:1432-0916 22084:1432-0916 22020:122630492 21963:119627708 21938:1403.4545 21895:0030-364X 21821:Footnotes 21814:Z N model 21794:t-J model 21688:physics. 21647:ϵ 21627:γ 21607:ϵ 21587:γ 21567:ϵ 21547:γ 21510:− 21505:γ 21497:⁡ 21491:− 21477:γ 21469:⁡ 21448:ϵ 21379:± 21353:σ 21290:. (Note: 21273:β 21270:− 21250:∑ 21048:∑ 21014:⁡ 20985:− 20965:⟩ 20942:⟨ 20909:γ 20888:ϵ 20868:ϵ 20865:β 20862:≡ 20829:≡ 20826:τ 20803:⁡ 20797:≡ 20774:γ 20741:τ 20725:⁡ 20711:γ 20685:∑ 20669:γ 20655:− 20652:γ 20646:γ 20631:⁡ 20622:⁡ 20604:γ 20596:γ 20581:⁡ 20569:β 20566:− 20472:melt pond 20386:∑ 20372:− 20277:∑ 20273:− 20228:∑ 20212:− 20159:∑ 20155:− 20129:in 1972. 20042:μ 20039:− 20021:∑ 20007:− 19979:⟩ 19967:⟨ 19963:∑ 19949:− 19852:μ 19843:∑ 19808:⟩ 19796:⟨ 19792:∑ 19778:− 19679:Magnetism 19627:⁡ 19618:⁡ 19549:⁡ 19540:⁡ 19528:Δ 19491:− 19483:− 19461:⁡ 19452:⁡ 19406:− 19398:− 19376:⁡ 19349:− 19344:− 19315:− 19307:− 19164:… 19102:∑ 19054:∑ 18987:ε 18983:∝ 18949:ε 18945:∝ 18918:Whatever 18904:⋯ 18876:ε 18852:∂ 18841:β 18835:∂ 18796:universal 18763:ε 18696:ε 18681:− 18672:ε 18660:≈ 18648:β 18642:⁡ 18552:β 18546:⁡ 18526:β 18520:− 18496:β 18490:− 18482:− 18467:− 18406:β 18400:− 18347:β 18327:− 18219:ε 18214:− 18153:ε 18145:λ 18112:λ 18109:− 18077:λ 18068:− 18065:ε 18053:λ 18049:λ 18013:τ 18001:τ 17955:τ 17949:∫ 17937:− 17916:changed. 17868:π 17843:Λ 17829:Λ 17809:π 17785:Λ 17775:π 17706:π 17677:Λ 17640:Λ 17636:∫ 17587:λ 17578:− 17563:λ 17559:λ 17532:λ 17523:− 17456:λ 17446:− 17440:λ 17437:δ 17398:λ 17388:− 17332:∫ 17316:λ 17306:− 17300:λ 17297:δ 17278:anomalous 17241:λ 17232:− 17215:δ 17131:Λ 17074:∫ 17068:− 17042:∫ 16971:π 16942:Λ 16905:Λ 16901:∫ 16878:δ 16805:λ 16800:− 16776:∇ 16643:λ 16597:∇ 16561:∫ 16545:function 16463:− 16440:− 16430:≈ 16373:− 16329:− 16316:τ 16310:− 16230:− 16206:τ 16197:− 16181:− 16173:τ 16167:∫ 16114:τ 16108:− 16102:τ 16083:− 16061:τ 16058:π 16040:τ 16034:∫ 15910:− 15872:− 15856:→ 15829:∇ 15770:− 15703:− 15613:− 15602:∫ 15549:⋅ 15546:∇ 15519:∇ 15416:∇ 15407:→ 15390:⟩ 15363:⟨ 15338:∇ 15334:− 15241:… 15162:∑ 15146:⟩ 15121:⋯ 15080:⟨ 14988:− 14953:π 14933:∫ 14921:− 14906:⟩ 14879:⟨ 14876:∝ 14873:⟩ 14846:⟨ 14735:λ 14699:λ 14669:∂ 14665:∂ 14589:⟩ 14567:⟨ 14564:− 14514:⟩ 14492:⟨ 14477:⟩ 14454:⟨ 14451:− 14307:∇ 14285:λ 14248:∫ 14143:∫ 14047:λ 14026:∇ 13981:∫ 13978:− 13963:∫ 13908:λ 13887:∇ 13842:∫ 13839:− 13824:∫ 13800:⋯ 13742:λ 13721:∇ 13676:∫ 13673:− 13658:∫ 13649:⟩ 13627:⋯ 13586:⟨ 13541:β 13538:− 13510:λ 13489:∇ 13444:∫ 13441:− 13433:∝ 13404:is lost. 13358:∂ 13344:∂ 13275:⋯ 13259:λ 13234:∂ 13200:∑ 13162:∫ 13153:β 13037:nonplanar 13027:In 2000, 12831:− 12812:β 12806:⁡ 12790:β 12784:⁡ 12772:− 12695:† 12691:ψ 12676:† 12672:ψ 12664:∗ 12641:ψ 12631:ψ 12616:− 12609:ψ 12596:† 12592:ψ 12584:∗ 12561:ψ 12548:† 12544:ψ 12521:σ 12471:ψ 12465:† 12456:ψ 12449:∑ 12383:σ 12368:σ 12347:σ 12334:∑ 12292:σ 12246:σ 12140:⟩ 12110:⟩ 12081:∑ 12074:⟩ 11943:∫ 11922:Δ 11866:Δ 11833:and time 11700:∑ 11689:⁡ 11671:∑ 11541:⋯ 11507:⁡ 11332:⁡ 11284:⁡ 11214:θ 11207:⁡ 11188:β 11179:⁡ 11173:− 11161:θ 11154:⁡ 11135:β 11126:⁡ 11120:− 11104:β 11095:⁡ 11076:β 11067:⁡ 11058:⁡ 11046:θ 11037:π 11025:∫ 11015:θ 11006:π 10994:∫ 10981:π 10962:⁡ 10950:β 10947:− 10779:β 10750:⁡ 10744:≈ 10738:β 10712:β 10683:β 10674:⁡ 10668:⁡ 10642:β 10614:β 10572:⁡ 10556:⁡ 10534:β 10476:⁡ 10418:⁡ 10402:⁡ 10364:⁡ 10319:⁡ 10313:⁡ 10269:⁡ 10223:σ 10210:σ 10197:⁡ 10177:σ 10167:σ 10130:⋯ 10112:σ 10099:σ 10086:⁡ 10074:⋅ 10056:σ 10043:σ 10030:⁡ 10016:σ 10012:∑ 9996:β 9963:⋯ 9951:σ 9938:σ 9914:β 9888:⋯ 9874:σ 9861:σ 9848:σ 9825:σ 9812:σ 9799:σ 9785:σ 9781:∑ 9765:β 9703:β 9667:β 9646:β 9570:∑ 9563:− 9516:… 9500:∑ 9493:− 9484:σ 9401:σ 9344:∏ 9326:σ 9279:σ 9269:∑ 9262:− 9242:σ 9227:σ 9215:… 9199:∑ 9192:− 9183:σ 9143:which is 9113:β 9107:− 9087:β 9081:− 9060:⁡ 9036:⁡ 9024:β 8947:∏ 8922:∑ 8907:∑ 8859:− 8845:− 8837:∝ 8834:⟩ 8811:⟨ 8767:β 8761:− 8744:− 8664:β 8658:− 8634:β 8631:⁡ 8617:β 8598:β 8595:⁡ 8584:β 8567:λ 8554:, while λ 8508:λ 8498:λ 8465:λ 8447:λ 8429:λ 8404:⁡ 8392:β 8347:− 8338:β 8335:− 8320:β 8317:− 8300:β 8297:− 8270:β 8225:σ 8212:β 8195:σ 8191:σ 8185:β 8175:σ 8163:β 8143:σ 8136:σ 8105:σ 8098:σ 8062:σ 8049:σ 8040:⋯ 8029:σ 8016:σ 7999:σ 7986:σ 7969:σ 7962:… 7950:σ 7945:∑ 7930:σ 7920:σ 7913:β 7897:σ 7890:β 7882:⋯ 7871:σ 7861:σ 7854:β 7838:σ 7831:β 7815:σ 7805:σ 7798:β 7782:σ 7775:β 7759:σ 7752:… 7740:σ 7735:∑ 7725:β 7661:β 7658:− 7642:β 7627:β 7624:− 7616:− 7608:β 7589:⟩ 7574:σ 7564:σ 7560:⟨ 7523:β 7520:− 7504:β 7491:⁡ 7483:β 7475:− 7460:β 7422:− 7405:β 7402:− 7386:β 7355:σ 7348:β 7329:σ 7324:∑ 7303:∏ 7285:σ 7276:− 7269:σ 7262:β 7254:⋯ 7243:σ 7233:σ 7226:β 7210:σ 7200:σ 7193:β 7177:σ 7170:… 7158:σ 7153:∑ 7143:β 7112:≥ 7097:− 7090:σ 7080:σ 7064:σ 7029:σ 7020:− 7013:σ 7006:⋯ 6994:σ 6984:σ 6972:− 6963:σ 6889:− 6875:β 6866:− 6855:β 6843:⟩ 6834:σ 6830:⟨ 6827:⟩ 6818:σ 6814:⟨ 6811:− 6808:⟩ 6799:σ 6789:σ 6785:⟨ 6750:β 6744:− 6720:β 6717:⁡ 6703:β 6684:β 6681:⁡ 6670:β 6657:⁡ 6649:β 6641:− 6632:β 6623:⁡ 6611:β 6600:∞ 6597:→ 6586:− 6571:β 6516:σ 6506:∑ 6499:− 6484:σ 6474:σ 6465:− 6456:… 6440:∑ 6433:− 6424:σ 6367:α 6364:− 6351:− 6340:∼ 6299:− 6286:− 6275:∼ 6233:α 6230:− 6217:− 6206:∼ 6163:α 6160:− 6147:− 6136:∼ 6019:μ 6011:− 6006:ν 5995:β 5992:− 5937:otherwise 5913:μ 5905:− 5900:ν 5875:μ 5867:− 5862:ν 5851:β 5848:− 5826:ν 5820:μ 5750:μ 5742:− 5737:ν 5726:β 5723:− 5706:μ 5700:ν 5686:ν 5680:μ 5639:μ 5631:− 5626:ν 5615:β 5612:− 5591:μ 5580:β 5577:− 5550:ν 5539:β 5536:− 5506:μ 5498:β 5485:ν 5477:β 5457:μ 5451:ν 5437:ν 5431:μ 5410:μ 5404:ν 5392:μ 5386:ν 5372:ν 5366:μ 5354:ν 5348:μ 5327:μ 5321:ν 5307:ν 5301:μ 5126:σ 5116:σ 5110:⟩ 5098:⟨ 5094:∑ 5087:− 5078:σ 5039:σ 5029:∑ 5022:− 5013:σ 5003:σ 4997:⟩ 4985:⟨ 4981:∑ 4974:− 4965:σ 4819:β 4813:− 4776:∑ 4667:ε 4638:∑ 4488:σ 4467:∑ 4288:⟩ 4279:σ 4269:σ 4265:⟨ 4262:⟩ 4253:σ 4243:σ 4239:⟨ 4231:∈ 4224:∑ 4220:≤ 4217:⟩ 4208:σ 4198:σ 4194:⟨ 4037:⟩ 4028:σ 4024:⟨ 4013:results. 3998:≥ 3995:⟩ 3986:σ 3982:⟨ 3962:∅ 3933:⟩ 3924:σ 3915:∈ 3908:∏ 3904:⟨ 3898:⟩ 3889:σ 3885:⟨ 3860:⟩ 3851:σ 3847:⟨ 3844:⟩ 3835:σ 3831:⟨ 3828:≥ 3825:⟩ 3816:σ 3806:σ 3802:⟨ 3774:σ 3747:σ 3642:β 3633:≥ 3628:β 3624:⟩ 3614:σ 3604:σ 3600:⟨ 3540:− 3526:β 3517:− 3509:⁡ 3500:≤ 3495:β 3491:⟩ 3481:σ 3471:σ 3467:⟨ 3356:Questions 3317:δ 3306:− 3276:∈ 3266:∑ 3256:σ 3207:δ 3169:− 3102:δ 3099:∈ 3089:∑ 3068:σ 3045:σ 3019:σ 2959:δ 2956:∈ 2946:∑ 2909:∈ 2899:∑ 2895:− 2851:δ 2848:∈ 2838:∑ 2811:− 2797:∈ 2787:∑ 2783:− 2746:∈ 2736:∑ 2732:− 2719:σ 2549:δ 2546:∈ 2536:∑ 2498:δ 2481:bipartite 2451:δ 2410:δ 2380:− 2328:σ 2292:δ 2270:− 2218:σ 2210:into two 2169:σ 2146:± 2134:σ 2104:σ 2094:σ 2063:∈ 2053:∑ 2049:− 2040:σ 1975:σ 1965:σ 1959:⟩ 1947:⟨ 1943:∑ 1936:− 1927:σ 1858:σ 1848:σ 1829:⟩ 1817:⟨ 1813:∑ 1809:− 1800:σ 1480:σ 1446:σ 1423:σ 1415:β 1362:σ 1354:β 1343:σ 1332:σ 1328:∑ 1319:β 1315:⟩ 1308:⟨ 1255:σ 1246:β 1243:− 1233:σ 1229:∑ 1220:β 1163:β 1134:β 1121:σ 1112:β 1109:− 1093:σ 1085:β 1055:≥ 1052:β 1021:μ 957:⟩ 948:⟨ 917:σ 897:∑ 893:μ 890:− 881:σ 871:σ 852:⟩ 843:⟨ 839:∑ 835:− 826:σ 794:σ 739:Λ 736:∈ 680:Λ 677:∈ 643:Λ 640:∈ 623:σ 612:σ 572:− 566:∈ 557:σ 530:σ 509:Λ 506:∈ 459:Λ 270:Ehrenfest 250:Boltzmann 130:Canonical 25748:Category 25632:Abstract 25166:Bühlmann 24772:Compound 24406:Archived 24001:(1963), 23968:Lenz, W. 23909:(1989), 23795:Z. Phys. 23775:(1987), 23738:(1999), 23572:citation 23526:16625187 23424:17870175 23286:27091987 23211:20133571 23159:18854281 23078:July 19, 23049:35625559 22934:39692193 22881:14538564 22873:26871060 22774:53710971 22382:17526526 22299:25314429 22206:55391558 22092:17543488 21809:XY model 21692:See also 20119:learning 20111:Hopfield 19608:′ 19252:larger. 18369:so that 16841:< (1+ 16130:This is 14834:Gaussian 11538:2.269185 10645:′ 10617:′ 10504:′ 10293:′ 10162:′ 9145:analytic 8698:Comments 8690:| < λ 8228:′ 8198:′ 8146:′ 8108:′ 7363:′ 7337:′ 7072:′ 6918:(β) and 5891:if 5270:states: 5156:Overview 4400:analytic 265:Einstein 212:Enthalpy 177:Einstein 25255:Ergodic 25143:Vašíček 24985:Poisson 24645:Meander 24310:Bibcode 24302:Physica 24282:Physica 24258:Bibcode 24250:Physica 24230:Bibcode 24214:0051740 24194:Bibcode 24175:7944336 24167:2114521 24095:0010315 24075:Bibcode 24045:0148406 24017:Bibcode 23932:Vol. II 23803:Bibcode 23766:1707309 23694:Bibcode 23657:0690578 23629:8231704 23553:Bibcode 23541:W Selke 23517:1785327 23496:Bibcode 23447:Bibcode 23404:Bibcode 23341:6953413 23309:Bibcode 23277:4983820 23246:Bibcode 23183:Bibcode 23175:Science 23139:Bibcode 23021:Bibcode 22968:Bibcode 22914:Bibcode 22853:Bibcode 22754:Bibcode 22701:Bibcode 22545:0690578 22490:Bibcode 22439:Bibcode 22373:1920246 22129:Bibcode 22064:Bibcode 22000:Bibcode 21943:Bibcode 20466:Sea ice 20103:neurons 19009:Onsager 18802:in the 18183:= ε/3. 16742:), the 16541:is the 13407:Since β 12163:steps: 8911:configs 8546:where λ 6043:Repeat. 5788:, then 4341:atomism 4335:One of 3680: ( 3584:ordered 1907:, 1499:, 1274:is the 751:has an 401: ( 389: ( 368:lattice 346:, is a 245:Maxwell 25595:Tanaka 25280:Mixing 25275:Markov 25148:Wilkie 25113:Ho–Lee 25108:Heston 24880:Super- 24625:Bridge 24573:Biased 24212: 24173: 24165: 24155: 24124: 24093: 24043: 24035: 23990: 23960:online 23953: 23928:Vol. I 23921: 23899: 23879: 23859: 23841: 23821: 23783: 23764: 23754: 23725: 23655: 23645: 23627: 23524: 23514: 23478:Nature 23422: 23339: 23332:346238 23329: 23284: 23274: 23266: 23217: 23209: 23201: 23157: 23047: 23039: 22986: 22932: 22879: 22871: 22818: 22772: 22719: 22627: 22586: 22543: 22533: 22508: 22457: 22408: 22380: 22370: 22332:GitHub 22305: 22297: 22246: 22204: 22196: 22155: 22147: 22090: 22082: 22018: 21961: 21903:170992 21901: 21893: 21109:where 20880:(with 20766:where 20315:where 19696:Larmor 19692:Ampère 19428:where 18789:Landau 16408:) is: 15572:since 15293:where 14813:. For 14349:Since 14342:where 13385:define 13053:planar 12864:where 12414:where 11976:where 10633:where 9900:where 8686:and |λ 6914:where 6537:where 5104: 4991: 3877:where 3586:and a 2597:where 2394:. The 2011:vertex 1953: 1823: 1155:where 783:energy 280:Tolman 166:Models 25448:Tools 25224:M/M/c 25219:M/M/1 25214:M/G/1 25204:Fluid 24870:Local 24338:arXiv 24171:S2CID 24137:(PDF) 24112:: 261 23819:S2CID 23593:arXiv 23486:arXiv 23420:S2CID 23375:arXiv 23215:S2CID 23155:S2CID 23129:arXiv 23098:(PDF) 23045:S2CID 23011:arXiv 22984:S2CID 22958:arXiv 22930:S2CID 22904:arXiv 22877:S2CID 22843:arXiv 22816:S2CID 22798:arXiv 22770:S2CID 22744:arXiv 22717:S2CID 22691:arXiv 22506:S2CID 22455:S2CID 22303:S2CID 22277:arXiv 22202:S2CID 22153:S2CID 22119:arXiv 22088:S2CID 22016:S2CID 21970:(PDF) 21959:S2CID 21933:arXiv 21921:(PDF) 21899:JSTOR 18991:0.125 18953:0.308 18617:For β 18275:is ±2 15810:When 14784:When 13312:' 12927:Szegő 11812:space 10704:When 6938:Proof 5781:> 4124:from 3948:With 3710:Szegő 2001:graph 1041:with 473:graph 434:graph 290:Fermi 285:Debye 260:Gibbs 187:Potts 182:Ising 172:Debye 25400:Lévy 25199:Bulk 25083:Chen 24875:Sub- 24833:Both 24306:117A 24286:119A 24153:ISBN 24122:ISBN 24033:ISSN 23988:ISBN 23951:ISBN 23930:and 23919:ISBN 23897:ISBN 23877:ISBN 23857:ISBN 23839:ISBN 23835:CNRS 23781:ISBN 23752:ISBN 23723:ISBN 23643:ISBN 23625:OCLC 23578:link 23522:PMID 23337:PMID 23282:PMID 23264:ISSN 23207:PMID 23199:ISSN 23108:(6). 23080:2015 23037:ISSN 23007:2015 22869:PMID 22740:2015 22687:1307 22683:JHEP 22625:ISSN 22584:ISBN 22531:ISBN 22406:ISBN 22378:PMID 22295:PMID 22244:ISBN 22194:ISSN 22145:ISSN 22080:ISSN 21891:ISSN 21859:See 21832:See 21011:cosh 20800:tanh 20628:cosh 20553:and 19624:cosh 19546:cosh 19458:cosh 19373:cosh 18934:is: 18922:and 18639:tanh 18543:tanh 17659:< 17643:< 16924:< 16908:< 16758:is 2 16512:The 15978:For 15724:and 14765:For 12978:and 12933:for 12891:and 12803:sinh 12781:sinh 12418:and 12047:and 11820:time 11329:sinh 11281:sinh 11176:sinh 11123:sinh 11092:cosh 11064:cosh 10910:and 10671:cosh 10569:cosh 10415:cosh 10316:cosh 10266:cosh 10194:cosh 10083:cosh 10027:cosh 9612:and 8886:The 8628:sinh 8592:cosh 6714:sinh 6678:cosh 6541:and 5918:> 5232:Let 5160:The 4947:The 4739:< 4351:and 3765:and 3704:for 3688:and 3682:1944 3648:> 2396:size 2367:and 2320:the 1712:< 1673:> 1569:< 1527:> 989:and 403:1944 391:1920 342:and 334:(or 330:The 255:Bose 24980:Cox 24377:", 24348:doi 24318:doi 24290:doi 24266:doi 24254:99A 24238:doi 24202:doi 24145:doi 24083:doi 24025:doi 23811:doi 23744:doi 23702:doi 23561:doi 23549:164 23512:PMC 23504:doi 23482:440 23455:doi 23443:108 23412:doi 23400:106 23327:PMC 23317:doi 23272:PMC 23254:doi 23242:113 23191:doi 23179:327 23147:doi 23029:doi 22976:doi 22954:157 22922:doi 22900:D86 22861:doi 22808:doi 22762:doi 22709:doi 22617:doi 22576:doi 22498:doi 22447:doi 22368:PMC 22360:doi 22287:doi 22184:doi 22180:153 22137:doi 22115:343 22072:doi 22008:doi 21951:doi 21929:157 21883:doi 21659:.” 21344:or 20930:.) 20517:or 20470:2D 20447:i,k 20073:= ( 19761:). 19449:log 18277:JNH 17272:So 17182:to 16254:in 13394:= δ 13049:NP, 13031:of 13019:). 12929:'s 12157:sum 11837:+ Δ 11686:exp 11204:cos 11151:cos 10839:Wu. 10247:as 9057:log 9033:log 8243:or 7707:If 6945:If 6590:lim 6557:is 5774:If 3945:. 3506:exp 3459:|: 3378:If 3060:in 2479:to 354:in 350:of 151:NPT 143:NPH 135:µVT 127:NVT 119:NVE 25765:: 25398:, 25394:, 25390:, 25386:, 25382:, 24462:- 24447:, 24346:, 24334:90 24332:, 24316:, 24304:, 24284:, 24264:, 24252:, 24236:, 24226:38 24224:, 24210:MR 24208:, 24200:, 24190:85 24169:, 24163:MR 24161:, 24151:, 24139:, 24108:, 24091:MR 24089:, 24081:, 24071:65 24041:MR 24039:, 24031:, 24023:, 24011:, 24005:, 23976:21 23974:, 23945:, 23817:, 23809:, 23799:31 23797:, 23762:MR 23760:, 23750:, 23700:. 23690:39 23688:. 23676:, 23670:, 23653:MR 23651:, 23574:}} 23570:{{ 23559:, 23547:, 23520:, 23510:, 23502:, 23494:, 23480:, 23476:, 23453:, 23441:, 23418:, 23410:, 23398:, 23358:. 23335:, 23325:, 23315:, 23305:79 23303:, 23280:. 23270:. 23262:. 23252:. 23240:. 23236:. 23213:. 23205:. 23197:. 23189:. 23177:. 23153:. 23145:. 23137:. 23125:58 23123:. 23106:33 23104:. 23100:. 23066:. 23043:. 23035:. 23027:. 23019:. 23005:. 22982:. 22974:. 22966:. 22952:. 22928:. 22920:. 22912:. 22898:. 22875:. 22867:. 22859:. 22851:. 22839:93 22837:. 22814:. 22806:. 22794:13 22792:. 22768:. 22760:. 22752:. 22738:. 22715:. 22707:. 22699:. 22685:. 22646:. 22623:. 22613:46 22611:. 22607:. 22582:. 22541:MR 22539:, 22504:. 22496:. 22486:84 22484:. 22480:. 22453:. 22445:. 22435:12 22433:. 22429:. 22390:^ 22376:. 22366:. 22356:35 22354:. 22350:. 22329:. 22301:. 22293:. 22285:. 22273:90 22271:. 22267:. 22214:^ 22200:. 22192:. 22178:. 22174:. 22151:. 22143:. 22135:. 22127:. 22113:. 22109:. 22086:. 22078:. 22070:. 22060:77 22058:. 22054:. 22028:^ 22014:. 22006:. 21996:32 21994:. 21957:. 21949:. 21941:. 21927:. 21923:. 21897:. 21889:. 21879:36 21877:. 21494:ln 21466:ln 20926:, 20851:, 20818:, 20722:ln 20619:ln 20578:ln 20526:10 20499:10 20075:zJ 19924:2. 19759:βμ 19755:μN 19727:JB 19665:J' 19615:ln 19537:ln 19185:ij 19011:. 18599:. 18447:. 18299:. 18297:JH 17205:: 16868:: 16760:bt 16488:. 16397:. 16142:. 15732:. 14653:, 14390:dx 13397:ij 13391:ij 13090:. 13047:≠ 12281:: 12223:. 11987:: 11841:: 11652:. 11595:, 11504:ln 11380:1. 11055:ln 10959:ln 10747:ln 10701:. 10665:ln 10553:ln 10473:ln 10399:ln 10361:ln 10310:ln 9911::= 9624:. 9620:= 8704:JL 8401:Tr 7488:ln 7115:2. 6654:ln 6620:ln 5278:. 5229:. 4944:. 4719:: 4076:. 3673:. 3651:0. 3416:ij 3410:, 2696:. 2122:. 1899:= 1896:ij 1890:ij 1458:. 1399:. 1070:: 603:, 443:. 429:. 413:. 382:. 25402:) 25378:( 24499:e 24492:t 24485:v 24350:: 24340:: 24320:: 24312:: 24292:: 24268:: 24260:: 24240:: 24232:: 24204:: 24196:: 24147:: 24128:. 24110:6 24085:: 24077:: 24027:: 24019:: 24013:4 23962:) 23934:) 23903:. 23885:. 23813:: 23805:: 23746:: 23708:. 23704:: 23696:: 23601:. 23595:: 23580:) 23563:: 23555:: 23528:. 23506:: 23498:: 23488:: 23461:. 23457:: 23449:: 23426:. 23414:: 23406:: 23383:. 23377:: 23360:C 23343:. 23319:: 23311:: 23288:. 23256:: 23248:: 23221:. 23193:: 23185:: 23161:. 23149:: 23141:: 23131:: 23082:. 23051:. 23031:: 23023:: 23013:: 22990:. 22978:: 22970:: 22960:: 22936:. 22924:: 22916:: 22906:: 22883:. 22863:: 22855:: 22845:: 22822:. 22810:: 22800:: 22776:. 22764:: 22756:: 22746:: 22723:. 22711:: 22703:: 22693:: 22670:. 22656:. 22631:. 22619:: 22592:. 22578:: 22512:. 22500:: 22492:: 22476:r 22461:. 22449:: 22441:: 22414:. 22384:. 22362:: 22315:. 22289:: 22279:: 22252:. 22208:. 22186:: 22159:. 22139:: 22131:: 22121:: 22094:. 22074:: 22066:: 22022:. 22010:: 22002:: 21979:. 21953:: 21945:: 21935:: 21905:. 21885:: 21775:) 21525:. 21519:] 21516:) 21513:1 21500:( 21488:) 21485:1 21482:+ 21472:( 21463:[ 21458:B 21454:k 21445:2 21439:= 21434:C 21430:T 21407:C 21403:T 21382:1 21357:i 21330:i 21326:S 21303:i 21299:s 21276:H 21266:e 21260:} 21257:s 21254:{ 21226:N 21222:Z 21198:N 21194:2 21171:l 21167:t 21163:) 21160:l 21157:( 21152:n 21149:m 21145:g 21122:b 21118:N 21095:l 21091:t 21087:) 21084:l 21081:( 21076:n 21073:m 21069:g 21063:z 21058:1 21055:= 21052:l 21042:N 21038:2 21030:b 21026:N 21021:] 21017:J 21008:[ 21005:) 21002:T 20999:, 20996:0 20993:( 20988:1 20978:N 20974:Z 20968:= 20960:n 20956:s 20950:m 20946:s 20922:( 20859:J 20837:2 20833:t 20806:J 20794:t 20744:) 20738:( 20733:i 20729:J 20715:i 20707:1 20700:z 20695:2 20692:= 20689:i 20678:) 20675:1 20672:+ 20666:( 20661:) 20658:1 20649:( 20640:+ 20637:) 20634:J 20625:( 20613:) 20610:1 20607:+ 20601:( 20593:2 20587:+ 20584:2 20575:= 20572:f 20530:5 20503:4 20460:p 20456:p 20452:p 20445:J 20441:S 20427:, 20422:k 20418:S 20411:i 20407:S 20400:k 20397:, 20394:i 20390:J 20380:2 20377:1 20369:= 20366:H 20331:j 20328:i 20324:J 20301:i 20297:S 20291:i 20287:h 20281:i 20268:j 20264:S 20258:i 20254:S 20248:j 20245:i 20241:J 20235:j 20232:i 20221:2 20218:1 20209:= 20206:E 20183:i 20179:S 20173:i 20169:h 20163:i 20152:= 20149:E 20083:z 20079:μ 20071:h 20053:i 20049:S 20045:) 20036:J 20033:4 20030:( 20025:i 20015:2 20012:1 20002:j 19998:S 19992:i 19988:S 19984:J 19976:j 19973:, 19970:i 19957:2 19954:1 19946:= 19943:E 19920:/ 19916:) 19913:1 19910:+ 19905:i 19901:S 19897:( 19894:= 19889:i 19885:B 19860:i 19856:B 19847:i 19839:+ 19834:j 19830:B 19824:i 19820:B 19816:J 19813:4 19805:j 19802:, 19799:i 19786:2 19783:1 19775:= 19772:E 19751:N 19739:j 19735:B 19731:i 19722:i 19720:B 19661:J 19645:. 19642:) 19639:] 19636:J 19633:4 19630:[ 19621:( 19612:= 19605:J 19601:3 19591:J 19587:J 19583:J 19567:. 19564:) 19561:] 19558:J 19555:4 19552:[ 19543:( 19534:= 19531:F 19505:. 19502:) 19499:] 19496:) 19487:N 19478:+ 19474:N 19470:( 19467:J 19464:[ 19455:( 19446:= 19443:F 19433:± 19430:N 19414:) 19411:] 19402:N 19393:+ 19389:N 19385:[ 19382:J 19379:( 19370:2 19367:= 19362:) 19357:+ 19353:N 19340:N 19336:( 19333:J 19329:e 19325:+ 19320:) 19311:N 19302:+ 19298:N 19294:( 19291:J 19287:e 19276:J 19269:t 19265:J 19261:J 19238:2 19222:j 19218:i 19214:j 19210:i 19206:j 19204:, 19202:i 19200:( 19194:j 19190:i 19183:J 19167:. 19159:l 19155:S 19149:k 19145:S 19139:j 19135:S 19129:i 19125:S 19119:l 19116:k 19113:j 19110:i 19106:J 19099:+ 19094:j 19090:S 19084:i 19080:S 19074:j 19071:i 19067:J 19061:j 19058:i 19050:= 19047:E 18980:H 18942:H 18928:F 18924:B 18920:A 18901:+ 18896:3 18892:H 18888:B 18885:+ 18882:H 18879:) 18873:A 18870:+ 18867:1 18864:( 18861:= 18855:h 18847:) 18844:F 18838:( 18829:= 18826:H 18816:H 18812:H 18808:H 18785:H 18781:H 18760:3 18755:= 18752:H 18742:H 18738:H 18720:3 18714:3 18710:H 18704:3 18700:) 18693:+ 18690:1 18687:( 18678:H 18675:) 18669:+ 18666:1 18663:( 18657:) 18654:H 18651:J 18645:( 18636:= 18633:H 18623:H 18619:J 18612:H 18608:H 18604:H 18597:N 18593:J 18585:H 18581:H 18577:J 18561:) 18558:H 18555:J 18549:( 18540:= 18532:H 18529:J 18523:2 18516:e 18512:+ 18509:1 18502:H 18499:J 18493:2 18486:e 18479:1 18473:= 18470:1 18464:p 18461:2 18458:= 18455:H 18445:H 18441:p 18437:p 18433:p 18412:H 18409:J 18403:2 18396:e 18392:+ 18389:1 18385:1 18380:= 18377:p 18353:H 18350:J 18344:2 18340:e 18336:= 18330:p 18324:1 18320:p 18308:p 18304:p 18293:J 18289:N 18285:N 18281:J 18273:H 18269:H 18228:) 18222:3 18211:1 18207:( 18200:2 18197:1 18181:B 18177:t 18160:B 18157:3 18148:= 18135:B 18115:B 18106:2 18103:= 18094:t 18090:t 18087:d 18074:B 18071:3 18062:= 18046:d 18010:t 18007:+ 17998:2 17992:2 17988:x 17981:e 17972:2 17969:d 17964:t 17960:1 17952:d 17946:= 17943:) 17940:y 17934:x 17931:( 17928:G 17905:t 17901:t 17897:t 17893:B 17872:2 17864:8 17860:1 17855:= 17847:4 17839:b 17835:1 17826:b 17817:4 17813:) 17806:2 17803:( 17799:1 17794:) 17789:3 17779:2 17771:2 17768:( 17765:= 17762:B 17752:b 17732:4 17728:k 17724:1 17714:4 17710:) 17703:2 17700:( 17695:k 17690:4 17686:d 17674:) 17671:b 17668:+ 17665:1 17662:( 17655:| 17651:k 17647:| 17632:= 17629:b 17626:B 17616:B 17596:b 17591:2 17584:B 17581:3 17572:= 17556:d 17546:b 17542:) 17536:2 17529:B 17520:2 17516:( 17512:= 17503:t 17499:t 17496:d 17465:b 17460:2 17452:B 17449:3 17443:= 17413:b 17408:2 17402:2 17394:3 17385:= 17377:2 17373:) 17369:t 17366:+ 17361:2 17357:k 17353:( 17349:1 17344:k 17341:d 17336:k 17326:2 17320:2 17312:3 17303:= 17285:H 17274:t 17258:t 17255:b 17251:) 17245:2 17238:B 17229:2 17225:( 17221:= 17218:t 17199:t 17195:t 17191:t 17184:t 17180:t 17176:H 17172:t 17168:t 17152:t 17149:b 17146:B 17143:+ 17140:b 17135:2 17127:A 17124:= 17118:) 17115:t 17112:+ 17107:2 17103:k 17099:( 17094:2 17090:k 17085:1 17080:k 17077:d 17071:t 17061:2 17057:k 17053:1 17048:k 17045:d 17012:) 17009:t 17006:+ 17001:2 16997:k 16993:( 16989:1 16979:4 16975:) 16968:2 16965:( 16960:k 16955:4 16951:d 16939:) 16936:b 16933:+ 16930:1 16927:( 16920:| 16916:k 16912:| 16897:H 16894:3 16891:= 16886:3 16882:H 16866:H 16862:H 16858:k 16854:H 16850:H 16843:b 16839:k 16823:. 16818:3 16814:H 16808:6 16797:= 16794:H 16791:t 16788:+ 16785:H 16780:2 16756:b 16752:t 16748:b 16744:t 16740:b 16733:k 16729:k 16721:H 16717:H 16704:H 16700:H 16693:Z 16667:] 16661:4 16657:H 16650:! 16647:4 16638:+ 16633:2 16629:H 16623:2 16620:t 16615:+ 16610:2 16605:| 16600:H 16593:| 16587:2 16584:Z 16578:[ 16574:x 16569:4 16565:d 16558:= 16555:F 16539:H 16527:ε 16507:H 16502:H 16498:H 16466:2 16460:d 16456:r 16450:r 16445:t 16436:e 16427:) 16424:r 16421:( 16418:G 16406:r 16404:( 16402:G 16383:r 16378:t 16369:e 16358:r 16340:2 16336:r 16332:t 16325:e 16321:= 16313:t 16306:e 16295:r 16293:( 16291:G 16283:t 16279:x 16275:x 16264:G 16260:k 16256:k 16252:t 16233:t 16225:2 16221:k 16216:1 16211:= 16203:) 16200:t 16192:2 16188:k 16184:( 16177:e 16170:d 16164:= 16161:) 16158:k 16155:( 16152:G 16140:k 16136:x 16132:G 16111:t 16099:4 16093:2 16089:x 16079:e 16071:d 16066:) 16055:2 16050:( 16045:1 16037:d 16031:= 16028:) 16025:x 16022:( 16019:G 15992:t 15980:r 15964:0 15961:= 15958:) 15955:r 15952:( 15949:G 15946:t 15943:+ 15939:) 15932:r 15929:d 15924:G 15921:d 15913:1 15907:d 15903:r 15898:( 15891:r 15888:d 15884:d 15875:1 15869:d 15865:r 15861:1 15853:0 15850:= 15847:G 15844:t 15841:+ 15838:G 15833:2 15816:H 15812:t 15804:r 15802:( 15800:G 15793:C 15773:2 15767:d 15763:r 15759:C 15754:= 15751:) 15748:r 15745:( 15742:G 15730:r 15726:G 15706:1 15700:d 15696:r 15692:C 15687:= 15684:E 15659:t 15656:n 15653:a 15650:t 15647:s 15644:n 15641:o 15638:c 15634:= 15629:r 15625:E 15621:S 15616:1 15610:d 15606:d 15590:d 15586:G 15582:E 15578:d 15574:G 15558:0 15555:= 15552:E 15522:G 15516:= 15513:E 15493:t 15486:H 15482:H 15462:0 15459:= 15452:) 15449:x 15446:( 15443:G 15440:t 15437:+ 15434:) 15431:x 15428:( 15425:G 15420:2 15400:0 15397:= 15387:) 15384:y 15381:( 15378:H 15375:) 15372:x 15369:( 15366:H 15359:) 15355:t 15352:+ 15347:2 15342:x 15330:( 15311:G 15299:G 15295:C 15279:) 15274:n 15271:j 15267:x 15263:, 15258:n 15255:i 15251:x 15247:( 15244:G 15238:) 15233:2 15230:j 15226:x 15222:, 15217:2 15214:i 15210:x 15206:( 15203:G 15200:) 15195:1 15192:j 15188:x 15184:, 15179:1 15176:i 15172:x 15168:( 15165:G 15157:n 15153:C 15149:= 15143:) 15138:n 15135:2 15131:x 15127:( 15124:S 15118:) 15113:2 15109:x 15105:( 15102:S 15099:) 15094:1 15090:x 15086:( 15083:S 15068:y 15064:x 15062:( 15060:G 15048:y 15044:x 15042:( 15040:G 15036:y 15032:x 15013:t 15010:+ 15005:2 15001:k 14994:) 14991:y 14985:x 14982:( 14979:k 14976:i 14972:e 14961:d 14957:) 14950:2 14947:( 14942:k 14939:d 14930:= 14927:) 14924:y 14918:x 14915:( 14912:G 14909:= 14903:) 14900:y 14897:( 14894:H 14891:) 14888:x 14885:( 14882:H 14870:) 14867:y 14864:( 14861:S 14858:) 14855:x 14852:( 14849:S 14826:H 14819:H 14815:d 14811:d 14807:H 14803:H 14799:d 14786:t 14775:t 14771:H 14767:t 14751:0 14748:= 14743:3 14739:H 14732:4 14729:+ 14726:H 14723:t 14720:2 14717:= 14713:) 14707:4 14703:H 14696:+ 14691:2 14687:H 14683:t 14679:( 14672:H 14651:t 14644:t 14640:t 14636:t 14632:λ 14628:t 14624:t 14620:t 14616:t 14598:2 14593:) 14584:2 14580:) 14576:x 14573:( 14570:H 14559:2 14555:) 14551:x 14548:( 14545:H 14541:( 14536:+ 14531:2 14527:) 14523:x 14520:( 14517:H 14509:2 14505:) 14501:x 14498:( 14495:H 14489:2 14486:+ 14481:2 14471:2 14467:) 14463:x 14460:( 14457:H 14448:= 14443:4 14439:) 14435:x 14432:( 14429:H 14417:H 14413:t 14409:H 14402:d 14398:d 14394:Z 14387:H 14383:d 14379:H 14375:H 14363:H 14359:H 14355:t 14351:t 14344:t 14328:, 14324:] 14318:2 14314:) 14310:H 14304:( 14301:Z 14298:+ 14293:4 14289:H 14282:+ 14277:2 14273:H 14269:t 14265:[ 14261:x 14256:d 14252:d 14245:= 14242:F 14230:A 14226:F 14222:A 14218:H 14214:A 14209:H 14201:A 14197:A 14190:H 14173:. 14168:2 14164:H 14160:A 14156:x 14151:d 14147:d 14140:= 14137:F 14125:H 14118:H 14114:F 14102:i 14094:H 14090:F 14086:H 14082:F 14078:H 14061:] 14055:4 14051:H 14044:+ 14039:2 14034:| 14029:H 14022:| 14018:Z 14015:+ 14010:2 14006:H 14002:A 13998:[ 13994:x 13989:d 13985:d 13974:e 13969:H 13966:D 13960:= 13957:Z 13949:: 13931:. 13922:] 13916:4 13912:H 13905:+ 13900:2 13895:| 13890:H 13883:| 13879:Z 13876:+ 13871:2 13867:H 13863:A 13859:[ 13855:x 13850:d 13846:d 13835:e 13830:H 13827:D 13819:) 13814:n 13810:x 13806:( 13803:H 13797:) 13792:2 13788:x 13784:( 13781:H 13778:) 13773:1 13769:x 13765:( 13762:H 13756:] 13750:4 13746:H 13739:+ 13734:2 13729:| 13724:H 13717:| 13713:Z 13710:+ 13705:2 13701:H 13697:A 13693:[ 13689:x 13684:d 13680:d 13669:e 13664:H 13661:D 13652:= 13646:) 13641:n 13637:x 13633:( 13630:H 13624:) 13619:2 13615:x 13611:( 13608:H 13605:) 13600:1 13596:x 13592:( 13589:H 13574:H 13558:. 13553:] 13550:H 13547:[ 13544:F 13534:e 13530:= 13524:] 13518:4 13514:H 13507:+ 13502:2 13497:| 13492:H 13485:| 13481:Z 13478:+ 13473:2 13469:H 13465:A 13461:[ 13457:x 13452:d 13448:d 13437:e 13430:) 13427:H 13424:( 13421:P 13409:F 13389:Z 13367:H 13362:j 13353:H 13348:i 13337:j 13334:i 13330:Z 13316:H 13308:i 13306:Z 13301:i 13299:Z 13283:. 13279:] 13272:+ 13267:4 13263:H 13256:+ 13251:2 13247:) 13243:H 13238:i 13230:( 13225:i 13221:Z 13215:d 13210:1 13207:= 13204:i 13196:+ 13191:2 13187:H 13183:A 13179:[ 13175:x 13170:d 13166:d 13159:= 13156:F 13141:H 13134:H 13130:F 13126:H 13122:H 13115:H 13111:H 13107:H 13105:( 13103:F 13099:H 13095:H 13088:H 13084:H 13080:H 13075:H 13045:P 12904:2 12900:J 12877:1 12873:J 12849:8 12846:1 12840:) 12834:2 12826:] 12820:2 12816:J 12809:2 12798:1 12794:J 12787:2 12777:[ 12769:1 12765:( 12760:= 12757:M 12747:M 12729:T 12713:. 12708:1 12705:+ 12702:i 12683:i 12660:C 12656:+ 12651:1 12648:+ 12645:i 12635:i 12627:C 12624:+ 12619:1 12613:i 12603:i 12580:D 12576:+ 12571:1 12568:+ 12565:i 12555:i 12538:D 12535:= 12530:x 12525:i 12480:. 12475:i 12460:i 12452:C 12437:T 12425:i 12420:B 12416:A 12398:z 12393:1 12390:+ 12387:i 12377:z 12372:i 12364:B 12361:+ 12356:x 12351:i 12343:A 12338:i 12321:T 12306:. 12301:z 12296:i 12279:i 12260:. 12255:x 12250:i 12228:T 12221:Z 12217:T 12201:. 12198:) 12193:N 12189:T 12185:( 12181:r 12178:t 12174:= 12171:Z 12161:N 12137:S 12133:| 12107:S 12103:| 12099:) 12096:S 12093:( 12090:A 12085:S 12077:= 12071:A 12067:| 12056:T 12052:2 12049:C 12045:1 12042:C 12038:T 12022:. 12015:2 12011:C 12005:1 12001:C 11996:T 11978:N 11960:L 11957:i 11953:e 11949:X 11946:D 11940:= 11935:N 11931:) 11925:t 11919:H 11916:i 11912:e 11908:( 11905:= 11900:N 11896:U 11869:t 11863:H 11860:i 11856:e 11852:= 11849:U 11839:t 11835:t 11831:t 11816:j 11808:i 11792:. 11787:) 11781:) 11775:j 11772:, 11769:1 11766:+ 11763:i 11759:S 11755:+ 11750:1 11747:+ 11744:j 11741:, 11738:i 11734:S 11729:( 11723:j 11720:, 11717:i 11713:S 11707:j 11704:i 11694:( 11681:} 11678:S 11675:{ 11636:0 11633:= 11630:h 11608:2 11604:J 11581:1 11577:J 11552:k 11549:J 11544:) 11535:( 11532:= 11526:) 11521:2 11516:+ 11513:1 11510:( 11501:k 11496:J 11493:2 11487:= 11482:c 11478:T 11453:c 11449:T 11428:J 11425:= 11420:2 11416:J 11412:= 11407:1 11403:J 11377:= 11373:) 11365:c 11361:T 11357:k 11350:2 11346:J 11342:2 11336:( 11325:) 11317:c 11313:T 11309:k 11302:1 11298:J 11294:2 11288:( 11257:c 11253:T 11229:. 11226:] 11223:) 11218:2 11210:( 11201:) 11196:2 11192:J 11185:2 11182:( 11170:) 11165:1 11157:( 11148:) 11143:1 11139:J 11132:2 11129:( 11117:) 11112:2 11108:J 11101:2 11098:( 11089:) 11084:1 11080:J 11073:2 11070:( 11061:[ 11050:2 11042:d 11034:2 11029:0 11019:1 11011:d 11003:2 10998:0 10985:2 10977:8 10973:1 10968:+ 10965:2 10956:= 10953:f 10923:2 10919:J 10896:1 10892:J 10871:0 10868:= 10865:h 10802:K 10782:) 10776:( 10773:f 10753:2 10741:) 10735:( 10732:f 10689:) 10686:J 10680:2 10677:( 10660:2 10657:1 10652:= 10649:J 10621:) 10610:( 10607:f 10602:2 10599:1 10594:+ 10590:) 10584:) 10581:K 10578:2 10575:( 10564:2 10560:( 10548:2 10545:1 10540:= 10537:) 10531:( 10528:f 10508:) 10501:K 10497:( 10492:2 10488:/ 10484:N 10480:Z 10467:2 10463:/ 10459:N 10455:1 10448:2 10445:1 10440:+ 10436:) 10430:) 10427:K 10424:2 10421:( 10410:2 10406:( 10394:2 10391:1 10386:= 10383:) 10380:K 10377:( 10372:N 10368:Z 10356:N 10353:1 10331:) 10328:K 10325:2 10322:( 10305:2 10302:1 10297:= 10290:K 10286:, 10281:) 10278:K 10275:2 10272:( 10261:2 10258:= 10255:A 10235:) 10232:) 10227:3 10219:+ 10214:1 10206:( 10203:K 10200:( 10191:2 10188:= 10181:3 10171:1 10159:K 10154:e 10150:A 10127:) 10124:) 10121:) 10116:5 10108:+ 10103:3 10095:( 10092:K 10089:( 10080:2 10077:( 10071:) 10068:) 10065:) 10060:3 10052:+ 10047:1 10039:( 10036:K 10033:( 10024:2 10021:( 10008:= 10005:) 10002:J 9999:, 9993:( 9988:N 9984:Z 9960:, 9955:4 9947:, 9942:2 9917:J 9908:K 9883:) 9878:5 9870:+ 9865:3 9857:( 9852:4 9844:K 9840:e 9834:) 9829:3 9821:+ 9816:1 9808:( 9803:2 9795:K 9791:e 9777:= 9774:) 9771:J 9768:, 9762:( 9757:N 9753:Z 9732:N 9712:) 9709:J 9706:, 9700:( 9695:N 9691:Z 9670:) 9664:( 9661:f 9658:= 9655:) 9652:0 9649:, 9643:( 9640:f 9622:h 9618:J 9614:J 9610:h 9594:. 9589:x 9584:i 9580:T 9574:i 9566:J 9558:z 9553:1 9550:+ 9547:i 9543:T 9537:z 9532:i 9528:T 9522:L 9519:, 9513:, 9510:1 9507:= 9504:i 9496:h 9490:= 9487:) 9481:( 9478:H 9454:. 9449:z 9444:1 9441:+ 9438:n 9434:T 9428:z 9423:n 9419:T 9415:= 9410:x 9405:n 9379:, 9374:x 9369:i 9365:T 9359:n 9354:1 9351:= 9348:i 9340:= 9335:z 9330:n 9313:h 9309:J 9293:. 9288:x 9283:i 9273:i 9265:h 9257:z 9252:1 9249:+ 9246:i 9236:z 9231:i 9221:L 9218:, 9212:, 9209:1 9206:= 9203:i 9195:J 9189:= 9186:) 9180:( 9177:H 9129:, 9125:) 9116:J 9110:2 9103:e 9099:+ 9096:1 9090:J 9084:2 9077:e 9071:+ 9068:1 9064:( 9054:= 9051:) 9048:p 9045:+ 9042:1 9039:( 9030:= 9027:f 9012:L 8996:. 8991:L 8987:) 8983:p 8980:+ 8977:1 8974:( 8971:= 8968:) 8965:p 8962:+ 8959:1 8956:( 8951:k 8943:= 8936:k 8932:S 8926:k 8917:e 8903:= 8900:Z 8872:. 8866:| 8862:j 8856:i 8852:| 8848:p 8841:e 8829:j 8825:S 8819:i 8815:S 8799:p 8775:. 8770:J 8764:2 8757:e 8753:= 8747:p 8741:1 8737:p 8723:p 8719:k 8715:k 8708:J 8692:1 8688:2 8674:, 8667:J 8661:2 8654:e 8650:+ 8645:2 8641:) 8637:h 8625:( 8620:J 8614:2 8610:e 8604:+ 8601:h 8587:J 8580:e 8576:= 8571:1 8556:2 8552:V 8548:1 8534:, 8530:] 8524:L 8519:) 8512:1 8502:2 8492:( 8487:+ 8484:1 8480:[ 8474:L 8469:1 8461:= 8456:L 8451:2 8443:+ 8438:L 8433:1 8425:= 8421:) 8416:L 8412:V 8408:( 8398:= 8395:) 8389:( 8386:Z 8366:. 8361:] 8353:) 8350:J 8344:h 8341:( 8331:e 8323:J 8313:e 8303:J 8293:e 8285:) 8282:J 8279:+ 8276:h 8273:( 8266:e 8259:[ 8254:= 8251:V 8219:2 8215:h 8205:e 8188:J 8181:e 8170:2 8166:h 8156:e 8152:= 8139:, 8132:V 8101:, 8094:V 8073:. 8066:1 8058:, 8053:L 8044:V 8033:3 8025:, 8020:2 8011:V 8003:2 7995:, 7990:1 7981:V 7973:L 7965:, 7959:, 7954:1 7941:= 7934:1 7924:L 7916:J 7909:e 7901:L 7893:h 7886:e 7875:3 7865:2 7857:J 7850:e 7842:2 7834:h 7827:e 7819:2 7809:1 7801:J 7794:e 7786:1 7778:h 7771:e 7763:L 7755:, 7749:, 7744:1 7731:= 7728:) 7722:( 7719:Z 7709:h 7702:T 7698:T 7682:, 7677:N 7672:] 7664:J 7654:e 7650:+ 7645:J 7638:e 7630:J 7620:e 7611:J 7604:e 7597:[ 7592:= 7584:N 7581:+ 7578:j 7568:j 7536:. 7532:] 7526:J 7516:e 7512:+ 7507:J 7500:e 7495:[ 7480:1 7472:= 7469:) 7466:0 7463:, 7457:( 7454:f 7430:. 7425:1 7419:L 7414:] 7408:J 7398:e 7394:+ 7389:J 7382:e 7377:[ 7372:2 7369:= 7359:j 7351:J 7344:e 7333:j 7318:L 7313:2 7310:= 7307:j 7299:2 7296:= 7289:L 7279:1 7273:L 7265:J 7258:e 7247:3 7237:2 7229:J 7222:e 7214:2 7204:1 7196:J 7189:e 7181:L 7173:, 7167:, 7162:1 7149:= 7146:) 7140:( 7137:Z 7109:j 7105:, 7100:1 7094:j 7084:j 7076:= 7068:j 7043:. 7039:) 7033:L 7023:1 7017:L 7009:+ 7003:+ 6998:2 6988:1 6979:( 6975:J 6969:= 6966:) 6960:( 6957:H 6947:h 6932:c 6928:T 6924:T 6920:c 6916:C 6902:, 6896:| 6892:j 6886:i 6882:| 6878:) 6872:( 6869:c 6862:e 6858:) 6852:( 6849:C 6846:= 6838:j 6822:i 6803:j 6793:i 6765:, 6761:) 6753:J 6747:2 6740:e 6736:+ 6731:2 6727:) 6723:h 6711:( 6706:J 6700:2 6696:e 6690:+ 6687:h 6673:J 6666:e 6661:( 6646:1 6638:= 6635:) 6629:( 6626:Z 6614:L 6607:1 6594:L 6583:= 6580:) 6577:h 6574:, 6568:( 6565:f 6551:h 6547:J 6543:h 6539:J 6525:, 6520:i 6510:i 6502:h 6494:1 6491:+ 6488:i 6478:i 6468:1 6462:L 6459:, 6453:, 6450:1 6447:= 6444:i 6436:J 6430:= 6427:) 6421:( 6418:H 6408:L 6395:T 6359:| 6354:j 6348:i 6344:| 6335:j 6332:i 6328:J 6302:2 6294:| 6289:j 6283:i 6279:| 6270:j 6267:i 6263:J 6225:| 6220:j 6214:i 6210:| 6201:j 6198:i 6194:J 6155:| 6150:j 6144:i 6140:| 6131:j 6128:i 6124:J 6097:J 6093:σ 6090:H 6086:L 6078:β 6075:P 6059:μ 6056:H 6052:ν 6049:H 6029:. 6024:) 6015:H 6002:H 5998:( 5988:e 5968:g 5941:. 5931:1 5924:, 5921:0 5909:H 5896:H 5885:, 5880:) 5871:H 5858:H 5854:( 5844:e 5837:{ 5832:= 5829:) 5823:, 5817:( 5814:A 5802:A 5798:A 5794:A 5790:A 5786:μ 5783:H 5779:ν 5776:H 5760:. 5755:) 5746:H 5733:H 5729:( 5719:e 5715:= 5709:) 5703:, 5697:( 5694:A 5689:) 5683:, 5677:( 5674:A 5649:. 5644:) 5635:H 5622:H 5618:( 5608:e 5604:= 5596:) 5587:H 5583:( 5573:e 5567:Z 5564:1 5555:) 5546:H 5542:( 5532:e 5526:Z 5523:1 5515:= 5509:) 5503:( 5494:P 5488:) 5482:( 5473:P 5466:= 5460:) 5454:, 5448:( 5445:A 5440:) 5434:, 5428:( 5425:A 5419:= 5413:) 5407:, 5401:( 5398:A 5395:) 5389:, 5383:( 5380:g 5375:) 5369:, 5363:( 5360:A 5357:) 5351:, 5345:( 5342:g 5336:= 5330:) 5324:, 5318:( 5315:P 5310:) 5304:, 5298:( 5295:P 5276:L 5272:g 5268:L 5264:L 5260:L 5234:c 5227:J 5225:4 5222:J 5218:ν 5215:H 5211:μ 5208:H 5189:g 5173:A 5169:g 5135:. 5130:j 5120:i 5107:j 5101:i 5090:J 5084:= 5081:) 5075:( 5072:H 5062:h 5048:. 5043:j 5033:j 5025:h 5017:j 5007:i 4994:j 4988:i 4977:J 4971:= 4968:) 4962:( 4959:H 4938:2 4930:S 4926:, 4924:L 4920:j 4915:j 4912:σ 4907:L 4891:β 4827:, 4822:L 4816:4 4809:e 4803:L 4800:2 4796:4 4790:2 4786:L 4780:L 4755:. 4750:L 4747:2 4743:4 4736:) 4733:L 4730:( 4727:N 4717:L 4713:L 4709:e 4705:L 4701:L 4697:L 4676:, 4671:L 4661:L 4657:) 4653:d 4650:2 4647:( 4642:L 4627:L 4620:d 4606:, 4601:L 4597:) 4593:d 4590:2 4587:( 4584:= 4581:) 4578:L 4575:( 4572:N 4562:d 4558:L 4551:L 4537:. 4535:M 4531:M 4524:M 4520:M 4497:. 4492:i 4482:N 4477:1 4474:= 4471:i 4461:N 4458:1 4453:= 4450:M 4393:e 4291:. 4283:y 4273:z 4257:z 4247:x 4234:S 4228:z 4212:y 4202:x 4172:y 4152:x 4132:y 4112:x 4092:S 4062:B 4058:J 4032:A 4001:0 3990:A 3959:= 3956:B 3928:j 3918:A 3912:j 3901:= 3893:A 3863:, 3855:B 3839:A 3820:B 3810:A 3778:B 3751:A 3645:) 3639:( 3636:c 3618:j 3608:i 3556:, 3552:) 3547:| 3543:j 3537:i 3533:| 3529:) 3523:( 3520:c 3513:( 3503:C 3485:j 3475:i 3457:j 3453:i 3448:j 3445:σ 3442:i 3433:d 3421:h 3412:J 3408:Z 3404:d 3380:β 3373:j 3369:i 3341:. 3337:| 3333:) 3328:+ 3324:V 3320:( 3313:| 3309:4 3301:j 3298:i 3294:W 3288:) 3285:G 3282:( 3279:E 3273:j 3270:i 3262:= 3259:) 3253:( 3250:H 3227:| 3223:) 3218:+ 3214:V 3210:( 3203:| 3180:j 3177:i 3173:J 3166:= 3161:j 3158:i 3154:W 3131:j 3128:i 3124:J 3118:) 3113:+ 3109:V 3105:( 3096:j 3093:i 3048:) 3042:( 3039:H 2993:, 2988:j 2985:i 2981:J 2975:) 2970:+ 2966:V 2962:( 2953:j 2950:i 2942:2 2939:+ 2934:j 2931:i 2927:J 2921:) 2918:G 2915:( 2912:E 2906:j 2903:i 2892:= 2880:j 2877:i 2873:J 2867:) 2862:+ 2858:V 2854:( 2845:j 2842:i 2834:+ 2829:j 2826:i 2822:J 2816:) 2807:V 2803:( 2800:E 2794:j 2791:i 2778:j 2775:i 2771:J 2765:) 2760:+ 2756:V 2752:( 2749:E 2743:j 2740:i 2729:= 2722:) 2716:( 2713:H 2682:i 2679:j 2675:W 2671:= 2666:j 2663:i 2659:W 2638:j 2635:i 2613:j 2610:i 2606:W 2583:, 2578:j 2575:i 2571:W 2565:) 2560:+ 2556:V 2552:( 2543:j 2540:i 2530:2 2527:1 2522:= 2518:| 2514:) 2509:+ 2505:V 2501:( 2494:| 2467:) 2462:+ 2458:V 2454:( 2430:| 2426:) 2421:+ 2417:V 2413:( 2406:| 2376:V 2353:+ 2349:V 2308:) 2303:+ 2299:V 2295:( 2266:V 2243:+ 2239:V 2198:) 2195:G 2192:( 2189:V 2149:1 2143:= 2138:i 2108:j 2098:i 2088:j 2085:i 2081:J 2075:) 2072:G 2069:( 2066:E 2060:j 2057:i 2046:= 2043:) 2037:( 2034:H 1984:. 1979:j 1969:i 1956:j 1950:i 1939:J 1933:= 1930:) 1924:( 1921:H 1909:j 1905:i 1901:J 1894:J 1883:h 1867:. 1862:j 1852:i 1842:j 1839:i 1835:J 1826:j 1820:i 1806:= 1803:) 1797:( 1794:H 1782:j 1778:h 1754:0 1751:= 1746:j 1742:h 1729:j 1715:0 1707:j 1703:h 1690:j 1676:0 1668:j 1664:h 1649:j 1645:H 1630:. 1614:0 1611:= 1606:j 1603:i 1599:J 1588:, 1572:0 1564:j 1561:i 1557:J 1546:, 1530:0 1522:j 1519:i 1515:J 1501:j 1497:i 1483:) 1477:( 1474:H 1426:) 1420:( 1411:P 1387:f 1365:) 1359:( 1350:P 1346:) 1340:( 1337:f 1324:= 1311:f 1286:f 1258:) 1252:( 1249:H 1239:e 1225:= 1216:Z 1193:) 1190:T 1185:B 1181:k 1177:( 1173:/ 1169:1 1166:= 1141:, 1130:Z 1124:) 1118:( 1115:H 1105:e 1099:= 1096:) 1090:( 1081:P 1058:0 997:j 977:i 954:j 951:i 926:, 921:j 911:j 907:h 901:j 885:j 875:i 865:j 862:i 858:J 849:j 846:i 832:= 829:) 823:( 820:H 767:j 763:h 733:j 711:j 708:i 704:J 674:j 671:, 668:i 637:k 633:} 627:k 619:{ 616:= 587:} 584:1 581:+ 578:, 575:1 569:{ 561:k 534:k 503:k 483:d 319:e 312:t 305:v

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