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principle, and comments that this is to some extent disappointing; she also points to the difficulty of finding a thermodynamically consistent form of entropy production. Another top expert offers an extensive discussion of the possibilities for principles of extrema of entropy production and of dissipation of energy: Chapter 12 of Grandy (2008) is very cautious, and finds difficulty in defining the 'rate of internal entropy production' in many cases, and finds that sometimes for the prediction of the course of a process, an extremum of the quantity called the rate of dissipation of energy may be more useful than that of the rate of entropy production; this quantity appeared in
Onsager's 1931 origination of this subject. Other writers have also felt that prospects for general global extremal principles are clouded. Such writers include Glansdorff and Prigogine (1971), Lebon, Jou and Casas-Vásquez (2008), and Šilhavý (1997). There is good experimental evidence that heat convection does not obey extremal principles for time rate of entropy production. Theoretical analysis shows that chemical reactions do not obey extremal principles for the second differential of time rate of entropy production. The development of a general extremal principle seems infeasible in the current state of knowledge.
2434:(see also Keizer (1987) means that conceptually, for study and analysis, the system can be spatially and temporally divided into 'cells' or 'micro-phases' of small (infinitesimal) size, in which classical thermodynamical equilibrium conditions for matter are fulfilled to good approximation. These conditions are unfulfilled, for example, in very rarefied gases, in which molecular collisions are infrequent; and in the boundary layers of a star, where radiation is passing energy to space; and for interacting fermions at very low temperature, where dissipative processes become ineffective. When these 'cells' are defined, one admits that matter and energy may pass freely between contiguous 'cells', slowly enough to leave the 'cells' in their respective individual local thermodynamic equilibria with respect to intensive variables.
1853:
remote from one another. In the classical irreversible thermodynamic approach, there is allowed spatial variation from infinitesimal volume element to adjacent infinitesimal volume element, but it is assumed that the global entropy of the system can be found by simple spatial integration of the local entropy density. This approach assumes spatial and temporal continuity and even differentiability of locally defined intensive variables such as temperature and internal energy density. While these demands may appear severely constrictive, it has been found that the assumptions of local equilibrium hold for a wide variety of systems, including reacting interfaces, on the surfaces of catalysts, in confined systems such as zeolites, under temperature gradients as large as
1789:
state variables, for example temperature and pressure, correspond closely with equilibrium state variables. It is necessary that measuring probes be small enough, and rapidly enough responding, to capture relevant non-uniformity. Further, the non-equilibrium state variables are required to be mathematically functionally related to one another in ways that suitably resemble corresponding relations between equilibrium thermodynamic state variables. In reality, these requirements, although strict, have been shown to be fulfilled even under extreme conditions, such as during phase transitions, at reacting interfaces, and in plasma droplets surrounded by ambient air. There are, however, situations where there are appreciable non-linear effects even at the local scale.
1849:
system effectively homogeneous, or well-mixed, or without an effective spatial structure. Even within the thought-frame of classical irreversible thermodynamics, care is needed in choosing the independent variables for systems. In some writings, it is assumed that the intensive variables of equilibrium thermodynamics are sufficient as the independent variables for the task (such variables are considered to have no 'memory', and do not show hysteresis); in particular, local flow intensive variables are not admitted as independent variables; local flows are considered as dependent on quasi-static local intensive variables.
36:
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significant level of probability. Fluctuations about stable stationary states are extremely small except near critical points (Kondepudi and
Prigogine 1998, page 323). The stable stationary state has a local maximum of entropy and is locally the most reproducible state of the system. There are theorems about the irreversible dissipation of fluctuations. Here 'local' means local with respect to the abstract space of thermodynamic coordinates of state of the system.
89:
1659:
3422:. In the case of chemically reacting substances, which was investigated by Prigogine, the internal variables appear to be measures of incompleteness of chemical reactions, that is measures of how much the considered system with chemical reactions is out of equilibrium. The theory can be generalised, to consider any deviation from the equilibrium state as an internal variable, so that we consider the set of internal variables
1736:. One fundamental difference between equilibrium thermodynamics and non-equilibrium thermodynamics lies in the behaviour of inhomogeneous systems, which require for their study knowledge of rates of reaction which are not considered in equilibrium thermodynamics of homogeneous systems. This is discussed below. Another fundamental and very important difference is the difficulty, in defining
3036:, when he and his collaborators investigated the systems of chemically reacting substances. The stationary states of such systems exists due to exchange both particles and energy with the environment. In section 8 of the third chapter of his book, Prigogine has specified three contributions to the variation of entropy of the considered system at the given volume and constant temperature
1982:. Equilibrium conditions of thermodynamic systems are related to the maximum property of the entropy. If the only extensive quantity that is allowed to fluctuate is the internal energy, all the other ones being kept strictly constant, the temperature of the system is measurable and meaningful. The system's properties are then most conveniently described using the thermodynamic potential
2442:. For example, in the atmosphere, the speed of sound is much greater than the wind speed; this favours the idea of local thermodynamic equilibrium of matter for atmospheric heat transfer studies at altitudes below about 60 km where sound propagates, but not above 100 km, where, because of the paucity of intermolecular collisions, sound does not propagate.
1785:
state variables, or by corresponding time and space derivatives, including fluxes of matter and energy. In general, non-equilibrium thermodynamic systems are spatially and temporally non-uniform, but their non-uniformity still has a sufficient degree of smoothness to support the existence of suitable time and space derivatives of non-equilibrium state variables.
1970:
vibrational and rotational molecular motion), the requirement for two component 'temperatures' in the one small region of space, precluding local thermodynamic equilibrium, which demands that only one temperature be needed. Damping of acoustic perturbations or shock waves are non-stationary non-equilibrium processes. Driven
1840:
quantities of matter, thermal radiation is weak and can be practically nearly ignored. But, for example, atmospheric physics is concerned with large amounts of matter, occupying cubic kilometers, that, taken as a whole, are not within the range of laboratory quantities; then thermal radiation cannot be ignored.
1725:
however, be considered to be in equilibrium locally, thus allowing description by currently known equilibrium thermodynamics. Nevertheless, some natural systems and processes remain beyond the scope of equilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of
4006:
One wants to take the analysis to the further stage of describing the behaviour of surface and volume integrals of non-stationary local quantities; these integrals are macroscopic fluxes and production rates. In general the dynamics of these integrals are not adequately described by linear equations,
1848:
The terms 'classical irreversible thermodynamics' and 'local equilibrium thermodynamics' are sometimes used to refer to a version of non-equilibrium thermodynamics that demands certain simplifying assumptions, as follows. The assumptions have the effect of making each very small volume element of the
1852:
Also it is assumed that the local entropy density is the same function of the other local intensive variables as in equilibrium; this is called the local thermodynamic equilibrium assumption (see also Keizer (1987)). Radiation is ignored because it is transfer of energy between regions, which can be
1784:
The suitable relationship that defines non-equilibrium thermodynamic state variables is as follows. When the system is in local equilibrium, non-equilibrium state variables are such that they can be measured locally with sufficient accuracy by the same techniques as are used to measure thermodynamic
2025:
Non-equilibrium systems are much more complex and they may undergo fluctuations of more extensive quantities. The boundary conditions impose on them particular intensive variables, like temperature gradients or distorted collective motions (shear motions, vortices, etc.), often called thermodynamic
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to have a very close connection with those of equilibrium thermodynamics. This conceptual issue is overcome under the assumption of local equilibrium, which entails that the relationships that hold between macroscopic state variables at equilibrium hold locally, also outside equilibrium. Throughout
2437:
One can think here of two 'relaxation times' separated by order of magnitude. The longer relaxation time is of the order of magnitude of times taken for the macroscopic dynamical structure of the system to change. The shorter is of the order of magnitude of times taken for a single 'cell' to reach
2401:
In thermodynamics one is often interested in a stationary state of a process, allowing that the stationary state include the occurrence of unpredictable and experimentally unreproducible fluctuations in the state of the system. The fluctuations are due to the system's internal sub-processes and to
1758:
Equilibrium thermodynamics restricts its considerations to processes that have initial and final states of thermodynamic equilibrium; the time-courses of processes are deliberately ignored. Non-equilibrium thermodynamics, on the other hand, attempting to describe continuous time-courses, needs its
1724:
Almost all systems found in nature are not in thermodynamic equilibrium, for they are changing or can be triggered to change over time, and are continuously and discontinuously subject to flux of matter and energy to and from other systems and to chemical reactions. Many systems and processes can,
4508:
Until recently, prospects for useful extremal principles in this area have seemed clouded. Nicolis (1999) concludes that one model of atmospheric dynamics has an attractor which is not a regime of maximum or minimum dissipation; she says this seems to rule out the existence of a global organizing
1740:
at an instant of time in macroscopic terms for systems not in thermodynamic equilibrium. However, it can be done locally, and the macroscopic entropy will then be given by the integral of the locally defined entropy density. It has been found that many systems far outside global equilibrium still
1788:
Because of the spatial non-uniformity, non-equilibrium state variables that correspond to extensive thermodynamic state variables have to be defined as spatial densities of the corresponding extensive equilibrium state variables. When the system is in local equilibrium, intensive non-equilibrium
2475:
It is pointed out by W.T. Grandy Jr, that entropy, though it may be defined for a non-equilibrium system is—when strictly considered—only a macroscopic quantity that refers to the whole system, and is not a dynamical variable and in general does not act as a local potential that describes local
2466:
of the matter of the 'cell'. Then it strictly obeys
Kirchhoff's law of equality of radiative emissivity and absorptivity, with a black body source function. The key to local thermodynamic equilibrium here is that the rate of collisions of ponderable matter particles such as molecules should far
2405:
If the stationary state of the process is stable, then the unreproducible fluctuations involve local transient decreases of entropy. The reproducible response of the system is then to increase the entropy back to its maximum by irreversible processes: the fluctuation cannot be reproduced with a
2388:
Intensities are global values, valid for the system as a whole. When boundaries impose to the system different local conditions, (e.g. temperature differences), there are intensive variables representing the average value and others representing gradients or higher moments. The latter are the
1969:
action is also a non-equilibrium process, but it depends on departure from local thermodynamic equilibrium and is thus beyond the scope of classical irreversible thermodynamics; here a strong temperature difference is maintained between two molecular degrees of freedom (with molecular laser,
1839:
According to Wildt (see also Essex), current versions of non-equilibrium thermodynamics ignore radiant heat; they can do so because they refer to laboratory quantities of matter under laboratory conditions with temperatures well below those of stars. At laboratory temperatures, in laboratory
3955:
For classical non-equilibrium studies, we will consider some new locally defined intensive macroscopic variables. We can, under suitable conditions, derive these new variables by locally defining the gradients and flux densities of the basic locally defined macroscopic quantities.
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1937:
depend not only on present values but also on past values of local equilibrium variables. Thus time comes into the picture more deeply than for time-dependent local equilibrium thermodynamics with memoryless materials, but fluxes are not independent variables of state.
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1767:
Thus, non-equilibrium thermodynamics provides a consistent framework for modelling not only the initial and final states of a system, but also the evolution of the system in time. Together with the concept of entropy production, this provides a powerful tool in
1754:
A profound difference separates equilibrium from non-equilibrium thermodynamics. Equilibrium thermodynamics ignores the time-courses of physical processes. In contrast, non-equilibrium thermodynamics attempts to describe their time-courses in continuous detail.
2476:
physical forces. Under special circumstances, however, one can metaphorically think as if the thermal variables behaved like local physical forces. The approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking.
2438:
local thermodynamic equilibrium. If these two relaxation times are not well separated, then the classical non-equilibrium thermodynamical concept of local thermodynamic equilibrium loses its meaning and other approaches have to be proposed, see for instance
3449:
in equation (1) to consist of the quantities defining not only degrees of completeness of all chemical reactions occurring in the system, but also the structure of the system, gradients of temperature, difference of concentrations of substances and so on.
3959:
Such locally defined gradients of intensive macroscopic variables are called 'thermodynamic forces'. They 'drive' flux densities, perhaps misleadingly often called 'fluxes', which are dual to the forces. These quantities are defined in the article on
4529:, which can be out of equilibrium in systems where catalysis and electrochemical conversion is involved. Also, ideas from non-equilibrium thermodynamics and the informatic theory of entropy have been adapted to describe general economic systems.
4002:
and others, when an open system is in conditions that allow it to reach a stable stationary thermodynamically non-equilibrium state, it organizes itself so as to minimize total entropy production defined locally. This is considered further below.
1712:
but can be described in terms of macroscopic quantities (non-equilibrium state variables) that represent an extrapolation of the variables used to specify the system in thermodynamic equilibrium. Non-equilibrium thermodynamics is concerned with
1912:
In other writings, local flow variables are considered; these might be considered as classical by analogy with the time-invariant long-term time-averages of flows produced by endlessly repeated cyclic processes; examples with flows are in the
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the past decades, the assumption of local equilibrium has been tested, and found to hold, under increasingly extreme conditions, such as in the shock front of violent explosions, on reacting surfaces, and under extreme thermal gradients.
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It may be shown that the
Legendre transformation changes the maximum condition of the entropy (valid at equilibrium) in a minimum condition of the extended Massieu function for stationary states, no matter whether at equilibrium or not.
2026:
forces. If free energies are very useful in equilibrium thermodynamics, it must be stressed that there is no general law defining stationary non-equilibrium properties of the energy as is the second law of thermodynamics for the
2409:
If the stationary state is unstable, then any fluctuation will almost surely trigger the virtually explosive departure of the system from the unstable stationary state. This can be accompanied by increased export of entropy.
2418:
The scope of present-day non-equilibrium thermodynamics does not cover all physical processes. A condition for the validity of many studies in non-equilibrium thermodynamics of matter is that they deal with what is known as
4183:
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in each small local 'cell'. He defined 'local thermodynamic equilibrium' in a 'cell' by requiring that it macroscopically absorb and spontaneously emit radiation as if it were in radiative equilibrium in a cavity at the
1977:
The mechanics of macroscopic systems depends on a number of extensive quantities. It should be stressed that all systems are permanently interacting with their surroundings, thereby causing unavoidable fluctuations of
2479:
This point of view shares many points in common with the concept and the use of entropy in continuum thermomechanics, which evolved completely independently of statistical mechanics and maximum-entropy principles.
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In stationary conditions, such forces and associated flux densities are by definition time invariant, as also are the system's locally defined entropy and rate of entropy production. Notably, according to
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3994:) may be coupled. The article on Onsager reciprocal relations considers the stable near-steady thermodynamically non-equilibrium regime, which has dynamics linear in the forces and flux densities.
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Following
Onsager (1931,I), let us extend our considerations to thermodynamically non-equilibrium systems. As a basis, we need locally defined versions of the extensive macroscopic quantities
4311:
1827:
One initial approach to non-equilibrium thermodynamics is sometimes called 'classical irreversible thermodynamics'. There are other approaches to non-equilibrium thermodynamics, for example
5073:
Costa e Silva, André; Ågren, John; Clavaguera-Mora, Maria Teresa; Djurovic, D.; Gomez-Acebo, Tomas; Lee, Byeong-Joo; Liu, Zi-Kui; Miodownik, Peter; Seifert, Hans
Juergen (2007-03-01).
2597:
6178:
Pokrovskii V.N. (2013) A derivation of the main relations of non-equilibrium thermodynamics. Hindawi
Publishing Corporation: ISRN Thermodynamics, vol. 2013, article ID 906136, 9 p.
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in equilibrium thermodynamics. That is why in such cases a more generalized
Legendre transformation should be considered. This is the extended Massieu potential. By definition, the
2551:
1925:
in the twentieth. These effects occur at metal junctions, which were originally effectively treated as two-dimensional surfaces, with no spatial volume, and no spatial variation.
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1949:
is a branch of non-equilibrium thermodynamics that goes outside the restriction to the local equilibrium hypothesis. The space of state variables is enlarged by including the
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1961:
There are many examples of stationary non-equilibrium systems, some very simple, like a system confined between two thermostats at different temperatures or the ordinary
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of mass, momentum and energy and eventually higher order fluxes. The formalism is well-suited for describing high-frequency processes and small-length scales materials.
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are equal to zero. The above equation is valid for small deviations from equilibrium; The dynamics of internal variables in general case is considered by
Pokrovskii.
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Barthélemy, Olivier; Margot, Joëlle; Laville, Stéphane; Vidal, François; Chaker, Mohamed; Le
Drogoff, Boris; Johnston, Tudor W.; Sabsabi, Mohamad (April 2005).
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The first term on the right hand side of the equation presents a stream of thermal energy into the system; the last term—a part of a stream of energy
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of the system, is their tending to disappear; the local law of disappearing can be written as relaxation equation for each internal variable
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2002:
of the energy. If, next to fluctuations of the energy, the macroscopic dimensions (volume) of the system are left fluctuating, we use the
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511:
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Attard, P. (2012). "Optimising Principle for Non-Equilibrium Phase Transitions and Pattern Formation with Results for Heat Convection".
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have been introduced. The equilibrium state is considered to be stable and the main property of the internal variables, as measures of
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Some concepts of particular importance for non-equilibrium thermodynamics include time rate of dissipation of energy (Rayleigh 1873,
1367:
5075:"Applications of computational thermodynamics — the extension from phase equilibrium to phase transformations and other properties"
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1400:
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3203:{\displaystyle T\,dS=\Delta Q-\sum _{j}\,\Xi _{j}\,\Delta \xi _{j}+\sum _{\alpha =1}^{k}\,\eta _{\alpha }\,\Delta N_{\alpha }.}
5588:
Essex, C. (1984c). "Radiation and the violation of bilinearity in the irreversible thermodynamics of irreversible processes".
1965:, a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls.
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Bazant, Martin Z. (22 March 2013). "Theory of Chemical Kinetics and Charge Transfer based on Nonequilibrium Thermodynamics".
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2746:{\displaystyle {\frac {d\xi _{i}}{dt}}=-{\frac {1}{\tau _{i}}}\,\left(\xi _{i}-\xi _{i}^{(0)}\right),\quad i=1,\,2,\ldots ,}
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Stephan Herminghaus' Dynamics of Complex Fluids Department at the Max Planck Institute for Dynamics and Self Organization
4421:{\displaystyle \sigma =\sum _{i,j}L_{ij}{\frac {\partial F_{i}}{\partial x_{i}}}{\frac {\partial F_{j}}{\partial x_{j}}}}
2095:(a restricted definition of intensive variable is used here by comparison to the definition given in this link) so that:
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57:
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Establishing the relation between such forces and flux densities is a problem in statistical mechanics. Flux densities (
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947:
401:
6854:
6374:
Kimizuka, Hideo; Kaibara, Kozue (September 1975). "Nonequilibrium thermodynamics of ion transport through membranes".
6317:"Qualms Regarding the Range of Validity of the Glansdorff-Prigogine Criterion for Stability of Non-Equilibrium States"
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A further extension of local equilibrium thermodynamics is to allow that materials may have "memory", so that their
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exchange of matter or energy with the system's surroundings that create the constraints that define the process.
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Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.
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5797:
Coleman, B.D.; Noll, W. (1963). "The thermodynamics of elastic materials with heat conduction and viscosity".
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4735:. Dover books on physics (Dover ed., 1. publ., unabridged, corr. republ ed.). New York, NY: Dover Publ.
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Following Section III of Rayleigh (1873), Onsager (1931, I) showed that in the regime where both the flows (
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168:
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To describe deviation of the thermodynamic system from equilibrium, in addition to constitutive variables
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The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by
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53:
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The generalization of the above equations for the rate of creation of entropy was given by Pokrovskii.
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Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics
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4860:"Criteria for validity of thermodynamic equations from non-equilibrium molecular dynamics simulations"
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Non-equilibrium thermodynamics has been successfully applied to describe biological processes such as
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1726:
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Nicolis, C. (1999). "Entropy production and dynamical complexity in a low-order atmospheric model".
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4973:"Investigation of the State of Local Thermodynamic Equilibrium of a Laser-Produced Aluminum Plasma"
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is a relaxation time of a corresponding variables. It is convenient to consider the initial value
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1999:
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Baranowski, B. (April 1991). "Non-equilibrium thermodynamics as applied to membrane transport".
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Wildt, R. (1972). "Thermodynamics of the gray atmosphere. IV. Entropy transfer and production".
4673:; Ahlers, Guenter (December 1992). "Experiments on three systems with non-variational aspects".
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3569:{\displaystyle dS={\frac {1}{T}}dU+{\frac {p}{T}}dV-\sum _{i=1}^{s}{\frac {\mu _{i}}{T}}dN_{i}}
2453:, thinking about stars, gave a definition of 'local thermodynamic equilibrium' in terms of the
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1312:
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150:
125:
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5742:
4910:"Irreversible thermodynamics—a tool to describe phase transitions far from global equilibrium"
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2022:), where the system's properties are determined both by the temperature and by the pressure.
1983:
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5747:. Series on Advances in Statistical Mechanics. Vol. 20 (2 ed.). WORLD SCIENTIFIC.
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1831:, and generalized thermodynamics, but they are hardly touched on in the present article.
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6857:- 2005 book by Dorion Sagan and Eric D. Schneider, on nonequilibrium thermodynamics and
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Statistical Mechanics of Nonequilibrium Processes: Relaxation and Hydrodynamic Processes
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Bodenschatz, Eberhard; Cannell, David S.; de Bruyn, John R.; Ecke, Robert; Hu, Yu-Chou;
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6597:, translated from the Hungarian (1967) by E. Gyarmati and W.F. Heinz, Springer, Berlin.
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that are used to fix the equilibrium state, as was described above, a set of variables
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88:
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Non-equilibrium Statistical Thermodynamics applied to Fluid Dynamics and Laser Physics
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Entropy of the system in non-equilibrium is a function of the total set of variables
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4937:
4887:
4837:
4829:
4736:
4694:
4649:
4559:
4463:
considerations involving microscopic reversibility of dynamics imply that the matrix
2454:
2003:
1835:
Quasi-radiationless non-equilibrium thermodynamics of matter in laboratory conditions
1773:
1639:
1600:
1590:
1162:
960:
788:
290:
280:
222:
6475:
6041:
5984:
5153:
5020:
4790:
Hafskjold, Bjørn; Bedeaux, Dick; Kjelstrup, Signe; Wilhelmsen, Øivind (2021-07-23).
1749:
7146:
7086:
7012:
6865:
6455:
6418:
6391:
6346:
6336:
6259:
6076:
6029:
5972:
5921:
5814:
5748:
5605:
5568:
5526:
5487:
5278:
5230:
5188:
5137:
5133:
5086:
5047:
4992:
4929:
4879:
4821:
4690:
4641:
4484:
2166:
1801:
1931, also), time rate of entropy production (Onsager 1931), thermodynamic fields,
1560:
1545:
1485:
1480:
1297:
1292:
942:
410:
275:
5409:
Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers
5090:
7166:
7076:
7022:
6922:
6729:
Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory
6600:
6507:
5673:
4883:
4670:
4584:
4574:
4518:
4102:
3786:
2604:
1510:
1358:
1012:
653:
476:
237:
204:
5999:
4825:
4792:"Theory and simulation of shock waves: Entropy production and energy conversion"
2389:
thermodynamic forces driving fluxes of extensive properties through the system.
7096:
7051:
7029:
6917:
6742:
6724:
6681:
6671:
6604:
6560:
6526:
6191:
5942:
5863:
4497:
3999:
1974:, turbulent systems and glasses are other examples of non-equilibrium systems.
1971:
1760:
1705:
1565:
1335:
435:
315:
252:
242:
110:
80:
5872:, translated from the Russian by P.J. Shepherd, New York, Consultants Bureau.
5234:
4933:
4178:{\displaystyle \sigma =\sum _{i}J_{i}{\frac {\partial F_{i}}{\partial x_{i}}}}
7227:
7106:
5926:
5899:
5200:
5145:
5114:"Rapid phase transformation under local non-equilibrium diffusion conditions"
5098:
5059:
5004:
4996:
4941:
4891:
4833:
4295:{\displaystyle J_{i}=\sum _{j}L_{ij}{\frac {\partial F_{j}}{\partial x_{j}}}}
1634:
952:
521:
482:
194:
6871:
7131:
6607:(1999). 'The physics and mathematics of the second law of thermodynamics',
6467:
6360:
6263:
5283:
5258:
5012:
4841:
4539:
4526:
4188:
and the flows are related to the gradient of the forces, parametrized by a
1962:
1922:
1809:
1798:
1585:
1570:
1520:
1003:
6341:
35:
6806:
6616:
6179:
6016:
5959:
5072:
4604:
3606:
2463:
1909:, and even in shock fronts moving at up to six times the speed of sound.
1540:
348:
6218:
Thermodynamics and Statistical Mechanics (Classical Theoretical Physics)
7151:
7141:
6848:
6595:
Non-equilibrium Thermodynamics. Field Theory and Variational Principles
6509:
The Unity of Science and Economics: A New Foundation of Economic Theory
5818:
5549:"Minimum entropy production in the steady state and radiative transfer"
2445:
1629:
1575:
6459:
1808:
One problem of interest is the thermodynamic study of non-equilibrium
7002:
6858:
3033:
227:
5218:
4858:
Kjelstrup, S.; Bedeaux, D.; Inzoli, I.; Simon, J. -M. (2008-08-01).
6057:"Time Evolution in Macroscopic Systems. III: Selected Applications"
5573:
5548:
5491:
4808:
4789:
3629:
3005:{\displaystyle S=S(T,x_{1},x_{2},,x_{n};\xi _{1},\xi _{2},\ldots )}
1343:
1260:
1052:
460:
232:
6792:
Modern Thermodynamics: From Heat Engines to Dissipative Structures
6450:
6300:
5841:
Modern Thermodynamics. From Heat Engines to Dissipative Structures
5752:
3458:
The fundamental relation of classical equilibrium thermodynamics
3253:
coming into the system with the stream of particles of substances
5900:"The effect of collisions on monochromatic radiative equilibrium"
4645:
3580:
3057:
2396:
2031:
2027:
1813:
1750:
Difference between equilibrium and non-equilibrium thermodynamics
1737:
449:
6161:
Thermodynamics of Materials with Memory: Theory and Applications
5169:"Deviation from local equilibrium at migrating phase interfaces"
4634:
Kjelstrup, S; Bedeaux, D; Johannessen, E; Gross, J (June 2010).
4633:
2856:{\displaystyle \tau _{i}=\tau _{i}(T,x_{1},x_{2},\ldots ,x_{n})}
5943:"Time Evolution in Macroscopic Systems. I. Equations of Motion"
5403:
5401:
5399:
3711:
2458:
2155:{\displaystyle I_{i}={\frac {\partial {S}}{\partial {E_{i}}}}.}
6762:
Atmospheric turbulence : a molecular dynamics perspective
6565:
Thermodynamic Theory of Structure, Stability, and Fluctuations
5383:
Thermodynamic Theory of Structure, Stability, and Fluctuations
4710:
Thermodynamics of Complex Systems: Principles and applications
4668:
2068:. Each extensive quantity has a conjugate intensive variable
1966:
1929:
Local equilibrium thermodynamics with materials with "memory"
6658:
Non-equilibrium Thermodynamics and the Production of Entropy
5396:
4970:
4498:
Speculated extremal principles for non-equilibrium processes
1917:
known as the Seebeck and the Peltier effects, considered by
5532:
10.1175/1520-0469(1984)041<1985:RATITO>2.0.CO;2
4857:
4525:. It is also used to give a description of the dynamics of
2375:{\displaystyle \ k_{\rm {B}}\,dM=\sum _{i}(E_{i}\,dI_{i}).}
1950:
1817:
425:
5507:"Radiation and the irreversible thermodynamics of climate"
4575:
Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations
3329:{\displaystyle \eta _{\alpha }=h_{\alpha }-\mu _{\alpha }}
2467:
exceed the rates of creation and annihilation of photons.
6676:
Introduction to Thermodynamics of Irreversible Processes
6533:, (1st edition 1960) 2nd edition 1985, Wiley, New York,
6196:
Introduction to Thermodynamics of Irreversible Processes
6000:"Time Evolution in Macroscopic Systems. II. The Entropy"
3028:
The essential contribution to the thermodynamics of the
6210:
5744:
Non-equilibrium Thermodynamics of Heterogeneous Systems
5658:
5656:
2248:{\displaystyle \ k_{\rm {B}}M=S-\sum _{i}(I_{i}E_{i}),}
1941:
6711:
Statistical Thermodynamics of Nonequilibrium Processes
6531:
Thermodynamics and an Introduction to Thermostatistics
5721:
Statistical Thermodynamics of Nonequilibrium Processes
5697:
Probability and Heat: Fundamentals of Thermostatistics
3605:
of a system as a function of the intensive quantities
6777:
Entropy and the Time Evolution of Macroscopic Systems
6580:
Entropy and the Time Evolution of Macroscopic Systems
6244:
Quarterly Journal of the Royal Meteorological Society
5640:
Equilibrium and Non-equilibrium Statistical Mechanics
5357:
Equilibrium and Non-equilibrium Statistical Mechanics
4769:
4767:
4765:
4555:
Extremal principles in non-equilibrium thermodynamics
4504:
Extremal principles in non-equilibrium thermodynamics
4469:
4441:
4314:
4221:
4198:
4114:
4081:
4054:
4027:
3973:
3931:
3911:
3891:
3864:
3844:
3824:
3794:
3762:
3742:
3719:
3710:
and of the differentials of the extensive quantities
3689:
3657:
3637:
3614:
3588:
3467:
3428:
3401:
3373:
3342:
3289:
3259:
3232:
3090:
3065:
3042:
2909:
2869:
2775:
2618:
2559:
2494:
2303:
2264:
2178:
2104:
2074:
2047:
1886:
1859:
1171:
1116:
1061:
1021:
895:
874:
848:
827:
799:
763:
742:
716:
695:
664:
628:
607:
581:
560:
532:
5859:
5857:
5653:
2446:
Milne's definition in terms of radiative equilibrium
5259:"Reciprocal relations in irreversible processes, I"
1843:
5741:Kjelstrup, Signe; Bedeaux, Dick (September 2020).
5036:"Local heat changes during aluminium electrolysis"
4762:
4475:
4447:
4420:
4294:
4204:
4177:
4093:
4067:
4040:
4007:though in special cases they can be so described.
3986:
3944:
3917:
3897:
3877:
3850:
3830:
3807:
3778:
3748:
3725:
3702:
3673:
3643:
3620:
3597:
3568:
3441:
3414:
3379:
3355:
3328:
3275:
3245:
3202:
3071:
3048:
3004:
2887:
2855:
2745:
2591:
2545:
2413:
2374:
2282:
2247:
2154:
2087:
2060:
1901:
1872:
1779:
1207:
1152:
1097:
1042:
904:
880:
857:
833:
808:
772:
748:
725:
701:
676:
637:
613:
590:
566:
541:
6495:. Springer, Dordrecht-Heidelberg-London-New York.
5905:Monthly Notices of the Royal Astronomical Society
5854:
5252:
5250:
5248:
5246:
5244:
5166:
1708:that deals with physical systems that are not in
7225:
5740:
5691:
5689:
5212:
5210:
5033:
4957:
4955:
4953:
4951:
4907:
6373:
5634:
5632:
5377:
5375:
5373:
5300:
5298:
5296:
5294:
4075:) vary slowly, the rate of creation of entropy
4010:
2385:The independent variables are the intensities.
6849:Nonequilibrium Thermodynamics of Small Systems
6492:Econodynamics. The Theory of Social Production
5715:
5713:
5431:Jou, D., Casas-Vázquez, J., Lebon, G. (1993).
5407:Lebon, G., Jou, D., Casas-Vázquez, J. (2008).
5241:
5223:Proceedings of the London Mathematical Society
5219:"Some General Theorems relating to Vibrations"
4908:Bedeaux, Dick; Kjelstrup, Signe (2004-01-01).
4731:Groot, Sybren Ruurds de; Mazur, Peter (1984).
3395:) due to the relaxation of internal variables
2470:
2397:Stationary states, fluctuations, and stability
6901:
6887:
6825:The Non-Equilibrium Thermodynamics, Perpetual
6216:W. Greiner, L. Neise, and H. Stöcker (1997),
6174:
6172:
6170:
5686:
5207:
4948:
2483:
1682:
6678:. 3rd edition, Wiley Interscience, New York.
6198:. 3rd edition, Wiley Interscience, New York.
6146:The Mechanics and Thermodynamics of Continua
5629:
5427:
5425:
5370:
5328:
5291:
4637:Non-Equilibrium Thermodynamics for Engineers
3885:and of the intensive macroscopic quantities
3079:can be calculated according to the formula
1772:, and provides a theoretical foundation for
6656:Kleidon, A., Lorenz, R.D., editors (2005).
6308:
6287:
6278:
5796:
5710:
5699:, Freidr. Vieweg & Sohn, Braunschweig,
5626:, Longmans, Green & Co, London, page 1.
5581:
5540:
5498:
6894:
6880:
6488:
6408:
6167:
5891:
5463:
5167:Hillert, M.; Rettenmayr, M. (2003-06-11).
4755:Glansdorff, P., Prigogine, I. (1971), Ch.
4730:
4707:
4565:Autocatalytic reactions and order creation
4048:) are small and the thermodynamic forces (
1689:
1675:
87:
6866:Thermodynamics "beyond" local equilibrium
6823:Ramiro Augusto Salazar La Rotta. (2011).
6687:Nonequilibrium Statistical Thermodynamics
6548:, Kluwer Academic Publishers, Dordrecht,
6449:
6350:
6340:
6314:
6299:
6113:
6015:
5958:
5925:
5869:Nonequilibrium Statistical Thermodynamics
5572:
5530:
5422:
5282:
4807:
3183:
3172:
3134:
3123:
3094:
2730:
2669:
2592:{\displaystyle \xi _{1},\xi _{2},\ldots }
2432:Local thermodynamic equilibrium of matter
2352:
2319:
6376:Journal of Colloid and Interface Science
6158:
5663:Mihalas, D., Weibel-Mihalas, B. (1984).
5306:Thermodynamics of Irreversible Processes
6241:
5349:
5256:
5111:
4192:of coefficients conventionally denoted
1741:obey the concept of local equilibrium.
14:
7226:
7092:Atomic, molecular, and optical physics
6435:
6293:
6180:https://dx.doi.org/10.1155/2013/906136
6143:
6128:
6054:
5997:
5940:
5665:Foundations of Radiation Hydrodynamics
5622:Prigogine, I., Defay, R. (1950/1954).
5587:
5546:
5504:
5381:Glansdorff, P., Prigogine, I. (1971).
5216:
5034:Kjelstrup Ratkje, Signe (1991-01-01).
1805:, and non-linear dynamical structure.
6875:
6794:. John Wiley & Sons, Chichester.
6790:Kondepudi, D., Prigogine, I. (1998).
5897:
5839:Kondepudi, D., Prigogine, I, (1998).
5777:Introduction to Modern Thermodynamics
5469:
5325:Gyarmati, I. (1967/1970), pages 4-14.
2546:{\displaystyle x_{1},x_{2},...,x_{n}}
2038:) is a function of the collection of
6845:- 1992- book by Xavier de Hemptinne.
6567:, Wiley-Interscience, London, 1971,
6505:
6315:Keizer, J.; Fox, R. (January 1974).
5433:Extended Irreversible Thermodynamics
5385:, Wiley-Interscience, London, 1971,
4903:
4901:
4853:
4851:
4785:
4783:
4629:
4627:
4625:
3081:
2900:
2609:
2440:Extended irreversible thermodynamics
2426:
1947:Extended irreversible thermodynamics
1942:Extended irreversible thermodynamics
1829:extended irreversible thermodynamics
29:
5667:, Oxford University Press, New York
5642:, John Wiley & Sons, New York,
5511:Journal of the Atmospheric Sciences
3453:
24:
6635:An introduction to Thermomechanics
6623:
6081:10.1023/B:FOOP.0000022187.45866.81
6034:10.1023/B:FOOP.0000012008.36856.c1
5977:10.1023/B:FOOP.0000012007.06843.ed
5336:An Introduction to Thermomechanics
4773:De Groot, S.R., Mazur, P. (1962).
4402:
4387:
4368:
4353:
4276:
4261:
4159:
4144:
3283:that can be positive or negative,
3276:{\displaystyle \Delta N_{\alpha }}
3260:
3184:
3135:
3125:
3104:
2313:
2274:
2188:
2131:
2121:
896:
849:
764:
717:
629:
582:
402:Intensive and extensive properties
25:
7250:
6831:
6159:Amendola, Giovambattista (2012).
4898:
4848:
4780:
4622:
3387:. The middle term in (1) depicts
1956:
1921:in the nineteenth century and by
6690:. New York, Consultants Bureau.
5359:, Wiley-Interscience, New York,
5118:Materials Science and Technology
1844:Local equilibrium thermodynamics
1658:
1657:
977:Table of thermodynamic equations
34:
7213:Timeline of physics discoveries
6745:, Morozov V., Ropke G. (1997):
6727:, Morozov V., Ropke G. (1996):
6499:
6482:
6429:
6402:
6367:
6235:
6201:
6185:
6152:
6137:
6122:
6107:
6095:
6048:
5991:
5934:
5833:
5790:
5769:
5734:
5616:
5454:
5319:
5160:
5112:Sobolev, S. L. (October 2015).
5105:
5066:
5027:
4964:
4712:. IOP Publishing, Bristol, UK.
4512:
2717:
2421:local thermodynamic equilibrium
2414:Local thermodynamic equilibrium
1780:Non-equilibrium state variables
1453:Maxwell's thermodynamic surface
7234:Non-equilibrium thermodynamics
6811:Non-Equilibrium Thermodynamics
5138:10.1179/1743284715Y.0000000051
4775:Non-equilibrium Thermodynamics
4749:
4733:Non-equilibrium thermodynamics
4724:
4701:
4675:Physica D: Nonlinear Phenomena
4662:
4088:
4082:
3356:{\displaystyle \mu _{\alpha }}
2999:
2919:
2850:
2799:
2704:
2698:
2366:
2339:
2239:
2216:
1820:are non-zero, but there is no
1702:Non-equilibrium thermodynamics
1187:
1175:
1132:
1120:
1077:
1065:
1037:
1025:
13:
1:
6713:, Springer-Verlag, New York,
6489:Pokrovskii, Vladimir (2011).
6438:Accounts of Chemical Research
6423:10.1016/S0376-7388(00)80675-4
6148:. Cambridge University Press.
5723:, Springer-Verlag, New York,
5411:, Springer-Verlag, Berlin, e-
5193:10.1016/S1359-6454(03)00085-5
5091:10.1016/j.calphad.2006.02.006
4708:Pokrovskii, Vladimir (2020).
4616:
4600:Spontaneous symmetry breaking
2283:{\displaystyle \ k_{\rm {B}}}
1354:Mechanical equivalent of heat
6637:. North Holland, Amsterdam.
6396:10.1016/0021-9797(75)90276-3
6220:,Springer-Verlag, New York,
6114:Truesdell, Clifford (1984).
5799:Arch. Ration. Mach. Analysis
5610:10.1016/0032-0633(84)90060-6
5338:, North-Holland, Amsterdam,
5052:10.1016/0013-4686(91)85155-Z
4914:Chemical Engineering Science
4884:10.1016/j.energy.2008.04.005
4695:10.1016/0167-2789(92)90150-L
4489:Onsager reciprocal relations
4433:second law of thermodynamics
4305:from which it follows that:
4017:Onsager reciprocal relations
4011:Onsager reciprocal relations
3962:Onsager reciprocal relations
2888:{\displaystyle \xi _{i}^{0}}
2165:We then define the extended
966:Onsager reciprocal relations
7:
7177:Quantum information science
6779:. Oxford University Press.
6764:. Oxford University Press.
6582:. Oxford University Press.
6411:Journal of Membrane Science
5843:, Wiley, Chichester, 1998,
5590:Planetary and Space Science
5367:, Section 3.2, pages 64-72.
4826:10.1103/PhysRevE.104.014131
4777:, North-Holland, Amsterdam.
4570:Self-organizing criticality
4532:
4523:transport through membranes
3246:{\displaystyle h_{\alpha }}
3216:
3018:
2759:
2471:Entropy in evolving systems
1792:
1458:Entropy as energy dispersal
1269:"Perpetual motion" machines
1208:{\displaystyle G(T,p)=H-TS}
1153:{\displaystyle A(T,V)=U-TS}
1098:{\displaystyle H(S,p)=U+pV}
47:to comply with Knowledge's
10:
7255:
7239:Branches of thermodynamics
7008:Classical electromagnetism
6593:Gyarmati, I. (1967/1970).
6519:
6144:Gurtin, Morton E. (2010).
6129:Maugin, Gérard A. (2002).
4501:
4487:. This fact is called the
4014:
2484:Entropy in non-equilibrium
1734:equilibrium thermodynamics
905:{\displaystyle \partial T}
858:{\displaystyle \partial V}
773:{\displaystyle \partial p}
726:{\displaystyle \partial V}
638:{\displaystyle \partial T}
591:{\displaystyle \partial S}
7185:
7122:
7050:
6966:
6938:
6910:
6749:. John Wiley & Sons.
6731:. John Wiley & Sons.
6131:Continuum Thermomechanics
6055:Grandy, W. T. Jr (2004).
5998:Grandy, W.T. Jr. (2004).
5941:Grandy, W.T. Jr. (2004).
4961:Gyarmati, I. (1967/1970).
4934:10.1016/j.ces.2003.09.028
4435:requires that the matrix
4094:{\displaystyle (\sigma )}
1710:thermodynamic equilibrium
1379:An Inquiry Concerning the
7114:Condensed matter physics
6775:Grandy, W.T. Jr (2008).
6760:Tuck, Adrian F. (2008).
6645:. Second edition (1983)
6578:Grandy, W.T. Jr (2008).
6284:Grandy, W.T., Jr (2008).
5779:, Wiley, Chichester UK,
4997:10.1366/0003702053641531
3945:{\displaystyle \mu _{i}}
3703:{\displaystyle \mu _{i}}
3579:expresses the change in
3442:{\displaystyle \xi _{j}}
3415:{\displaystyle \xi _{j}}
1915:thermoelectric phenomena
1744:
1392:Heterogeneous Substances
809:{\displaystyle \alpha =}
677:{\displaystyle \beta =-}
60:may contain suggestions.
45:may need to be rewritten
27:Branch of thermodynamics
6118:(2 ed.). Springer.
6116:Rational Thermodynamics
5624:Chemical Thermodynamics
5602:1984P&SS...32.1035E
5235:10.1112/plms/s1-4.1.357
4610:Maximum power principle
3380:{\displaystyle \alpha }
3030:non-equilibrium systems
2000:Legendre transformation
1873:{\displaystyle 10^{12}}
1824:of physical variables.
7198:Nobel Prize in Physics
7060:Relativistic mechanics
6264:10.1002/qj.49712555718
6061:Foundations of Physics
6004:Foundations of Physics
5947:Foundations of Physics
5927:10.1093/mnras/88.6.493
5775:Kondepudi, D. (2008).
5304:Lavenda, B.H. (1978).
5284:10.1103/PhysRev.37.405
5217:Strutt, J. W. (1871).
4477:
4449:
4422:
4296:
4206:
4179:
4095:
4069:
4042:
3988:
3946:
3919:
3899:
3879:
3852:
3832:
3809:
3780:
3779:{\displaystyle i^{th}}
3750:
3727:
3704:
3675:
3674:{\displaystyle i^{th}}
3645:
3622:
3599:
3570:
3535:
3443:
3416:
3381:
3357:
3330:
3277:
3247:
3204:
3171:
3073:
3050:
3006:
2889:
2857:
2747:
2593:
2547:
2376:
2284:
2249:
2156:
2089:
2062:
1935:constitutive equations
1903:
1874:
1717:and with the rates of
1209:
1154:
1099:
1044:
1043:{\displaystyle U(S,V)}
906:
882:
859:
835:
810:
774:
750:
727:
703:
678:
639:
615:
592:
568:
543:
522:Specific heat capacity
126:Quantum thermodynamics
7203:Philosophy of physics
6342:10.1073/pnas.71.1.192
6101:Grandy 2004 see also
5695:Schloegl, F. (1989).
5553:Astrophysical Journal
5472:Astrophysical Journal
5334:Ziegler, H., (1983).
5308:, Macmillan, London,
4478:
4461:Statistical mechanics
4450:
4423:
4297:
4207:
4180:
4096:
4070:
4068:{\displaystyle F_{i}}
4043:
4041:{\displaystyle J_{i}}
3989:
3987:{\displaystyle J_{i}}
3947:
3920:
3900:
3880:
3878:{\displaystyle N_{i}}
3853:
3833:
3810:
3808:{\displaystyle N_{i}}
3781:
3751:
3728:
3705:
3676:
3646:
3623:
3600:
3571:
3515:
3444:
3417:
3382:
3358:
3331:
3278:
3248:
3205:
3151:
3074:
3051:
3007:
2890:
2858:
2748:
2594:
2548:
2377:
2285:
2250:
2157:
2090:
2088:{\displaystyle I_{i}}
2063:
2061:{\displaystyle E_{i}}
1984:Helmholtz free energy
1904:
1902:{\displaystyle ^{-1}}
1875:
1803:dissipative structure
1390:On the Equilibrium of
1210:
1155:
1108:Helmholtz free energy
1100:
1045:
907:
883:
860:
836:
811:
775:
751:
728:
704:
679:
640:
616:
593:
569:
544:
7162:Mathematical physics
6660:, Springer, Berlin.
6222:P85, 91, 101,108,116
5898:Milne, E.A. (1928).
5638:Balescu, R. (1975).
5435:, Springer, Berlin,
5355:Balescu, R. (1975).
5257:Onsager, L. (1931).
4977:Applied Spectroscopy
4640:. WORLD SCIENTIFIC.
4467:
4439:
4312:
4219:
4196:
4112:
4079:
4052:
4025:
3971:
3929:
3909:
3889:
3862:
3842:
3822:
3792:
3760:
3740:
3717:
3687:
3655:
3635:
3612:
3586:
3465:
3426:
3399:
3371:
3340:
3287:
3257:
3230:
3088:
3063:
3040:
2907:
2867:
2773:
2616:
2557:
2492:
2301:
2262:
2176:
2102:
2072:
2045:
2040:extensive quantities
1980:extensive quantities
1884:
1857:
1816:production and some
1770:process optimisation
1403:Motive Power of Fire
1169:
1114:
1059:
1019:
971:Bridgman's equations
948:Fundamental relation
893:
872:
846:
825:
797:
761:
740:
714:
693:
662:
626:
605:
579:
558:
530:
7137:Atmospheric physics
6976:Classical mechanics
6904:branches of physics
6859:evolutionary theory
6709:Keizer, J. (1987).
6506:Chen, Jing (2015).
6388:1975JCIS...52..516K
6333:1974PNAS...71..192K
6256:1999QJRMS.125.1859N
6073:2004FoPh...34..771G
6026:2004FoPh...34...21G
5969:2004FoPh...34....1G
5918:1928MNRAS..88..493M
5811:1963ArRMA..13..167C
5719:Keizer, J. (1987).
5565:1984ApJ...285..279E
5547:Essex, C. (1984b).
5523:1984JAtS...41.1985E
5505:Essex, C. (1984a).
5484:1972ApJ...174...69W
5275:1931PhRv...37..405O
5185:2003AcMat..51.2803H
5130:2015MatST..31.1607S
5040:Electrochimica Acta
4989:2005ApSpe..59..529B
4926:2004ChEnS..59..109B
4876:2008Ene....33.1185K
4818:2021PhRvE.104a4131H
4718:2020tcsp.book.....P
4687:1992PhyD...61...77B
4595:Information entropy
3056:. The increment of
2884:
2708:
1715:transport processes
1381:Source ... Friction
1313:Loschmidt's paradox
505:Material properties
383:Conjugate variables
7193:History of physics
6851:- PhysicsToday.org
6674:(1955/1961/1967).
6250:(557): 1859–1878.
6194:(1955/1961/1967).
5819:10.1007/bf01262690
5672:2011-10-08 at the
4580:Boltzmann equation
4550:Entropy production
4545:Dissipative system
4473:
4445:
4418:
4336:
4292:
4244:
4202:
4175:
4130:
4091:
4065:
4038:
3984:
3942:
3915:
3895:
3875:
3848:
3828:
3805:
3776:
3746:
3723:
3700:
3682:chemical potential
3671:
3641:
3618:
3598:{\displaystyle dS}
3595:
3566:
3439:
3412:
3393:entropy production
3389:energy dissipation
3377:
3365:chemical potential
3353:
3326:
3273:
3243:
3200:
3122:
3069:
3046:
3002:
2885:
2870:
2853:
2743:
2688:
2601:internal variables
2589:
2543:
2372:
2338:
2292:Boltzmann constant
2280:
2245:
2215:
2152:
2085:
2058:
1899:
1870:
1719:chemical reactions
1645:Order and disorder
1401:Reflections on the
1308:Heat death paradox
1205:
1150:
1095:
1040:
902:
878:
855:
831:
806:
770:
746:
723:
699:
674:
635:
611:
588:
564:
542:{\displaystyle c=}
539:
512:Property databases
488:Reduced properties
472:Chemical potential
436:Functions of state
359:Thermal efficiency
95:Carnot heat engine
7221:
7220:
7208:Physics education
7157:Materials science
7124:Interdisciplinary
7082:Quantum mechanics
6785:978-0-19-954617-6
6770:978-0-19-923653-4
6704:978-0-306-10895-2
6588:978-0-19-954617-6
6544:Eu, B.C. (2002).
6460:10.1021/ar300145c
5886:978-0-306-10895-2
5785:978-0-470-01598-8
5762:978-981-12-1676-3
5517:(12): 1985–1991.
5417:978-3-540-74252-4
5179:(10): 2803–2809.
5124:(13): 1607–1617.
4796:Physical Review E
4742:978-0-486-64741-8
4655:978-981-4322-15-7
4560:Self-organization
4476:{\displaystyle L}
4457:positive definite
4448:{\displaystyle L}
4416:
4382:
4321:
4290:
4235:
4205:{\displaystyle L}
4173:
4121:
3918:{\displaystyle p}
3898:{\displaystyle T}
3851:{\displaystyle V}
3831:{\displaystyle U}
3749:{\displaystyle V}
3726:{\displaystyle U}
3644:{\displaystyle p}
3621:{\displaystyle T}
3551:
3504:
3485:
3224:
3223:
3113:
3072:{\displaystyle S}
3049:{\displaystyle T}
3026:
3025:
2767:
2766:
2667:
2644:
2455:thermal radiation
2427:Ponderable matter
2329:
2306:
2267:
2206:
2181:
2147:
2004:Gibbs free energy
1699:
1698:
1640:Self-organization
1465:
1464:
1163:Gibbs free energy
961:Maxwell relations
919:
918:
915:
914:
881:{\displaystyle V}
834:{\displaystyle 1}
789:Thermal expansion
783:
782:
749:{\displaystyle V}
702:{\displaystyle 1}
648:
647:
614:{\displaystyle N}
567:{\displaystyle T}
495:
494:
411:Process functions
397:Property diagrams
376:System properties
366:
365:
331:Endoreversibility
223:Equation of state
75:
74:
49:quality standards
16:(Redirected from
7246:
7147:Chemical physics
7087:Particle physics
7013:Classical optics
6896:
6889:
6882:
6873:
6872:
6559:Glansdorff, P.,
6514:
6513:
6503:
6497:
6496:
6486:
6480:
6479:
6453:
6444:(5): 1144–1160.
6433:
6427:
6426:
6417:(2–3): 119–159.
6406:
6400:
6399:
6371:
6365:
6364:
6354:
6344:
6312:
6306:
6305:
6303:
6291:
6285:
6282:
6276:
6275:
6239:
6233:
6214:
6208:
6205:
6199:
6189:
6183:
6176:
6165:
6164:
6156:
6150:
6149:
6141:
6135:
6134:
6126:
6120:
6119:
6111:
6105:
6099:
6093:
6092:
6052:
6046:
6045:
6019:
6017:cond-mat/0303291
5995:
5989:
5988:
5962:
5960:cond-mat/0303290
5938:
5932:
5931:
5929:
5895:
5889:
5861:
5852:
5837:
5831:
5830:
5794:
5788:
5787:, pages 333-338.
5773:
5767:
5766:
5738:
5732:
5717:
5708:
5693:
5684:
5660:
5651:
5636:
5627:
5620:
5614:
5613:
5596:(8): 1035–1043.
5585:
5579:
5578:
5576:
5544:
5538:
5536:
5534:
5502:
5496:
5495:
5467:
5461:
5460:Eu, B.C. (2002).
5458:
5452:
5429:
5420:
5405:
5394:
5379:
5368:
5353:
5347:
5332:
5326:
5323:
5317:
5302:
5289:
5288:
5286:
5254:
5239:
5238:
5214:
5205:
5204:
5164:
5158:
5157:
5109:
5103:
5102:
5070:
5064:
5063:
5031:
5025:
5024:
4968:
4962:
4959:
4946:
4945:
4905:
4896:
4895:
4870:(8): 1185–1196.
4855:
4846:
4845:
4811:
4787:
4778:
4771:
4760:
4758:
4753:
4747:
4746:
4728:
4722:
4721:
4705:
4699:
4698:
4671:Lerman, Kristina
4666:
4660:
4659:
4631:
4482:
4480:
4479:
4474:
4454:
4452:
4451:
4446:
4427:
4425:
4424:
4419:
4417:
4415:
4414:
4413:
4400:
4399:
4398:
4385:
4383:
4381:
4380:
4379:
4366:
4365:
4364:
4351:
4349:
4348:
4335:
4301:
4299:
4298:
4293:
4291:
4289:
4288:
4287:
4274:
4273:
4272:
4259:
4257:
4256:
4243:
4231:
4230:
4211:
4209:
4208:
4203:
4184:
4182:
4181:
4176:
4174:
4172:
4171:
4170:
4157:
4156:
4155:
4142:
4140:
4139:
4129:
4103:linearly related
4100:
4098:
4097:
4092:
4074:
4072:
4071:
4066:
4064:
4063:
4047:
4045:
4044:
4039:
4037:
4036:
3993:
3991:
3990:
3985:
3983:
3982:
3951:
3949:
3948:
3943:
3941:
3940:
3924:
3922:
3921:
3916:
3904:
3902:
3901:
3896:
3884:
3882:
3881:
3876:
3874:
3873:
3857:
3855:
3854:
3849:
3837:
3835:
3834:
3829:
3814:
3812:
3811:
3806:
3804:
3803:
3785:
3783:
3782:
3777:
3775:
3774:
3755:
3753:
3752:
3747:
3732:
3730:
3729:
3724:
3709:
3707:
3706:
3701:
3699:
3698:
3680:
3678:
3677:
3672:
3670:
3669:
3650:
3648:
3647:
3642:
3627:
3625:
3624:
3619:
3604:
3602:
3601:
3596:
3575:
3573:
3572:
3567:
3565:
3564:
3552:
3547:
3546:
3537:
3534:
3529:
3505:
3497:
3486:
3478:
3454:Flows and forces
3448:
3446:
3445:
3440:
3438:
3437:
3421:
3419:
3418:
3413:
3411:
3410:
3386:
3384:
3383:
3378:
3362:
3360:
3359:
3354:
3352:
3351:
3335:
3333:
3332:
3327:
3325:
3324:
3312:
3311:
3299:
3298:
3282:
3280:
3279:
3274:
3272:
3271:
3252:
3250:
3249:
3244:
3242:
3241:
3218:
3209:
3207:
3206:
3201:
3196:
3195:
3182:
3181:
3170:
3165:
3147:
3146:
3133:
3132:
3121:
3082:
3078:
3076:
3075:
3070:
3055:
3053:
3052:
3047:
3020:
3011:
3009:
3008:
3003:
2992:
2991:
2979:
2978:
2966:
2965:
2950:
2949:
2937:
2936:
2901:
2894:
2892:
2891:
2886:
2883:
2878:
2862:
2860:
2859:
2854:
2849:
2848:
2830:
2829:
2817:
2816:
2798:
2797:
2785:
2784:
2761:
2752:
2750:
2749:
2744:
2713:
2709:
2707:
2696:
2684:
2683:
2668:
2666:
2665:
2653:
2645:
2643:
2635:
2634:
2633:
2620:
2610:
2599:that are called
2598:
2596:
2595:
2590:
2582:
2581:
2569:
2568:
2552:
2550:
2549:
2544:
2542:
2541:
2517:
2516:
2504:
2503:
2381:
2379:
2378:
2373:
2365:
2364:
2351:
2350:
2337:
2318:
2317:
2316:
2304:
2289:
2287:
2286:
2281:
2279:
2278:
2277:
2265:
2254:
2252:
2251:
2246:
2238:
2237:
2228:
2227:
2214:
2193:
2192:
2191:
2179:
2167:Massieu function
2161:
2159:
2158:
2153:
2148:
2146:
2145:
2144:
2143:
2129:
2128:
2119:
2114:
2113:
2094:
2092:
2091:
2086:
2084:
2083:
2067:
2065:
2064:
2059:
2057:
2056:
1908:
1906:
1905:
1900:
1898:
1897:
1879:
1877:
1876:
1871:
1869:
1868:
1691:
1684:
1677:
1661:
1660:
1368:Key publications
1349:
1348:("living force")
1298:Brownian ratchet
1293:Entropy and life
1288:Entropy and time
1239:
1238:
1214:
1212:
1211:
1206:
1159:
1157:
1156:
1151:
1104:
1102:
1101:
1096:
1049:
1047:
1046:
1041:
943:Clausius theorem
938:Carnot's theorem
911:
909:
908:
903:
887:
885:
884:
879:
864:
862:
861:
856:
840:
838:
837:
832:
819:
818:
815:
813:
812:
807:
779:
777:
776:
771:
755:
753:
752:
747:
732:
730:
729:
724:
708:
706:
705:
700:
687:
686:
683:
681:
680:
675:
644:
642:
641:
636:
620:
618:
617:
612:
597:
595:
594:
589:
573:
571:
570:
565:
552:
551:
548:
546:
545:
540:
518:
517:
391:
390:
210:
209:
91:
77:
76:
70:
67:
61:
38:
30:
21:
7254:
7253:
7249:
7248:
7247:
7245:
7244:
7243:
7224:
7223:
7222:
7217:
7181:
7167:Medical physics
7118:
7077:Nuclear physics
7046:
7040:Non-equilibrium
6962:
6934:
6906:
6900:
6834:
6829:
6805:de Groot S.R.,
6626:
6624:Further reading
6621:
6609:Physics Reports
6522:
6517:
6504:
6500:
6487:
6483:
6434:
6430:
6407:
6403:
6372:
6368:
6313:
6309:
6292:
6288:
6283:
6279:
6240:
6236:
6215:
6211:
6206:
6202:
6190:
6186:
6177:
6168:
6157:
6153:
6142:
6138:
6127:
6123:
6112:
6108:
6100:
6096:
6053:
6049:
5996:
5992:
5939:
5935:
5896:
5892:
5862:
5855:
5838:
5834:
5795:
5791:
5774:
5770:
5763:
5739:
5735:
5718:
5711:
5694:
5687:
5674:Wayback Machine
5661:
5654:
5637:
5630:
5621:
5617:
5586:
5582:
5545:
5541:
5503:
5499:
5468:
5464:
5459:
5455:
5430:
5423:
5406:
5397:
5380:
5371:
5354:
5350:
5333:
5329:
5324:
5320:
5303:
5292:
5263:Physical Review
5255:
5242:
5215:
5208:
5173:Acta Materialia
5165:
5161:
5110:
5106:
5071:
5067:
5032:
5028:
4969:
4965:
4960:
4949:
4906:
4899:
4856:
4849:
4788:
4781:
4772:
4763:
4756:
4754:
4750:
4743:
4729:
4725:
4706:
4702:
4667:
4663:
4656:
4632:
4623:
4619:
4614:
4590:Maxwell's demon
4585:Vlasov equation
4535:
4521:/unfolding and
4519:protein folding
4515:
4506:
4500:
4468:
4465:
4464:
4440:
4437:
4436:
4409:
4405:
4401:
4394:
4390:
4386:
4384:
4375:
4371:
4367:
4360:
4356:
4352:
4350:
4341:
4337:
4325:
4313:
4310:
4309:
4283:
4279:
4275:
4268:
4264:
4260:
4258:
4249:
4245:
4239:
4226:
4222:
4220:
4217:
4216:
4197:
4194:
4193:
4166:
4162:
4158:
4151:
4147:
4143:
4141:
4135:
4131:
4125:
4113:
4110:
4109:
4080:
4077:
4076:
4059:
4055:
4053:
4050:
4049:
4032:
4028:
4026:
4023:
4022:
4019:
4013:
3978:
3974:
3972:
3969:
3968:
3936:
3932:
3930:
3927:
3926:
3910:
3907:
3906:
3890:
3887:
3886:
3869:
3865:
3863:
3860:
3859:
3843:
3840:
3839:
3823:
3820:
3819:
3799:
3795:
3793:
3790:
3789:
3787:particle number
3767:
3763:
3761:
3758:
3757:
3741:
3738:
3737:
3718:
3715:
3714:
3694:
3690:
3688:
3685:
3684:
3662:
3658:
3656:
3653:
3652:
3636:
3633:
3632:
3613:
3610:
3609:
3587:
3584:
3583:
3560:
3556:
3542:
3538:
3536:
3530:
3519:
3496:
3477:
3466:
3463:
3462:
3456:
3433:
3429:
3427:
3424:
3423:
3406:
3402:
3400:
3397:
3396:
3372:
3369:
3368:
3347:
3343:
3341:
3338:
3337:
3320:
3316:
3307:
3303:
3294:
3290:
3288:
3285:
3284:
3267:
3263:
3258:
3255:
3254:
3237:
3233:
3231:
3228:
3227:
3191:
3187:
3177:
3173:
3166:
3155:
3142:
3138:
3128:
3124:
3117:
3089:
3086:
3085:
3064:
3061:
3060:
3041:
3038:
3037:
3032:was brought by
2987:
2983:
2974:
2970:
2961:
2957:
2945:
2941:
2932:
2928:
2908:
2905:
2904:
2879:
2874:
2868:
2865:
2864:
2844:
2840:
2825:
2821:
2812:
2808:
2793:
2789:
2780:
2776:
2774:
2771:
2770:
2697:
2692:
2679:
2675:
2674:
2670:
2661:
2657:
2652:
2636:
2629:
2625:
2621:
2619:
2617:
2614:
2613:
2605:non-equilibrium
2577:
2573:
2564:
2560:
2558:
2555:
2554:
2537:
2533:
2512:
2508:
2499:
2495:
2493:
2490:
2489:
2486:
2473:
2451:Edward A. Milne
2448:
2429:
2416:
2399:
2360:
2356:
2346:
2342:
2333:
2312:
2311:
2307:
2302:
2299:
2298:
2273:
2272:
2268:
2263:
2260:
2259:
2233:
2229:
2223:
2219:
2210:
2187:
2186:
2182:
2177:
2174:
2173:
2139:
2135:
2134:
2130:
2124:
2120:
2118:
2109:
2105:
2103:
2100:
2099:
2079:
2075:
2073:
2070:
2069:
2052:
2048:
2046:
2043:
2042:
1959:
1944:
1931:
1890:
1887:
1885:
1882:
1881:
1864:
1860:
1858:
1855:
1854:
1846:
1837:
1795:
1782:
1774:exergy analysis
1761:state variables
1752:
1747:
1704:is a branch of
1695:
1650:
1649:
1625:
1617:
1616:
1615:
1475:
1467:
1466:
1445:
1431:
1406:
1402:
1395:
1391:
1384:
1380:
1347:
1340:
1322:
1303:Maxwell's demon
1265:
1236:
1235:
1219:
1218:
1217:
1170:
1167:
1166:
1165:
1115:
1112:
1111:
1110:
1060:
1057:
1056:
1055:
1020:
1017:
1016:
1015:
1013:Internal energy
1008:
993:
983:
982:
957:
932:
922:
921:
920:
894:
891:
890:
873:
870:
869:
847:
844:
843:
826:
823:
822:
798:
795:
794:
762:
759:
758:
741:
738:
737:
715:
712:
711:
694:
691:
690:
663:
660:
659:
654:Compressibility
627:
624:
623:
606:
603:
602:
580:
577:
576:
559:
556:
555:
531:
528:
527:
507:
497:
496:
477:Particle number
430:
389:
378:
368:
367:
326:Irreversibility
238:State of matter
205:Isolated system
190:
180:
179:
178:
153:
143:
142:
138:Non-equilibrium
130:
105:
97:
71:
65:
62:
52:
39:
28:
23:
22:
18:Non-equilibrium
15:
12:
11:
5:
7252:
7242:
7241:
7236:
7219:
7218:
7216:
7215:
7210:
7205:
7200:
7195:
7189:
7187:
7183:
7182:
7180:
7179:
7174:
7169:
7164:
7159:
7154:
7149:
7144:
7139:
7134:
7128:
7126:
7120:
7119:
7117:
7116:
7111:
7110:
7109:
7104:
7099:
7089:
7084:
7079:
7074:
7073:
7072:
7067:
7056:
7054:
7048:
7047:
7045:
7044:
7043:
7042:
7037:
7030:Thermodynamics
7027:
7026:
7025:
7020:
7010:
7005:
7000:
6999:
6998:
6993:
6988:
6983:
6972:
6970:
6964:
6963:
6961:
6960:
6959:
6958:
6948:
6942:
6940:
6936:
6935:
6933:
6932:
6931:
6930:
6920:
6914:
6912:
6908:
6907:
6899:
6898:
6891:
6884:
6876:
6870:
6869:
6862:
6852:
6846:
6840:
6833:
6832:External links
6830:
6828:
6827:
6821:
6803:
6788:
6773:
6758:
6740:
6722:
6707:
6679:
6669:
6654:
6627:
6625:
6622:
6620:
6619:
6617:See also this.
6598:
6591:
6576:
6557:
6542:
6523:
6521:
6518:
6516:
6515:
6498:
6481:
6428:
6401:
6382:(3): 516–525.
6366:
6327:(1): 192–196.
6307:
6286:
6277:
6234:
6209:
6200:
6184:
6166:
6151:
6136:
6121:
6106:
6094:
6047:
5990:
5933:
5912:(6): 493–502.
5890:
5853:
5832:
5805:(1): 167–178.
5789:
5768:
5761:
5733:
5709:
5685:
5652:
5628:
5615:
5580:
5574:10.1086/162504
5539:
5497:
5492:10.1086/151469
5462:
5453:
5421:
5395:
5369:
5348:
5327:
5318:
5290:
5269:(4): 405–426.
5240:
5206:
5159:
5104:
5065:
5046:(3): 661–665.
5026:
4983:(4): 529–536.
4963:
4947:
4920:(1): 109–118.
4897:
4847:
4779:
4761:
4748:
4741:
4723:
4700:
4681:(1–4): 77–93.
4661:
4654:
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4615:
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4607:
4602:
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4587:
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4572:
4567:
4562:
4557:
4552:
4547:
4542:
4536:
4534:
4531:
4514:
4511:
4502:Main article:
4499:
4496:
4472:
4444:
4429:
4428:
4412:
4408:
4404:
4397:
4393:
4389:
4378:
4374:
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4363:
4359:
4355:
4347:
4344:
4340:
4334:
4331:
4328:
4324:
4320:
4317:
4303:
4302:
4286:
4282:
4278:
4271:
4267:
4263:
4255:
4252:
4248:
4242:
4238:
4234:
4229:
4225:
4201:
4186:
4185:
4169:
4165:
4161:
4154:
4150:
4146:
4138:
4134:
4128:
4124:
4120:
4117:
4105:to the flows:
4090:
4087:
4084:
4062:
4058:
4035:
4031:
4015:Main article:
4012:
4009:
4000:Ilya Prigogine
3981:
3977:
3939:
3935:
3914:
3894:
3872:
3868:
3847:
3827:
3802:
3798:
3773:
3770:
3766:
3745:
3722:
3697:
3693:
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3665:
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3640:
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3518:
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3511:
3508:
3503:
3500:
3495:
3492:
3489:
3484:
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3476:
3473:
3470:
3455:
3452:
3436:
3432:
3409:
3405:
3376:
3350:
3346:
3323:
3319:
3315:
3310:
3306:
3302:
3297:
3293:
3270:
3266:
3262:
3240:
3236:
3222:
3221:
3212:
3210:
3199:
3194:
3190:
3186:
3180:
3176:
3169:
3164:
3161:
3158:
3154:
3150:
3145:
3141:
3137:
3131:
3127:
3120:
3116:
3112:
3109:
3106:
3103:
3100:
3097:
3093:
3068:
3045:
3024:
3023:
3014:
3012:
3001:
2998:
2995:
2990:
2986:
2982:
2977:
2973:
2969:
2964:
2960:
2956:
2953:
2948:
2944:
2940:
2935:
2931:
2927:
2924:
2921:
2918:
2915:
2912:
2882:
2877:
2873:
2852:
2847:
2843:
2839:
2836:
2833:
2828:
2824:
2820:
2815:
2811:
2807:
2804:
2801:
2796:
2792:
2788:
2783:
2779:
2765:
2764:
2755:
2753:
2742:
2739:
2736:
2733:
2729:
2726:
2723:
2720:
2716:
2712:
2706:
2703:
2700:
2695:
2691:
2687:
2682:
2678:
2673:
2664:
2660:
2656:
2651:
2648:
2642:
2639:
2632:
2628:
2624:
2588:
2585:
2580:
2576:
2572:
2567:
2563:
2540:
2536:
2532:
2529:
2526:
2523:
2520:
2515:
2511:
2507:
2502:
2498:
2485:
2482:
2472:
2469:
2447:
2444:
2428:
2425:
2415:
2412:
2398:
2395:
2383:
2382:
2371:
2368:
2363:
2359:
2355:
2349:
2345:
2341:
2336:
2332:
2328:
2325:
2322:
2315:
2310:
2276:
2271:
2256:
2255:
2244:
2241:
2236:
2232:
2226:
2222:
2218:
2213:
2209:
2205:
2202:
2199:
2196:
2190:
2185:
2163:
2162:
2151:
2142:
2138:
2133:
2127:
2123:
2117:
2112:
2108:
2082:
2078:
2055:
2051:
1972:complex fluids
1958:
1957:Basic concepts
1955:
1943:
1940:
1930:
1927:
1896:
1893:
1889:
1867:
1863:
1845:
1842:
1836:
1833:
1822:time variation
1794:
1791:
1781:
1778:
1751:
1748:
1746:
1743:
1706:thermodynamics
1697:
1696:
1694:
1693:
1686:
1679:
1671:
1668:
1667:
1666:
1665:
1652:
1651:
1648:
1647:
1642:
1637:
1632:
1626:
1623:
1622:
1619:
1618:
1614:
1613:
1608:
1603:
1598:
1593:
1588:
1583:
1578:
1573:
1568:
1563:
1558:
1553:
1548:
1543:
1538:
1533:
1528:
1523:
1518:
1513:
1508:
1503:
1498:
1493:
1488:
1483:
1477:
1476:
1473:
1472:
1469:
1468:
1463:
1462:
1461:
1460:
1455:
1447:
1446:
1444:
1443:
1440:
1436:
1433:
1432:
1430:
1429:
1424:
1422:Thermodynamics
1418:
1415:
1414:
1410:
1409:
1408:
1407:
1398:
1396:
1387:
1385:
1376:
1371:
1370:
1364:
1363:
1362:
1361:
1356:
1351:
1339:
1338:
1336:Caloric theory
1332:
1329:
1328:
1324:
1323:
1321:
1320:
1315:
1310:
1305:
1300:
1295:
1290:
1284:
1281:
1280:
1274:
1273:
1272:
1271:
1264:
1263:
1258:
1253:
1247:
1244:
1243:
1237:
1234:
1233:
1230:
1226:
1225:
1224:
1221:
1220:
1216:
1215:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1160:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1105:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1050:
1039:
1036:
1033:
1030:
1027:
1024:
1009:
1007:
1006:
1001:
995:
994:
989:
988:
985:
984:
981:
980:
973:
968:
963:
956:
955:
950:
945:
940:
934:
933:
928:
927:
924:
923:
917:
916:
913:
912:
901:
898:
888:
877:
866:
865:
854:
851:
841:
830:
816:
805:
802:
792:
785:
784:
781:
780:
769:
766:
756:
745:
734:
733:
722:
719:
709:
698:
684:
673:
670:
667:
657:
650:
649:
646:
645:
634:
631:
621:
610:
599:
598:
587:
584:
574:
563:
549:
538:
535:
525:
516:
515:
514:
508:
503:
502:
499:
498:
493:
492:
491:
490:
485:
480:
469:
458:
439:
438:
432:
431:
429:
428:
423:
417:
414:
413:
407:
406:
405:
404:
399:
380:
379:
374:
373:
370:
369:
364:
363:
362:
361:
356:
351:
343:
342:
336:
335:
334:
333:
328:
323:
318:
316:Free expansion
313:
308:
303:
298:
293:
288:
283:
278:
270:
269:
263:
262:
261:
260:
255:
253:Control volume
250:
245:
243:Phase (matter)
240:
235:
230:
225:
217:
216:
208:
207:
202:
197:
191:
186:
185:
182:
181:
177:
176:
171:
166:
161:
155:
154:
149:
148:
145:
144:
141:
140:
129:
128:
123:
118:
113:
107:
106:
103:
102:
99:
98:
93:The classical
92:
84:
83:
81:Thermodynamics
73:
72:
42:
40:
33:
26:
9:
6:
4:
3:
2:
7251:
7240:
7237:
7235:
7232:
7231:
7229:
7214:
7211:
7209:
7206:
7204:
7201:
7199:
7196:
7194:
7191:
7190:
7188:
7184:
7178:
7175:
7173:
7172:Ocean physics
7170:
7168:
7165:
7163:
7160:
7158:
7155:
7153:
7150:
7148:
7145:
7143:
7140:
7138:
7135:
7133:
7130:
7129:
7127:
7125:
7121:
7115:
7112:
7108:
7107:Modern optics
7105:
7103:
7100:
7098:
7095:
7094:
7093:
7090:
7088:
7085:
7083:
7080:
7078:
7075:
7071:
7068:
7066:
7063:
7062:
7061:
7058:
7057:
7055:
7053:
7049:
7041:
7038:
7036:
7033:
7032:
7031:
7028:
7024:
7021:
7019:
7016:
7015:
7014:
7011:
7009:
7006:
7004:
7001:
6997:
6994:
6992:
6989:
6987:
6984:
6982:
6979:
6978:
6977:
6974:
6973:
6971:
6969:
6965:
6957:
6956:Computational
6954:
6953:
6952:
6949:
6947:
6944:
6943:
6941:
6937:
6929:
6926:
6925:
6924:
6921:
6919:
6916:
6915:
6913:
6909:
6905:
6897:
6892:
6890:
6885:
6883:
6878:
6877:
6874:
6867:
6863:
6860:
6856:
6855:Into the Cool
6853:
6850:
6847:
6844:
6841:
6839:
6836:
6835:
6826:
6822:
6820:
6819:0-486-64741-2
6816:
6812:
6808:
6804:
6801:
6800:0-471-97393-9
6797:
6793:
6789:
6786:
6782:
6778:
6774:
6771:
6767:
6763:
6759:
6756:
6755:3-527-40084-2
6752:
6748:
6744:
6743:Zubarev D. N.
6741:
6738:
6737:3-05-501708-0
6734:
6730:
6726:
6725:Zubarev D. N.
6723:
6720:
6719:0-387-96501-7
6716:
6712:
6708:
6705:
6701:
6697:
6696:0-306-10895-X
6693:
6689:
6688:
6683:
6682:Zubarev D. N.
6680:
6677:
6673:
6672:Prigogine, I.
6670:
6667:
6666:3-540-22495-5
6663:
6659:
6655:
6652:
6651:0-444-86503-9
6648:
6644:
6643:0-444-11080-1
6640:
6636:
6632:
6631:Ziegler, Hans
6629:
6628:
6618:
6614:
6610:
6606:
6602:
6599:
6596:
6592:
6589:
6585:
6581:
6577:
6574:
6573:0-471-30280-5
6570:
6566:
6562:
6561:Prigogine, I.
6558:
6555:
6554:1-4020-0788-4
6551:
6547:
6543:
6540:
6539:0-471-86256-8
6536:
6532:
6529:(1960/1985).
6528:
6525:
6524:
6511:
6510:
6502:
6494:
6493:
6485:
6477:
6473:
6469:
6465:
6461:
6457:
6452:
6447:
6443:
6439:
6432:
6424:
6420:
6416:
6412:
6405:
6397:
6393:
6389:
6385:
6381:
6377:
6370:
6362:
6358:
6353:
6348:
6343:
6338:
6334:
6330:
6326:
6322:
6318:
6311:
6302:
6297:
6290:
6281:
6273:
6269:
6265:
6261:
6257:
6253:
6249:
6245:
6238:
6231:
6230:0-387-94299-8
6227:
6223:
6219:
6213:
6204:
6197:
6193:
6192:Prigogine, I.
6188:
6181:
6175:
6173:
6171:
6162:
6155:
6147:
6140:
6132:
6125:
6117:
6110:
6103:
6098:
6090:
6086:
6082:
6078:
6074:
6070:
6066:
6062:
6058:
6051:
6043:
6039:
6035:
6031:
6027:
6023:
6018:
6013:
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5879:
5878:0-306-10895-X
5875:
5871:
5870:
5865:
5864:Zubarev D. N.
5860:
5858:
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5849:0-471-97394-7
5846:
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5808:
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5793:
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5764:
5758:
5754:
5753:10.1142/11729
5750:
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5729:0-387-96501-7
5726:
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5706:
5705:3-528-06343-2
5702:
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5681:0-19-503437-6
5678:
5675:
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5659:
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5648:0-471-04600-0
5645:
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5449:0-387-55874-8
5446:
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5441:3-540-55874-8
5438:
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5410:
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5400:
5392:
5391:0-471-30280-5
5388:
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5374:
5366:
5365:0-471-04600-0
5362:
5358:
5352:
5345:
5344:0-444-86503-9
5341:
5337:
5331:
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5315:
5314:0-333-21616-4
5311:
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5119:
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4835:
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4827:
4823:
4819:
4815:
4810:
4805:
4802:(1): 014131.
4801:
4797:
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4786:
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4776:
4770:
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4766:
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4556:
4553:
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4546:
4543:
4541:
4538:
4537:
4530:
4528:
4527:nanoparticles
4524:
4520:
4510:
4505:
4495:
4492:
4490:
4486:
4470:
4462:
4458:
4442:
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3461:
3460:
3459:
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3434:
3430:
3407:
3403:
3394:
3390:
3374:
3367:of substance
3366:
3348:
3344:
3321:
3317:
3313:
3308:
3304:
3300:
3295:
3291:
3268:
3264:
3238:
3234:
3220:
3213:
3211:
3197:
3192:
3188:
3178:
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3167:
3162:
3159:
3156:
3152:
3148:
3143:
3139:
3129:
3118:
3114:
3110:
3107:
3101:
3098:
3095:
3091:
3084:
3083:
3080:
3066:
3059:
3043:
3035:
3031:
3022:
3015:
3013:
2996:
2993:
2988:
2984:
2980:
2975:
2971:
2967:
2962:
2958:
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2946:
2942:
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2933:
2929:
2925:
2922:
2916:
2913:
2910:
2903:
2902:
2899:
2896:
2880:
2875:
2871:
2845:
2841:
2837:
2834:
2831:
2826:
2822:
2818:
2813:
2809:
2805:
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2794:
2790:
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2777:
2763:
2756:
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2731:
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2721:
2718:
2714:
2710:
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2693:
2689:
2685:
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2671:
2662:
2658:
2654:
2649:
2646:
2640:
2637:
2630:
2626:
2622:
2612:
2611:
2608:
2606:
2602:
2586:
2583:
2578:
2574:
2570:
2565:
2561:
2538:
2534:
2530:
2527:
2524:
2521:
2518:
2513:
2509:
2505:
2500:
2496:
2481:
2477:
2468:
2465:
2460:
2456:
2452:
2443:
2441:
2435:
2433:
2424:
2422:
2411:
2407:
2403:
2394:
2390:
2386:
2369:
2361:
2357:
2353:
2347:
2343:
2334:
2330:
2326:
2323:
2320:
2308:
2297:
2296:
2295:
2293:
2269:
2242:
2234:
2230:
2224:
2220:
2211:
2207:
2203:
2200:
2197:
2194:
2183:
2172:
2171:
2170:
2168:
2149:
2140:
2136:
2125:
2115:
2110:
2106:
2098:
2097:
2096:
2080:
2076:
2053:
2049:
2041:
2037:
2033:
2029:
2023:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1975:
1973:
1968:
1964:
1954:
1952:
1948:
1939:
1936:
1926:
1924:
1920:
1916:
1910:
1894:
1891:
1888:
1865:
1861:
1850:
1841:
1832:
1830:
1825:
1823:
1819:
1815:
1811:
1810:steady states
1806:
1804:
1800:
1790:
1786:
1777:
1775:
1771:
1765:
1762:
1756:
1742:
1739:
1735:
1730:
1728:
1722:
1720:
1716:
1711:
1707:
1703:
1692:
1687:
1685:
1680:
1678:
1673:
1672:
1670:
1669:
1664:
1656:
1655:
1654:
1653:
1646:
1643:
1641:
1638:
1636:
1635:Self-assembly
1633:
1631:
1628:
1627:
1621:
1620:
1612:
1609:
1607:
1606:van der Waals
1604:
1602:
1599:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1531:von Helmholtz
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1478:
1471:
1470:
1459:
1456:
1454:
1451:
1450:
1449:
1448:
1441:
1438:
1437:
1435:
1434:
1428:
1425:
1423:
1420:
1419:
1417:
1416:
1412:
1411:
1405:
1404:
1397:
1394:
1393:
1386:
1383:
1382:
1375:
1374:
1373:
1372:
1369:
1366:
1365:
1360:
1357:
1355:
1352:
1350:
1346:
1342:
1341:
1337:
1334:
1333:
1331:
1330:
1326:
1325:
1319:
1316:
1314:
1311:
1309:
1306:
1304:
1301:
1299:
1296:
1294:
1291:
1289:
1286:
1285:
1283:
1282:
1279:
1276:
1275:
1270:
1267:
1266:
1262:
1259:
1257:
1254:
1252:
1249:
1248:
1246:
1245:
1241:
1240:
1231:
1228:
1227:
1223:
1222:
1202:
1199:
1196:
1193:
1190:
1184:
1181:
1178:
1172:
1164:
1161:
1147:
1144:
1141:
1138:
1135:
1129:
1126:
1123:
1117:
1109:
1106:
1092:
1089:
1086:
1083:
1080:
1074:
1071:
1068:
1062:
1054:
1051:
1034:
1031:
1028:
1022:
1014:
1011:
1010:
1005:
1002:
1000:
997:
996:
992:
987:
986:
979:
978:
974:
972:
969:
967:
964:
962:
959:
958:
954:
953:Ideal gas law
951:
949:
946:
944:
941:
939:
936:
935:
931:
926:
925:
899:
889:
875:
868:
867:
852:
842:
828:
821:
820:
817:
803:
800:
793:
790:
787:
786:
767:
757:
743:
736:
735:
720:
710:
696:
689:
688:
685:
671:
668:
665:
658:
655:
652:
651:
632:
622:
608:
601:
600:
585:
575:
561:
554:
553:
550:
536:
533:
526:
523:
520:
519:
513:
510:
509:
506:
501:
500:
489:
486:
484:
483:Vapor quality
481:
479:
478:
473:
470:
468:
467:
462:
459:
456:
452:
451:
446:
443:
442:
441:
440:
437:
434:
433:
427:
424:
422:
419:
418:
416:
415:
412:
409:
408:
403:
400:
398:
395:
394:
393:
392:
388:
384:
377:
372:
371:
360:
357:
355:
352:
350:
347:
346:
345:
344:
341:
338:
337:
332:
329:
327:
324:
322:
321:Reversibility
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
279:
277:
274:
273:
272:
271:
268:
265:
264:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
224:
221:
220:
219:
218:
215:
212:
211:
206:
203:
201:
198:
196:
195:Closed system
193:
192:
189:
184:
183:
175:
172:
170:
167:
165:
162:
160:
157:
156:
152:
147:
146:
139:
135:
132:
131:
127:
124:
122:
119:
117:
114:
112:
109:
108:
101:
100:
96:
90:
86:
85:
82:
79:
78:
69:
66:December 2018
59:
55:
50:
46:
43:This article
41:
37:
32:
31:
19:
7132:Astrophysics
7039:
6946:Experimental
6824:
6810:
6791:
6776:
6761:
6746:
6728:
6710:
6685:
6675:
6657:
6634:
6612:
6608:
6605:Yngvason, J.
6594:
6579:
6564:
6545:
6530:
6527:Callen, H.B.
6508:
6501:
6491:
6484:
6441:
6437:
6431:
6414:
6410:
6404:
6379:
6375:
6369:
6324:
6320:
6310:
6289:
6280:
6247:
6243:
6237:
6221:
6217:
6212:
6203:
6195:
6187:
6160:
6154:
6145:
6139:
6130:
6124:
6115:
6109:
6097:
6064:
6060:
6050:
6007:
6003:
5993:
5950:
5946:
5936:
5909:
5903:
5893:
5867:
5840:
5835:
5802:
5798:
5792:
5776:
5771:
5743:
5736:
5720:
5696:
5664:
5639:
5623:
5618:
5593:
5589:
5583:
5556:
5552:
5542:
5514:
5510:
5500:
5475:
5471:
5465:
5456:
5432:
5408:
5382:
5356:
5351:
5335:
5330:
5321:
5305:
5266:
5262:
5226:
5222:
5176:
5172:
5162:
5121:
5117:
5107:
5085:(1): 53–74.
5082:
5078:
5068:
5043:
5039:
5029:
4980:
4976:
4966:
4917:
4913:
4867:
4863:
4799:
4795:
4774:
4751:
4732:
4726:
4709:
4703:
4678:
4674:
4664:
4646:10.1142/7869
4636:
4540:Time crystal
4516:
4513:Applications
4507:
4493:
4488:
4430:
4304:
4187:
4020:
4005:
3996:
3966:
3958:
3954:
3817:
3578:
3457:
3225:
3214:
3027:
3016:
2897:
2768:
2757:
2600:
2487:
2478:
2474:
2449:
2436:
2431:
2430:
2420:
2417:
2408:
2404:
2400:
2391:
2387:
2384:
2257:
2169:as follows:
2164:
2035:
2024:
2019:
2015:
2011:
2007:
1995:
1991:
1987:
1976:
1963:Couette flow
1960:
1946:
1945:
1932:
1923:Lars Onsager
1911:
1851:
1847:
1838:
1826:
1807:
1796:
1787:
1783:
1766:
1757:
1753:
1731:
1723:
1701:
1700:
1496:Carathéodory
1427:Heat engines
1399:
1388:
1377:
1359:Motive power
1344:
1004:Free entropy
975:
475:
474: /
464:
463: /
455:introduction
448:
447: /
386:
349:Heat engines
137:
136: /
63:
54:You can help
44:
7035:Statistical
6951:Theoretical
6928:Engineering
6512:. Springer.
6163:. Springer.
5559:: 279–293.
5229:: 357–368.
4605:Autopoiesis
3607:temperature
2464:temperature
1812:, in which
1727:free energy
1318:Synergetics
999:Free energy
445:Temperature
306:Quasistatic
301:Isenthalpic
258:Instruments
248:Equilibrium
200:Open system
134:Equilibrium
116:Statistical
7228:Categories
7152:Geophysics
7142:Biophysics
6986:Analytical
6939:Approaches
6601:Lieb, E.H.
6067:(5): 771.
4809:2102.09019
4759:,§ 2.
4617:References
3336:, where
1630:Nucleation
1474:Scientists
1278:Philosophy
991:Potentials
354:Heat pumps
311:Polytropic
296:Isentropic
286:Isothermal
7102:Molecular
7003:Acoustics
6996:Continuum
6991:Celestial
6981:Newtonian
6968:Classical
6911:Divisions
6813:(Dover).
6451:1208.1587
6301:1208.5105
6272:121536072
6133:. Kluwer.
6089:119406182
6010:(1): 21.
5866:,(1974).
5827:189793830
5478:: 69–77.
5201:1359-6454
5146:0267-0836
5099:0364-5916
5060:0013-4686
5005:0003-7028
4942:0009-2509
4892:0360-5442
4834:2470-0045
4485:symmetric
4403:∂
4388:∂
4369:∂
4354:∂
4323:∑
4316:σ
4277:∂
4262:∂
4237:∑
4160:∂
4145:∂
4123:∑
4116:σ
4086:σ
3934:μ
3692:μ
3540:μ
3517:∑
3513:−
3431:ξ
3404:ξ
3375:α
3349:α
3345:μ
3322:α
3318:μ
3314:−
3309:α
3296:α
3292:η
3269:α
3261:Δ
3239:α
3193:α
3185:Δ
3179:α
3175:η
3157:α
3153:∑
3140:ξ
3136:Δ
3126:Ξ
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1413:Timelines
1197:−
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930:Equations
897:∂
850:∂
801:α
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718:∂
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291:Adiabatic
281:Isochoric
267:Processes
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1601:Thompson
1511:Clausius
1491:Bridgman
1345:Vis viva
1327:Theories
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1053:Enthalpy
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276:Isobaric
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104:Branches
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2457:of the
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2032:entropy
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