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760:(but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.). There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.
1718:. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. These ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's
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simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.
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simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.
796:: An ergodic system is one that evolves over time to explore "all accessible" states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.
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by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation–dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical
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In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system
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All of these processes occur over time with characteristic rates. These rates are important in engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to
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Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a
503:
of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency
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that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from
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examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are
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The
Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the "interesting" information is immediately (after just one collision) scrambled up into subtle
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However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The
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Using these two concepts, the state at any other time, past or future, can in principle be calculated. There is however a disconnect between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the
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is a state with a balance of forces that has ceased to evolve.) The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of
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Although some problems in statistical physics can be solved analytically using approximations and expansions, most current research utilizes the large processing power of modern computers to simulate or approximate solutions. A common approach to statistical problems is to use a
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of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna
Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his
702:(quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.
1742:, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as
562:"In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus."
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for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle
820:
Other fundamental postulates for statistical mechanics have also been proposed. For example, recent studies shows that the theory of statistical mechanics can be built without the equal a priori probability postulate. One such formalism is based on the
457:. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as
651:
over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a
938:), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of
1504:
For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical
1725:
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.
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describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and
732:
of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in
578:, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework
1312:
713:. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. (By contrast,
1842:
Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in
2035:
Analytical and computational techniques derived from statistical physics of disordered systems, can be extended to large-scale problems, including machine learning, e.g., to analyze the weight space of deep
2811:
Gao, Xiang; Gallicchio, Emilio; Roitberg, Adrian E. (July 21, 2019). "The generalized
Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy".
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describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise
1813:(Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.
886:
bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.
682:
the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner (
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is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.
1479:
916:. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.
1214:
2032:, and more. Statistical physics also plays a role in materials science, nuclear physics, astrophysics, chemistry, biology and medicine (e.g. study of the spread of infectious diseases).
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correlations, which essentially restricts them to rarefied gases. The
Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped
808:: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest
1710:
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as
1821:(a.k.a. NEGF—non-equilibrium Green functions): A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach is often used in electronic
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is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.
572:"Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics" is firmly entrenched. Shortly before his death, Gibbs published in 1902
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Altshuler, B L; Aronov, A G; Khmelnitsky, D E (December 30, 1982). "Effects of electron-electron collisions with small energy transfers on quantum localisation".
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The ensemble formalism can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:
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3718:
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1738:(random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from
1822:
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averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions.
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in 1884. According to Gibbs, the term "statistical", in the context of mechanics, i.e. statistical mechanics, was first used by the
Scottish physicist
2485:"Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another,"
837:
330:
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Advani, Madhu; Lahiri, Subhaneil; Ganguli, Surya (March 12, 2013). "Statistical mechanics of complex neural systems and high dimensional data".
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A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few
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Gyenis, Balazs (2017). "Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium".
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influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.
1220:
2020:. Many experimental studies of matter are entirely based on the statistical description of a system. These include the scattering of cold
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Mashaghi, Alireza; Ramezanpour, Abolfazl (March 16, 2018). "Statistical physics of medical diagnostics: Study of a probabilistic model".
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axes. In quantum statistical mechanics, the ensemble is a probability distribution over pure states and can be compactly summarized as a
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Some large systems consist of many separable microscopic systems, and each of the subsystems can be analysed independently. Notably,
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calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)
511:
Statistical mechanics was initiated in the 1870s with the work of
Boltzmann, much of which was collectively published in his 1896
3686:
3196:
Hill, R; Healy, B; Holloway, L; Kuncic, Z; Thwaites, D; Baldock, C (March 2014). "Advances in kilovoltage x-ray beam dosimetry".
1809:: In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The
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1760:: An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of
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647:, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a
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391:. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion.
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Touchette, Hugo (2015). "Equivalence and
Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels".
1923:) in the conductance of an electronic system is the use of the Green–Kubo relations, with the inclusion of stochastic
4073:
2471:"Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres,"
1592:
508:, a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further.
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The probability density function is proportional to some function of the ensemble parameters and random variables.
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The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the
865:
802:: In the absence of any further information, we can only assign equal probabilities to each compatible situation.
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has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.
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For an isolated system with an exactly known energy and exactly known composition, the system can be found with
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Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the
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For dense fluids, another approximate approach is based on reduced distribution functions, in particular the
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One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as
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These two meanings are equivalent for many purposes, and will be used interchangeably in this article.
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On the
Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics
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phenomenon, which has applications in many areas of science, from functional analysis to methods of
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described below. There are various arguments in favour of the equal a priori probability postulate:
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Ramezanpour, Abolfazl; Beam, Andrew L.; Chen, Jonathan H.; Mashaghi, Alireza (November 19, 2020).
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Philosophical
Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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for granted as a complete randomization, the motions of particles in a gas would follow a simple
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had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas.
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Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.
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410:—in terms of microscopic parameters that fluctuate about average values and are characterized by
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3414:"Statistical Physics for Medical Diagnostics: Learning, Inference, and Optimization Algorithms"
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Many physical phenomena involve quasi-thermodynamic processes out of equilibrium, for example:
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612:. For both types of mechanics, the standard mathematical approach is to consider two concepts:
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744:(steady state). Statistical equilibrium does not mean that the particles have stopped moving (
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The founding of the field of statistical mechanics is generally credited to three physicists:
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Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials.
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is a classic Monte Carlo method which was initially used to sample the canonical ensemble.
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Whereas statistical mechanics proper involves dynamics, here the attention is focussed on
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3049:"Blessing of dimensionality: mathematical foundations of the statistical physics of data"
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1579:, where they are used to model radiation transport for radiation dosimetry calculations.
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The complete state of the mechanical system at a given time, mathematically encoded as a
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582:, however they were of such generality that they were found to adapt easily to the later
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515:. Boltzmann's original papers on the statistical interpretation of thermodynamics, the
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There are three equilibrium ensembles with a simple form that can be defined for any
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Thermodynamic state functions are described by ensemble averages of random variables.
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3247:
Rogers, D W O (2006). "Fifty years of Monte Carlo simulations for medical physics".
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The term "statistical mechanics" was coined by the
American mathematical physicist
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111:
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Berger, Adam L.; Pietra, Vincent J. Della; Pietra, Stephen A. Della (March 1996).
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As is usual for probabilities, the ensemble can be interpreted in different ways:
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The equal a priori probability postulate therefore provides a motivation for the
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Cohen, Doron (2011). "Lecture Notes in Statistical Mechanics and Mesoscopics".
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SklogWiki is particularly orientated towards liquids and soft condensed matter.
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of non-interacting particles have this property, allowing exact derivations of
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Jaynes, E. T. (May 15, 1957). "Information Theory and Statistical Mechanics".
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Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems
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an ensemble can be taken to represent the various possible states that a
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by interactions between various electrons by use of the Keldysh method.
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One approach to non-equilibrium statistical mechanics is to incorporate
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1989:
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At infinite temperature, all the microstates have the same probability.
352:
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586:, and still form the foundation of statistical mechanics to this day.
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A few of the theoretical tools used to make this connection include:
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1307:{\displaystyle {\mathcal {Z}}=\sum _{k}e^{-(E_{k}-\mu N_{k})/k_{B}T}}
909:
595:
533:
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388:
372:
3611:
Flamm, Dieter (1998). "History and outlook of statistical physics".
1915:. As an example, one approach to compute quantum coherence effects (
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367:, its applications include many problems in the fields of physics,
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2040:. Statistical physics is thus finding applications in the area of
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An advanced approach uses a combination of stochastic methods and
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that are driven by imbalances. Examples of such processes include
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2021:
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electric currents carried by the motion of charges in a conductor
481:, who developed models of probability distribution of such states
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368:
344:
2508:
The Man Who Changed Everything – the Life of James Clerk Maxwell
2093:
describing the quantum system. This can be shown under various
2008:. In solid state physics, statistical physics aids the study of
491:
In 1859, after reading a paper on the diffusion of molecules by
2428:(in Portuguese). Rio de Janeiro: CiĂŞncia Moderna. p. 156.
2001:
1859:
This provides an indirect avenue for obtaining numbers such as
1795:), where the electrons are indeed analogous to a rarefied gas.
1575:, and related fields, and have diverse applications including
627:
An equation of motion which carries the state forward in time:
2454:
Compendium of the foundations of classical statistical physics
1780:
that would rapidly restore a gas to an equilibrium state (see
1587:
included, the errors are reduced to an arbitrarily low level.
359:
to large assemblies of microscopic entities. Sometimes called
2342:"How can statistical mechanics contribute to social science?"
2117:
List of textbooks in thermodynamics and statistical mechanics
2025:
1993:
1949:
106:
3642:
3141:"GPU-based high-performance computing for radiation therapy"
2596:. Series on Advances in Statistical Mechanics. Vol. 8.
849:
where the third postulate can be replaced by the following:
3411:
3018:. Mathematical Surveys and Monographs. Vol. 89. 2005.
2271:"A maximum entropy approach to natural language processing"
458:
417:
While classical thermodynamics is primarily concerned with
3594:
Statistical Physics: Statics, Dynamics and Renormalization
3327:
2142:
Teschendorff, Andrew E.; Feinberg, Andrew P. (July 2021).
1980:
Statistical physics explains and quantitatively describes
1619:
to include the effect of weak interactions, leading to a
763:
A common approach found in many textbooks is to take the
604:
In physics, two types of mechanics are usually examined:
3365:"Inelastic scattering time for conductance fluctuations"
2771:(1957). "Information Theory and Statistical Mechanics".
1647:
3195:
2426:
FĂsica EstatĂstica do EquilĂbrio: um curso introdutĂłrio
2193:
Journal of Statistical Mechanics: Theory and Experiment
425:
to the issues of microscopically modeling the speed of
589:
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Statistical mechanics arose out of the development of
3363:
Aleiner, I. L.; Blanter, Ya. M. (February 28, 2002).
3305:
Equilibrium and Non-Equilibrium Statistical Mechanics
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For rarefied non-ideal gases, approaches such as the
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Jia, Xun; Ziegenhein, Peter; Jiang, Steve B (2014).
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heat transport by the internal motions in a material
3462:
2659:(London, England: Longmans, Green, and Co., 1871),
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Studies in History and Philosophy of Modern Physics
2190:
904:describes a system of fixed composition that is in
748:), rather, only that the ensemble is not evolving.
3138:
2930:
2268:
1500:There are some cases which allow exact solutions.
1474:{\displaystyle \Omega =-k_{B}T\log {\mathcal {Z}}}
1473:
1420:
1371:
1306:
1208:
1136:
1092:
1059:
1026:
953:Important cases where the thermodynamic ensembles
471:, who developed the fundamental interpretation of
3671:Videos of lecture series in statistical mechanics
2144:"Statistical mechanics meets single-cell biology"
4412:
3691:this article in the web archive on 2012 April 28
2047:
859:
686:), in the limit of an infinite number of trials.
504:towards equilibrium. Five years later, in 1864,
3522:Fundamentals of Statistical and Thermal Physics
3047:Gorban, A. N.; Tyukin, I. Y. (April 28, 2018).
2933:Fundamentals of Statistical and Thermal Physics
2346:Proceedings of the National Academy of Sciences
825:together with the following set of postulates:
3687:Configuration integral (statistical mechanics)
3538:
3362:
3116:Exactly solved models in statistical mechanics
2718:Elementary Principles in Statistical Mechanics
2591:
1636:computer simulations can be used to calculate
575:Elementary Principles in Statistical Mechanics
461:is simply the kinetic energy of their motion.
4088:
4074:
3712:
2095:mathematical formalisms for quantum mechanics
1740:hypothetical situations involving black holes
1601:, also used to sample the canonical ensemble.
723:
546:Cover of Gibbs' text on statistical mechanics
324:
3046:
1847:. A remarkable result, as formalized by the
1797:A quantum technique related in theme is the
1209:{\displaystyle Z=\sum _{k}e^{-E_{k}/k_{B}T}}
421:, statistical mechanics has been applied in
1837:
1554:Monte Carlo method in statistical mechanics
967:Large systems with long-range interactions.
934:For systems containing many particles (the
445:In 1738, Swiss physicist and mathematician
4081:
4067:
3719:
3705:
3298:
3296:
3294:
2865:
2863:
2861:
2806:
2804:
2802:
2592:Ebeling, Werner; Sokolov, Igor M. (2005).
1563:to yield insight into the properties of a
487:, who coined the name of the field in 1884
331:
317:
42:
3726:
3660:
3616:
3573:"Statistical Physics and other resources"
3476:
3439:
3429:
3380:
3330:Journal of Physics C: Solid State Physics
3172:
3090:
3064:
2975:
2961:
2905:
2887:
2825:
2738:
2736:
2734:
2732:
2552:
2375:
2365:
2340:Durlauf, Steven N. (September 14, 1999).
2204:
2167:
1772:subsequently showed that, by taking this
1695:systems being pumped by external forces (
351:is a mathematical framework that applies
3649:Thermodynamics and Statistical Mechanics
3591:
2872:"The Mathematics of the Ensemble Theory"
2707:
2705:
2703:
2701:
2699:
2697:
2695:
2447:
2445:
2238:Statistical Mechanics of Neural Networks
2067:(probability distribution over possible
751:
541:
3302:
3291:
3016:The Concentration of Measure Phenomenon
2858:
2799:
2745:The Principles of Statistical Mechanics
2671:
2423:
2403:(2nd ed.). CRC Press. p. 15.
2339:
1567:. Monte Carlo methods are important in
840:matches with the entropy as defined in
475:in terms of a collection of microstates
14:
4413:
4279:Atomic, molecular, and optical physics
3246:
3113:
2767:
2742:
2729:
2538:
2451:
2304:
1702:and irreversible processes in general.
1486:
4062:
3700:
3654:
3610:
2711:
2692:
2675:The enigma of probability and physics
2632:
2505:
2442:
2398:
2235:
2097:. One such formalism is provided by
1729:
1648:Non-equilibrium statistical mechanics
433:and flows of particles and heat. The
423:non-equilibrium statistical mechanics
27:Physics of many interacting particles
3689:, 2008. this wiki site is down; see
3570:
3539:MĂĽller-Kirsten, Harald J W. (2013).
3519:
2928:
2922:
2399:Huang, Kerson (September 21, 2009).
1678:driven by a decrease in free energy,
1665:, driven by a temperature imbalance,
964:Large systems at a phase transition.
765:equal a priori probability postulate
240:Grand potential / Landau free energy
3637:Stanford Encyclopedia of Philosophy
3633:Philosophy of Statistical Mechanics
2869:
2401:Introduction to Statistical Physics
694:ensemble evolution is given by the
590:Principles: mechanics and ensembles
24:
3635:article by Lawrence Sklar for the
3513:
1466:
1436:
1226:
823:fundamental thermodynamic relation
25:
4437:
3626:
2000:, and the structural features of
1906:
3755:
2223:10.1088/1742-5468/2013/03/P03014
1671:, driven by a voltage imbalance,
957:give identical results include:
709:and their condition is known as
620:(classical mechanics) or a pure
4400:Timeline of physics discoveries
3456:
3405:
3356:
3321:
3249:Physics in Medicine and Biology
3240:
3198:Physics in Medicine and Biology
3189:
3145:Physics in Medicine and Biology
3132:
3107:
3040:
3008:
2955:
2814:The Journal of Chemical Physics
2761:
2665:
2649:
2626:
2585:
2532:
2499:
2460:
1930:
1876:Fluctuation–dissipation theorem
1849:fluctuation–dissipation theorem
1714:or its quantum equivalent, the
1421:{\displaystyle F=-k_{B}T\log Z}
866:Ensemble (mathematical physics)
780:consistent with that knowledge.
435:fluctuation–dissipation theorem
2964:Journal of Statistical Physics
2743:Tolman, Richard Chace (1979).
2417:
2392:
2333:
2298:
2262:
2229:
2184:
2135:
1654:Non-equilibrium thermodynamics
1547:
1281:
1252:
13:
1:
3542:Basics of Statistical Physics
2128:
2112:Quantum statistical mechanics
2053:Quantum statistical mechanics
2048:Quantum statistical mechanics
1593:Metropolis–Hastings algorithm
1372:{\displaystyle S=k_{B}\log W}
860:Three thermodynamic ensembles
767:. This postulate states that
698:(classical mechanics) or the
631:(classical mechanics) or the
453:which laid the basis for the
3742:Principle of maximum entropy
1881:Onsager reciprocal relations
1778:Boltzmann transport equation
1757:Boltzmann transport equation
1628:radial distribution function
1531:. Some examples include the
1514:Maxwell–Boltzmann statistics
1114:Canonical partition function
7:
4364:Quantum information science
3431:10.3390/diagnostics10110972
3350:10.1088/0022-3719/15/36/018
3269:10.1088/0031-9155/51/13/R17
3218:10.1088/0031-9155/59/6/R183
3165:10.1088/0031-9155/59/4/R151
2571:10.1016/j.shpsb.2017.01.001
2104:
2086:operator of trace 1 on the
2078:, which is a non-negative,
1891:Landauer–Büttiker formalism
806:Maximum information entropy
82:Indistinguishable particles
10:
4442:
4195:Classical electromagnetism
3766:Statistical thermodynamics
3495:10.1103/PhysRevE.97.032118
3399:10.1103/PhysRevB.65.115317
3114:Baxter, Rodney J. (1982).
2907:10.1016/j.rinp.2022.105230
2452:Uffink, Jos (March 2006).
2160:10.1038/s41576-021-00341-z
2063:. In quantum mechanics, a
2061:quantum mechanical systems
2004:. It underlies the modern
1992:, collective phenomena in
1940:propagation of uncertainty
1799:random phase approximation
1651:
1551:
1537:square-lattice Ising model
863:
836:The entropy as defined by
724:Statistical thermodynamics
679:, a form of knowledge), or
593:
440:
365:statistical thermodynamics
18:Statistical thermodynamics
4372:
4309:
4237:
4153:
4125:
4097:
4001:
3963:
3928:
3883:
3825:
3764:
3753:
3734:
3592:Kadanoff, Leo P. (2000).
2994:10.1007/s10955-015-1212-2
2870:Gao, Xiang (March 2022).
2678:. Springer. p. 174.
2278:Computational Linguistics
2246:10.1007/978-981-16-7570-6
1599:Path integral Monte Carlo
1317:
1103:
800:Principle of indifference
735:thermodynamic equilibrium
419:thermodynamic equilibrium
412:probability distributions
4301:Condensed matter physics
4026:Condensed matter physics
4009:Statistical field theory
2367:10.1073/pnas.96.19.10582
1921:conductance fluctuations
1838:Near-equilibrium methods
1605:
1522:Bose–Einstein statistics
1495:
1119:Grand partition function
1093:{\displaystyle T,\mu ,V}
976:Thermodynamic ensembles
940:concentration of measure
920:Grand canonical ensemble
878:Grand canonical ensemble
842:classical thermodynamics
730:classical thermodynamics
649:probability distribution
396:classical thermodynamics
3884:Mathematical approaches
3873:Lennard-Jones potential
3789:thermodynamic potential
2937:. McGraw–Hill. p.
2793:10.1103/PhysRev.106.620
2747:. Courier Corporation.
2723:Charles Scribner's Sons
2672:Mayants, Lazar (1984).
2327:10.1103/PhysRev.106.620
2236:Huang, Haiping (2021).
2148:Nature Reviews Genetics
1638:microcanonical ensemble
944:artificial intelligence
891:Microcanonical ensemble
870:Microcanonical ensemble
852:
787:microcanonical ensemble
742:statistical equilibrium
711:statistical equilibrium
455:kinetic theory of gases
125:Thermodynamic ensembles
77:Spin–statistics theorem
4385:Nobel Prize in Physics
4247:Relativistic mechanics
3920:conformal field theory
3651:by Richard Fitzpatrick
3303:Balescu, Radu (1975).
3118:. Academic Press Inc.
3083:10.1098/rsta.2017.0237
2510:. Hoboken, NJ: Wiley.
2488:Philosophical Magazine
2474:Philosophical Magazine
1913:linear response theory
1896:Mori–Zwanzig formalism
1845:linear response theory
1561:Monte Carlo simulation
1518:Fermi–Dirac statistics
1475:
1422:
1373:
1308:
1210:
1138:
1094:
1061:
1028:
746:mechanical equilibrium
720:non-isolated systems.
716:mechanical equilibrium
570:
547:
513:Lectures on Gas Theory
427:irreversible processes
4421:Statistical mechanics
4390:Philosophy of physics
3835:Ferromagnetism models
3728:Statistical mechanics
2713:Gibbs, Josiah Willard
2655:James Clerk Maxwell ,
2633:Gibbs, J. W. (1885).
2506:Mahon, Basil (2003).
2483:Maxwell, J.C. (1860)
2469:Maxwell, J.C. (1860)
2057:statistical mechanics
1569:computational physics
1476:
1423:
1374:
1327:Helmholtz free energy
1318:Macroscopic function
1309:
1211:
1139:
1104:Microscopic features
1095:
1062:
1060:{\displaystyle T,N,V}
1029:
1027:{\displaystyle E,N,V}
838:Gibbs entropy formula
752:Fundamental postulate
707:equilibrium ensembles
684:empirical probability
677:epistemic probability
560:
545:
495:, Scottish physicist
349:statistical mechanics
230:Helmholtz free energy
159:Isoenthalpic–isobaric
36:Statistical mechanics
4349:Mathematical physics
3596:. World Scientific.
2424:Germano, R. (2022).
2071:) is described by a
2065:statistical ensemble
1956:ensemble forecasting
1886:Green–Kubo relations
1865:thermal conductivity
1716:von Neumann equation
1712:Liouville's equation
1433:
1384:
1341:
1221:
1149:
1128:
1072:
1039:
1006:
961:Microscopic systems.
700:von Neumann equation
658:canonical coordinate
645:statistical ensemble
633:Schrödinger equation
629:Hamilton's equations
624:(quantum mechanics).
622:quantum state vector
600:Statistical ensemble
501:Maxwell distribution
485:Josiah Willard Gibbs
4324:Atmospheric physics
4163:Classical mechanics
4091:branches of physics
4014:elementary particle
3779:partition functions
3487:2018PhRvE..97c2118M
3391:2002PhRvB..65k5317A
3342:1982JPhC...15.7367A
3261:2006PMB....51R.287R
3210:2014PMB....59R.183H
3157:2014PMB....59R.151J
3075:2018RSPTA.37670237G
2986:2015JSP...159..987T
2898:2022ResPh..3405230G
2836:2019JChPh.151c4113G
2785:1957PhRv..106..620J
2602:2005stst.book.....E
2563:2017SHPMP..57...53G
2358:1999PNAS...9610582D
2352:(19): 10582–10584.
2319:1957PhRv..106..620J
2215:2013JSMTE..03..014A
2042:medical diagnostics
1946:regression analysis
1766:James Clerk Maxwell
1690:quantum decoherence
1617:perturbation theory
1487:Calculation methods
977:
936:thermodynamic limit
927:chemical potentials
906:thermal equilibrium
814:information entropy
635:(quantum mechanics)
606:classical mechanics
580:classical mechanics
556:James Clerk Maxwell
525:thermal equilibrium
497:James Clerk Maxwell
479:James Clerk Maxwell
361:statistical physics
353:statistical methods
167:Isothermal–isobaric
70:Particle statistics
4380:History of physics
4041:information theory
3948:correlation length
3943:Critical exponents
3930:Critical phenomena
3911:stochastic process
3891:Boltzmann equation
3784:equations of state
3579:on August 12, 2021
3524:. Waveland Press.
3059:(2118): 20170237.
2876:Results in Physics
2018:critical phenomena
1861:ohmic conductivity
1830:Liouville equation
1730:Stochastic methods
1676:chemical reactions
1634:Molecular dynamics
1584:Monte Carlo method
1573:physical chemistry
1541:hard hexagon model
1471:
1418:
1369:
1304:
1243:
1206:
1167:
1134:
1090:
1057:
1024:
975:
900:Canonical ensemble
874:Canonical ensemble
794:Ergodic hypothesis
696:Liouville equation
548:
431:chemical reactions
385:information theory
357:probability theory
107:Anyonic statistics
4408:
4407:
4395:Physics education
4344:Materials science
4311:Interdisciplinary
4269:Quantum mechanics
4056:
4055:
4046:Boltzmann machine
3916:mean-field theory
3817:Maxwell relations
3603:978-981-02-3764-6
3571:Kadanoff, Leo P.
3563:978-981-4449-53-3
3531:978-1-4786-1005-2
3520:Reif, F. (2009).
3465:Physical Review E
3369:Physical Review B
3336:(36): 7367–7386.
3314:978-0-471-04600-4
3255:(13): R287–R301.
3125:978-0-12-083180-7
3033:978-0-8218-3792-4
2948:978-0-07-051800-1
2929:Reif, F. (1965).
2844:10.1063/1.5111333
2754:978-0-486-63896-6
2685:978-90-277-1674-3
2619:978-981-02-1382-4
2517:978-0-470-86171-4
2435:978-65-5842-144-3
2410:978-1-4200-7902-9
2255:978-981-16-7569-0
2122:Laplace transform
2014:phase transitions
1982:superconductivity
1968:bounded-rational
1948:of gravitational
1917:weak localization
1901:GENERIC formalism
1823:quantum transport
1818:Keldysh formalism
1613:cluster expansion
1484:
1483:
1322:Boltzmann entropy
1234:
1158:
1137:{\displaystyle W}
774:equal probability
610:quantum mechanics
584:quantum mechanics
529:equation of state
341:
340:
235:Gibbs free energy
87:Maxwell–Boltzmann
16:(Redirected from
4433:
4334:Chemical physics
4274:Particle physics
4200:Classical optics
4083:
4076:
4069:
4060:
4059:
3938:Phase transition
3759:
3758:
3721:
3714:
3707:
3698:
3697:
3680:Leonard Susskind
3672:
3666:
3664:
3622:
3620:
3607:
3588:
3586:
3584:
3575:. Archived from
3567:
3547:
3535:
3507:
3506:
3480:
3460:
3454:
3453:
3443:
3433:
3409:
3403:
3402:
3384:
3382:cond-mat/0105436
3360:
3354:
3353:
3325:
3319:
3318:
3300:
3289:
3288:
3244:
3238:
3237:
3204:(6): R183–R231.
3193:
3187:
3186:
3176:
3151:(4): R151–R182.
3136:
3130:
3129:
3111:
3105:
3104:
3094:
3068:
3044:
3038:
3037:
3024:10.1090/surv/089
3012:
3006:
3005:
2979:
2959:
2953:
2952:
2936:
2926:
2920:
2919:
2909:
2891:
2867:
2856:
2855:
2829:
2808:
2797:
2796:
2765:
2759:
2758:
2740:
2727:
2726:
2709:
2690:
2689:
2669:
2663:
2653:
2647:
2646:
2630:
2624:
2623:
2589:
2583:
2582:
2556:
2536:
2530:
2529:
2503:
2497:
2464:
2458:
2457:
2449:
2440:
2439:
2421:
2415:
2414:
2396:
2390:
2389:
2379:
2369:
2337:
2331:
2330:
2302:
2296:
2295:
2290:
2275:
2266:
2260:
2259:
2233:
2227:
2226:
2208:
2188:
2182:
2181:
2171:
2139:
2073:density operator
1770:Ludwig Boltzmann
1621:virial expansion
1480:
1478:
1477:
1472:
1470:
1469:
1454:
1453:
1427:
1425:
1424:
1419:
1405:
1404:
1378:
1376:
1375:
1370:
1359:
1358:
1313:
1311:
1310:
1305:
1303:
1302:
1298:
1297:
1288:
1280:
1279:
1264:
1263:
1242:
1230:
1229:
1215:
1213:
1212:
1207:
1205:
1204:
1200:
1199:
1190:
1185:
1184:
1166:
1143:
1141:
1140:
1135:
1099:
1097:
1096:
1091:
1066:
1064:
1063:
1058:
1033:
1031:
1030:
1025:
1000:Fixed variables
978:
974:
568:
567:J. Clerk Maxwell
552:J. Willard Gibbs
521:transport theory
506:Ludwig Boltzmann
469:Ludwig Boltzmann
447:Daniel Bernoulli
381:computer science
333:
326:
319:
112:Braid statistics
46:
32:
31:
21:
4441:
4440:
4436:
4435:
4434:
4432:
4431:
4430:
4411:
4410:
4409:
4404:
4368:
4354:Medical physics
4305:
4264:Nuclear physics
4233:
4227:Non-equilibrium
4149:
4121:
4093:
4087:
4057:
4052:
3997:
3959:
3924:
3906:BBGKY hierarchy
3901:Vlasov equation
3879:
3868:depletion force
3861:Particles with
3821:
3760:
3756:
3751:
3730:
3725:
3670:
3629:
3618:physics/9803005
3604:
3582:
3580:
3564:
3545:
3532:
3516:
3514:Further reading
3511:
3510:
3461:
3457:
3410:
3406:
3361:
3357:
3326:
3322:
3315:
3301:
3292:
3245:
3241:
3194:
3190:
3137:
3133:
3126:
3112:
3108:
3045:
3041:
3034:
3014:
3013:
3009:
2970:(5): 987–1016.
2960:
2956:
2949:
2927:
2923:
2868:
2859:
2809:
2800:
2773:Physical Review
2766:
2762:
2755:
2741:
2730:
2710:
2693:
2686:
2670:
2666:
2654:
2650:
2631:
2627:
2620:
2590:
2586:
2537:
2533:
2518:
2504:
2500:
2465:
2461:
2450:
2443:
2436:
2422:
2418:
2411:
2397:
2393:
2338:
2334:
2307:Physical Review
2303:
2299:
2288:
2273:
2267:
2263:
2256:
2234:
2230:
2189:
2185:
2140:
2136:
2131:
2126:
2107:
2050:
2038:neural networks
2010:liquid crystals
1970:potential games
1963:neural networks
1933:
1909:
1840:
1835:
1811:BBGKY hierarchy
1806:BBGKY hierarchy
1774:molecular chaos
1732:
1697:optical pumping
1656:
1650:
1608:
1577:medical physics
1556:
1550:
1539:in zero field,
1510:idealized gases
1498:
1489:
1465:
1464:
1449:
1445:
1434:
1431:
1430:
1400:
1396:
1385:
1382:
1381:
1354:
1350:
1342:
1339:
1338:
1332:Grand potential
1293:
1289:
1284:
1275:
1271:
1259:
1255:
1248:
1244:
1238:
1225:
1224:
1222:
1219:
1218:
1195:
1191:
1186:
1180:
1176:
1172:
1168:
1162:
1150:
1147:
1146:
1129:
1126:
1125:
1073:
1070:
1069:
1040:
1037:
1036:
1007:
1004:
1003:
994:Grand canonical
884:isolated system
880:
864:Main articles:
862:
857:
847:
754:
726:
602:
594:Main articles:
592:
569:
566:
499:formulated the
493:Rudolf Clausius
443:
337:
308:
307:
253:
245:
244:
220:Internal energy
215:
205:
204:
180:
172:
171:
151:Grand canonical
127:
117:
116:
72:
28:
23:
22:
15:
12:
11:
5:
4439:
4429:
4428:
4426:Thermodynamics
4423:
4406:
4405:
4403:
4402:
4397:
4392:
4387:
4382:
4376:
4374:
4370:
4369:
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4230:
4229:
4224:
4217:Thermodynamics
4214:
4213:
4212:
4207:
4197:
4192:
4187:
4186:
4185:
4180:
4175:
4170:
4159:
4157:
4151:
4150:
4148:
4147:
4146:
4145:
4135:
4129:
4127:
4123:
4122:
4120:
4119:
4118:
4117:
4107:
4101:
4099:
4095:
4094:
4086:
4085:
4078:
4071:
4063:
4054:
4053:
4051:
4050:
4049:
4048:
4043:
4038:
4031:Complex system
4028:
4023:
4022:
4021:
4016:
4005:
4003:
3999:
3998:
3996:
3995:
3990:
3985:
3980:
3975:
3969:
3967:
3961:
3960:
3958:
3957:
3956:
3955:
3950:
3940:
3934:
3932:
3926:
3925:
3923:
3922:
3913:
3908:
3903:
3898:
3893:
3887:
3885:
3881:
3880:
3878:
3877:
3876:
3875:
3870:
3859:
3858:
3857:
3852:
3847:
3842:
3831:
3829:
3823:
3822:
3820:
3819:
3814:
3813:
3812:
3807:
3802:
3797:
3786:
3781:
3776:
3770:
3768:
3762:
3761:
3754:
3752:
3750:
3749:
3747:ergodic theory
3744:
3738:
3736:
3732:
3731:
3724:
3723:
3716:
3709:
3701:
3695:
3694:
3683:
3667:
3652:
3646:
3640:
3628:
3627:External links
3625:
3624:
3623:
3608:
3602:
3589:
3568:
3562:
3536:
3530:
3515:
3512:
3509:
3508:
3455:
3404:
3375:(11): 115317.
3355:
3320:
3313:
3290:
3239:
3188:
3131:
3124:
3106:
3039:
3032:
3007:
2954:
2947:
2921:
2857:
2798:
2779:(4): 620–630.
2760:
2753:
2728:
2691:
2684:
2664:
2657:Theory of Heat
2648:
2625:
2618:
2584:
2531:
2516:
2498:
2496:
2495:
2494: : 21–37.
2490:, 4th series,
2481:
2480: : 19–32.
2476:, 4th series,
2459:
2441:
2434:
2416:
2409:
2391:
2332:
2313:(4): 620–630.
2297:
2261:
2254:
2228:
2183:
2154:(7): 459–476.
2133:
2132:
2130:
2127:
2125:
2124:
2119:
2114:
2108:
2106:
2103:
2069:quantum states
2049:
2046:
1978:
1977:
1976:and economics.
1966:
1959:
1953:
1943:
1932:
1929:
1908:
1907:Hybrid methods
1905:
1904:
1903:
1898:
1893:
1888:
1883:
1878:
1839:
1836:
1834:
1833:
1826:
1814:
1802:
1789:semiconductors
1762:kinetic theory
1752:
1731:
1728:
1704:
1703:
1700:
1693:
1679:
1672:
1666:
1649:
1646:
1645:
1644:
1641:
1631:
1624:
1607:
1604:
1603:
1602:
1596:
1565:complex system
1552:Main article:
1549:
1546:
1545:
1544:
1525:
1506:
1497:
1494:
1488:
1485:
1482:
1481:
1468:
1463:
1460:
1457:
1452:
1448:
1444:
1441:
1438:
1428:
1417:
1414:
1411:
1408:
1403:
1399:
1395:
1392:
1389:
1379:
1368:
1365:
1362:
1357:
1353:
1349:
1346:
1335:
1334:
1329:
1324:
1319:
1315:
1314:
1301:
1296:
1292:
1287:
1283:
1278:
1274:
1270:
1267:
1262:
1258:
1254:
1251:
1247:
1241:
1237:
1233:
1228:
1216:
1203:
1198:
1194:
1189:
1183:
1179:
1175:
1171:
1165:
1161:
1157:
1154:
1144:
1133:
1122:
1121:
1116:
1111:
1105:
1101:
1100:
1089:
1086:
1083:
1080:
1077:
1067:
1056:
1053:
1050:
1047:
1044:
1034:
1023:
1020:
1017:
1014:
1011:
1001:
997:
996:
991:
986:
984:Microcanonical
981:
969:
968:
965:
962:
932:
931:
922:
917:
902:
897:
893:
861:
858:
856:
855:
851:
846:
845:
834:
831:
827:
818:
817:
803:
797:
783:
782:
753:
750:
725:
722:
688:
687:
680:
662:density matrix
637:
636:
625:
591:
588:
564:
489:
488:
482:
476:
442:
439:
339:
338:
336:
335:
328:
321:
313:
310:
309:
306:
305:
300:
295:
290:
285:
280:
275:
270:
265:
260:
254:
251:
250:
247:
246:
243:
242:
237:
232:
227:
222:
216:
211:
210:
207:
206:
203:
202:
197:
192:
187:
181:
178:
177:
174:
173:
170:
169:
161:
153:
145:
137:
135:Microcanonical
128:
123:
122:
119:
118:
115:
114:
109:
104:
102:Parastatistics
99:
94:
89:
84:
79:
73:
68:
67:
64:
63:
62:
61:
59:Kinetic theory
56:
54:Thermodynamics
48:
47:
39:
38:
26:
9:
6:
4:
3:
2:
4438:
4427:
4424:
4422:
4419:
4418:
4416:
4401:
4398:
4396:
4393:
4391:
4388:
4386:
4383:
4381:
4378:
4377:
4375:
4371:
4365:
4362:
4360:
4359:Ocean physics
4357:
4355:
4352:
4350:
4347:
4345:
4342:
4340:
4337:
4335:
4332:
4330:
4327:
4325:
4322:
4320:
4317:
4316:
4314:
4312:
4308:
4302:
4299:
4295:
4294:Modern optics
4292:
4290:
4287:
4285:
4282:
4281:
4280:
4277:
4275:
4272:
4270:
4267:
4265:
4262:
4258:
4255:
4253:
4250:
4249:
4248:
4245:
4244:
4242:
4240:
4236:
4228:
4225:
4223:
4220:
4219:
4218:
4215:
4211:
4208:
4206:
4203:
4202:
4201:
4198:
4196:
4193:
4191:
4188:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4165:
4164:
4161:
4160:
4158:
4156:
4152:
4144:
4143:Computational
4141:
4140:
4139:
4136:
4134:
4131:
4130:
4128:
4124:
4116:
4113:
4112:
4111:
4108:
4106:
4103:
4102:
4100:
4096:
4092:
4084:
4079:
4077:
4072:
4070:
4065:
4064:
4061:
4047:
4044:
4042:
4039:
4037:
4034:
4033:
4032:
4029:
4027:
4024:
4020:
4019:superfluidity
4017:
4015:
4012:
4011:
4010:
4007:
4006:
4004:
4000:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3971:
3970:
3968:
3966:
3962:
3954:
3951:
3949:
3946:
3945:
3944:
3941:
3939:
3936:
3935:
3933:
3931:
3927:
3921:
3917:
3914:
3912:
3909:
3907:
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3888:
3886:
3882:
3874:
3871:
3869:
3866:
3865:
3864:
3860:
3856:
3853:
3851:
3848:
3846:
3843:
3841:
3838:
3837:
3836:
3833:
3832:
3830:
3828:
3824:
3818:
3815:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3792:
3790:
3787:
3785:
3782:
3780:
3777:
3775:
3772:
3771:
3769:
3767:
3763:
3748:
3745:
3743:
3740:
3739:
3737:
3733:
3729:
3722:
3717:
3715:
3710:
3708:
3703:
3702:
3699:
3692:
3688:
3685:Vu-Quoc, L.,
3684:
3681:
3677:
3673:
3668:
3663:
3658:
3653:
3650:
3647:
3644:
3641:
3638:
3634:
3631:
3630:
3619:
3614:
3609:
3605:
3599:
3595:
3590:
3578:
3574:
3569:
3565:
3559:
3555:
3551:
3544:
3543:
3537:
3533:
3527:
3523:
3518:
3517:
3504:
3500:
3496:
3492:
3488:
3484:
3479:
3474:
3471:(3): 032118.
3470:
3466:
3459:
3451:
3447:
3442:
3437:
3432:
3427:
3423:
3419:
3415:
3408:
3400:
3396:
3392:
3388:
3383:
3378:
3374:
3370:
3366:
3359:
3351:
3347:
3343:
3339:
3335:
3331:
3324:
3316:
3310:
3306:
3299:
3297:
3295:
3286:
3282:
3278:
3274:
3270:
3266:
3262:
3258:
3254:
3250:
3243:
3235:
3231:
3227:
3223:
3219:
3215:
3211:
3207:
3203:
3199:
3192:
3184:
3180:
3175:
3170:
3166:
3162:
3158:
3154:
3150:
3146:
3142:
3135:
3127:
3121:
3117:
3110:
3102:
3098:
3093:
3088:
3084:
3080:
3076:
3072:
3067:
3062:
3058:
3054:
3050:
3043:
3035:
3029:
3025:
3021:
3017:
3011:
3003:
2999:
2995:
2991:
2987:
2983:
2978:
2973:
2969:
2965:
2958:
2950:
2944:
2940:
2935:
2934:
2925:
2917:
2913:
2908:
2903:
2899:
2895:
2890:
2885:
2881:
2877:
2873:
2866:
2864:
2862:
2853:
2849:
2845:
2841:
2837:
2833:
2828:
2823:
2820:(3): 034113.
2819:
2815:
2807:
2805:
2803:
2794:
2790:
2786:
2782:
2778:
2774:
2770:
2764:
2756:
2750:
2746:
2739:
2737:
2735:
2733:
2724:
2720:
2719:
2714:
2708:
2706:
2704:
2702:
2700:
2698:
2696:
2687:
2681:
2677:
2676:
2668:
2662:
2658:
2652:
2644:
2640:
2636:
2629:
2621:
2615:
2611:
2607:
2603:
2599:
2595:
2588:
2580:
2576:
2572:
2568:
2564:
2560:
2555:
2550:
2546:
2542:
2535:
2527:
2523:
2519:
2513:
2509:
2502:
2493:
2489:
2486:
2482:
2479:
2475:
2472:
2468:
2467:
2463:
2455:
2448:
2446:
2437:
2431:
2427:
2420:
2412:
2406:
2402:
2395:
2387:
2383:
2378:
2373:
2368:
2363:
2359:
2355:
2351:
2347:
2343:
2336:
2328:
2324:
2320:
2316:
2312:
2308:
2301:
2293:
2287:
2283:
2279:
2272:
2265:
2257:
2251:
2247:
2243:
2239:
2232:
2224:
2220:
2216:
2212:
2207:
2202:
2199:(3): P03014.
2198:
2194:
2187:
2179:
2175:
2170:
2165:
2161:
2157:
2153:
2149:
2145:
2138:
2134:
2123:
2120:
2118:
2115:
2113:
2110:
2109:
2102:
2100:
2099:quantum logic
2096:
2092:
2089:
2088:Hilbert space
2085:
2081:
2077:
2074:
2070:
2066:
2062:
2058:
2054:
2045:
2043:
2039:
2033:
2031:
2030:visible light
2027:
2023:
2019:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1987:
1986:superfluidity
1983:
1975:
1971:
1967:
1964:
1960:
1957:
1954:
1951:
1947:
1944:
1941:
1938:
1937:
1936:
1928:
1926:
1922:
1918:
1914:
1902:
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1882:
1879:
1877:
1874:
1873:
1872:
1869:
1866:
1862:
1857:
1856:equilibrium.
1854:
1850:
1846:
1831:
1827:
1825:calculations.
1824:
1820:
1819:
1815:
1812:
1808:
1807:
1803:
1800:
1796:
1794:
1790:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1758:
1754:
1753:
1751:
1749:
1745:
1741:
1737:
1727:
1723:
1721:
1720:Gibbs entropy
1717:
1713:
1708:
1701:
1698:
1694:
1691:
1687:
1683:
1680:
1677:
1673:
1670:
1667:
1664:
1661:
1660:
1659:
1655:
1642:
1639:
1635:
1632:
1629:
1625:
1622:
1618:
1614:
1610:
1609:
1600:
1597:
1594:
1590:
1589:
1588:
1585:
1580:
1578:
1574:
1570:
1566:
1562:
1555:
1542:
1538:
1534:
1530:
1526:
1523:
1519:
1515:
1511:
1507:
1503:
1502:
1501:
1493:
1461:
1458:
1455:
1450:
1446:
1442:
1439:
1429:
1415:
1412:
1409:
1406:
1401:
1397:
1393:
1390:
1387:
1380:
1366:
1363:
1360:
1355:
1351:
1347:
1344:
1337:
1336:
1333:
1330:
1328:
1325:
1323:
1320:
1316:
1299:
1294:
1290:
1285:
1276:
1272:
1268:
1265:
1260:
1256:
1249:
1245:
1239:
1235:
1231:
1217:
1201:
1196:
1192:
1187:
1181:
1177:
1173:
1169:
1163:
1159:
1155:
1152:
1145:
1131:
1124:
1123:
1120:
1117:
1115:
1112:
1110:
1106:
1102:
1087:
1084:
1081:
1078:
1075:
1068:
1054:
1051:
1048:
1045:
1042:
1035:
1021:
1018:
1015:
1012:
1009:
1002:
999:
998:
995:
992:
990:
987:
985:
982:
980:
979:
973:
966:
963:
960:
959:
958:
956:
951:
949:
945:
941:
937:
928:
923:
921:
918:
915:
912:of a precise
911:
907:
903:
901:
898:
894:
892:
889:
888:
887:
885:
879:
875:
871:
867:
853:
850:
843:
839:
835:
832:
829:
828:
826:
824:
815:
811:
810:Gibbs entropy
807:
804:
801:
798:
795:
792:
791:
790:
788:
781:
779:
773:
770:
769:
768:
766:
761:
759:
749:
747:
743:
738:
736:
731:
721:
718:
717:
712:
708:
703:
701:
697:
691:
685:
681:
678:
675:could be in (
674:
673:single system
670:
669:
668:
665:
663:
659:
655:
650:
646:
641:
634:
630:
626:
623:
619:
615:
614:
613:
611:
607:
601:
597:
587:
585:
581:
577:
576:
563:
559:
557:
553:
544:
540:
538:
536:
530:
526:
522:
518:
514:
509:
507:
502:
498:
494:
486:
483:
480:
477:
474:
470:
467:
466:
465:
462:
460:
456:
452:
451:Hydrodynamica
448:
438:
436:
432:
428:
424:
420:
415:
413:
409:
408:heat capacity
405:
401:
397:
392:
390:
386:
382:
378:
374:
370:
366:
362:
358:
354:
350:
346:
334:
329:
327:
322:
320:
315:
314:
312:
311:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
274:
271:
269:
266:
264:
261:
259:
256:
255:
249:
248:
241:
238:
236:
233:
231:
228:
226:
223:
221:
218:
217:
214:
209:
208:
201:
198:
196:
193:
191:
188:
186:
183:
182:
176:
175:
168:
165:
162:
160:
157:
154:
152:
149:
146:
144:
141:
138:
136:
133:
130:
129:
126:
121:
120:
113:
110:
108:
105:
103:
100:
98:
95:
93:
92:Bose–Einstein
90:
88:
85:
83:
80:
78:
75:
74:
71:
66:
65:
60:
57:
55:
52:
51:
50:
49:
45:
41:
40:
37:
34:
33:
30:
19:
4319:Astrophysics
4221:
4133:Experimental
4002:Applications
3953:size scaling
3727:
3593:
3581:. Retrieved
3577:the original
3554:10.1142/8709
3541:
3521:
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