Non-equilibrium thermodynamics

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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

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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.

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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.

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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.

36: 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 89: 1659: 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.

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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

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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.

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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

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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,

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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

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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

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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

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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

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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

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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

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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

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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,

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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

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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

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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

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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

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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

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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

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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

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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,

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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

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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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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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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

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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.

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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.

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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

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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.

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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.

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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

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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.

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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

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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.

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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

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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

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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

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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

2380: 3334: 2253: 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. 3087: 2615: 3818:

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

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One initial approach to non-equilibrium thermodynamics is sometimes called 'classical irreversible thermodynamics'. There are other approaches to non-equilibrium thermodynamics, for example

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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).

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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

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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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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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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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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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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,

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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. 6436:

Bazant, Martin Z. (22 March 2013). "Theory of Chemical Kinetics and Charge Transfer based on Nonequilibrium Thermodynamics".

2175: 1605: 6842: 2746:{\displaystyle {\frac {d\xi _{i}}{dt}}=-{\frac {1}{\tau _{i}}}\,\left(\xi _{i}-\xi _{i}^{(0)}\right),\quad i=1,\,2,\ldots ,} 1500: 4599: 1580: 1353: 330: 6838:

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: 6879: 4432: 4016: 3967:

Establishing the relation between such forces and flux densities is a problem in statistical mechanics. Flux densities (

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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" 4569: 173: 163: 1933:

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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Coleman, B.D.; Noll, W. (1963). "The thermodynamics of elastic materials with heat conduction and viscosity".

4735:. Dover books on physics (Dover ed., 1. publ., unabridged, corr. republ ed.). New York, NY: Dover Publ. 4609: 4021:

Following Section III of Rayleigh (1873), Onsager (1931, I) showed that in the regime where both the flows (

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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

7202: 7171: 7034: 6955: 4460: 3256: 1983: 1934: 1914: 1802: 1530: 1107: 420: 266: 115: 4494:

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

5168: 5074: 4909: 4860:"Criteria for validity of thermodynamic equations from non-equilibrium molecular dynamics simulations" 4859: 4517:

Non-equilibrium thermodynamics has been successfully applied to describe biological processes such as

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Nicolis, C. (1999). "Entropy production and dynamical complexity in a low-order atmospheric model".

3339: 7136: 6995: 6990: 6975: 6950: 6927: 6903: 4973:"Investigation of the State of Local Thermodynamic Equilibrium of a Laser-Produced Aluminum Plasma" 4594: 2450: 2261: 1821: 1714: 1555: 1550: 1317: 339: 305: 300: 213: 2866: 2863:

is a relaxation time of a corresponding variables. It is convenient to consider the initial value

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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". 4589: 1302: 892: 845: 760: 713: 625: 578: 4078: 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 7207: 7156: 7101: 7081: 6967: 6814: 6795: 6780: 6765: 6750: 6732: 6714: 6699: 6691: 6661: 6646: 6638: 6583: 6568: 6549: 6534: 6463: 6356: 6225: 6102: 5881: 5873: 5844: 5780: 5756: 5742: 5724: 5700: 5676: 5643: 5444: 5436: 5412: 5386: 5360: 5339: 5309: 5196: 5141: 5094: 5055: 5008: 5000: 4937: 4910:"Irreversible thermodynamics—a tool to describe phase transitions far from global equilibrium" 4887: 4837: 4829: 4736: 4649: 4559: 3928: 3686: 3425: 3398: 2454: 2003: 1773: 1639: 1600: 1590: 1162: 960: 796: 788: 661: 290: 280: 222: 5662: 4635: 3370: 2022:), where the system's properties are determined both by the temperature and by the pressure. 1856: 7146: 7086: 7012: 6455: 6418: 6391: 6346: 6336: 6259: 6076: 6029: 5972: 5921: 5814: 5748: 5747:. Series on Advances in Statistical Mechanics. Vol. 20 (2 ed.). WORLD SCIENTIFIC. 5605: 5601: 5568: 5526: 5487: 5278: 5230: 5188: 5133: 5086: 5047: 4992: 4929: 4879: 4821: 4690: 4641: 4484: 3759: 3654: 2166: 1560: 1545: 1485: 1480: 1297: 1292: 1018: 942: 410: 275: 4972: 4051: 4024: 3970: 3861: 3791: 2071: 2044: 1883: 7166: 7076: 7022: 6922: 6600: 5673: 4670: 4584: 4574: 4518: 4102: 3786: 2604: 1510: 1358: 1012: 653: 476: 237: 204: 6080: 6033: 5976: 1831:, and generalized thermodynamics, but they are hardly touched on in the present article. 6857:- 2005 book by Dorion Sagan and Eric D. Schneider, on nonequilibrium thermodynamics and 6747:

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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that are used to fix the equilibrium state, as was described above, a set of variables

1971: 1760: 1705: 1565: 1335: 871: 824: 739: 692: 604: 557: 435: 315: 252: 242: 110: 80: 6843:

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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considerations involving microscopic reversibility of dynamics imply that the matrix

1835:

Quasi-radiationless non-equilibrium thermodynamics of matter in laboratory conditions

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Hafskjold, Bjørn; Bedeaux, Dick; Kjelstrup, Signe; Wilhelmsen, Øivind (2021-07-23).

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1931, also), time rate of entropy production (Onsager 1931), thermodynamic fields,

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Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers

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Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory

6507: 4883: 6806: 5999: 4825: 4792:"Theory and simulation of shock waves: Entropy production and energy conversion" 4604: 3606: 2463: 2389:

thermodynamic forces driving fluxes of extensive properties through the system.

1540: 348: 5942: 1974:, turbulent systems and glasses are other examples of non-equilibrium systems. 7151: 7141: 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}}}} 1629: 1575: 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}}}} 7002: 6871: 6858: 3033: 227: 6607:(1999). 'The physics and mathematics of the second law of thermodynamics', 6467: 6360: 6263: 5283: 5258: 5012: 4841: 4188:

and the flows are related to the gradient of the forces, parametrized by a

6341: 6616: 6179: 6016: 5959: 3629: 1909:, and even in shock fronts moving at up to six times the speed of sound. 1343: 1260: 1052: 460: 232: 6218:

Thermodynamics and Statistical Mechanics (Classical Theoretical Physics)

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Non-equilibrium Thermodynamics. Field Theory and Variational Principles

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The Unity of Science and Economics: A New Foundation of Economic Theory

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One problem of interest is the thermodynamic study of non-equilibrium

7106: 3711: 2458: 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: 3005:{\displaystyle S=S(T,x_{1},x_{2},,x_{n};\xi _{1},\xi _{2},\ldots )} 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

1966: 5900:"The effect of collisions on monochromatic radiative equilibrium" 4645: 1750:

Difference between equilibrium and non-equilibrium thermodynamics

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).

2856:{\displaystyle \tau _{i}=\tau _{i}(T,x_{1},x_{2},\ldots ,x_{n})} 1950: 1817: 425: 6875: 5943:"Time Evolution in Macroscopic Systems. I. Equations of Motion" 5403: 5401: 5399: 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

29: 2068:. Each extensive quantity has a conjugate intensive variable 1929:

Local equilibrium thermodynamics with materials with "memory"

6658:

Non-equilibrium Thermodynamics and the Production of Entropy

4498:

Speculated extremal principles for non-equilibrium processes

2445: 1917:

known as the Seebeck and the Peltier effects, considered by

5532:

10.1175/1520-0469(1984)041<1985:RATITO>2.0.CO;2

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}).} 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

5744:

Non-equilibrium Thermodynamics of Heterogeneous Systems

5658: 5656: 2248:{\displaystyle \ k_{\rm {B}}M=S-\sum _{i}(I_{i}E_{i}),} 53: 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: 2446:

Milne's definition in terms of radiative equilibrium

7185: 7122: 7050: 6966: 6938: 6910: 5259:"Reciprocal relations in irreversible processes, I" 5741:Kjelstrup, Signe; Bedeaux, Dick (September 2020). 5036:"Local heat changes during aluminium electrolysis" 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: 2374: 2282: 2247: 2154: 2087: 2060: 1901: 1872: 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 5252: 5250: 5248: 5246: 5244: 1708:that deals with physical systems that are not in 5691: 5689: 5212: 5210: 4957: 4955: 4953: 4951: 5634: 5632: 5377: 5375: 5373: 5300: 5298: 5296: 5294: 4075:) vary slowly, the rate of creation of entropy 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). 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 2397:Stationary states, fluctuations, and stability 6887: 6825:The Non-Equilibrium Thermodynamics, Perpetual 6216:W. Greiner, L. Neise, and H. Stöcker (1997), 6174: 6172: 6170: 1682: 8: 6678:. 3rd edition, Wiley Interscience, New York. 6198:. 3rd edition, Wiley Interscience, New York. 6146:The Mechanics and Thermodynamics of Continua 5427: 5425: 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). 5699:, Freidr. Vieweg & Sohn, Braunschweig, 5626:, Longmans, Green & Co, London, page 1. 6894: 6880: 6872: 5167:Hillert, M.; Rettenmayr, M. (2003-06-11). 4755:Glansdorff, P., Prigogine, I. (1971), Ch. 4565:Autocatalytic reactions and order creation 4048:) are small and the thermodynamic forces ( 1689: 1675: 1238: 390: 209: 87: 76: 6866:Thermodynamics "beyond" local equilibrium 6823:Ramiro Augusto Salazar La Rotta. (2011). 6687:Nonequilibrium Statistical Thermodynamics 6548:, Kluwer Academic Publishers, Dordrecht, 6449: 6350: 6340: 6299: 6015: 5958: 5925: 5869:Nonequilibrium Statistical Thermodynamics 5572: 5530: 5282: 4807: 4468: 4440: 4409: 4394: 4384: 4375: 4360: 4350: 4341: 4325: 4313: 4283: 4268: 4258: 4249: 4239: 4226: 4220: 4197: 4166: 4151: 4141: 4135: 4125: 4113: 4080: 4059: 4053: 4032: 4026: 3978: 3972: 3936: 3930: 3910: 3890: 3869: 3863: 3843: 3823: 3799: 3793: 3767: 3761: 3741: 3718: 3694: 3688: 3662: 3656: 3636: 3613: 3587: 3560: 3542: 3536: 3530: 3519: 3496: 3477: 3466: 3433: 3427: 3406: 3400: 3372: 3347: 3341: 3320: 3307: 3294: 3288: 3267: 3258: 3237: 3231: 3191: 3183: 3177: 3172: 3166: 3155: 3142: 3134: 3128: 3123: 3117: 3094: 3089: 3064: 3041: 2987: 2974: 2961: 2945: 2932: 2908: 2879: 2874: 2868: 2844: 2825: 2812: 2793: 2780: 2774: 2730: 2697: 2692: 2679: 2669: 2661: 2652: 2629: 2619: 2617: 2592:{\displaystyle \xi _{1},\xi _{2},\ldots } 2577: 2564: 2558: 2537: 2512: 2499: 2493: 2432:Local thermodynamic equilibrium of matter 2360: 2352: 2346: 2333: 2319: 2312: 2311: 2302: 2273: 2272: 2263: 2233: 2223: 2210: 2187: 2186: 2177: 2139: 2134: 2124: 2118: 2109: 2103: 2079: 2073: 2052: 2046: 1890: 1885: 1864: 1858: 1170: 1115: 1060: 1020: 894: 873: 847: 826: 798: 762: 741: 715: 694: 663: 627: 606: 580: 559: 531: 6376:Journal of Colloid and Interface Science 5663:Mihalas, D., Weibel-Mihalas, B. (1984). 5306:Thermodynamics of Irreversible Processes 4621: 4192:of coefficients conventionally denoted 1741:obey the concept of local equilibrium. 1435: 1412: 1366: 1326: 1276: 1241: 434: 409: 338: 265: 212: 79: 7092:Atomic, molecular, and optical physics 6180:https://dx.doi.org/10.1155/2013/906136 5665:Foundations of Radiation Hydrodynamics 5622:Prigogine, I., Defay, R. (1950/1954). 5381:Glansdorff, P., Prigogine, I. (1971). 5034:Kjelstrup Ratkje, Signe (1991-01-01). 1805:, and non-linear dynamical structure. 6794:. John Wiley & Sons, Chichester. 6790:Kondepudi, D., Prigogine, I. (1998). 5839:Kondepudi, D., Prigogine, I, (1998). 5777:Introduction to Modern Thermodynamics 5325:Gyarmati, I. (1967/1970), pages 4-14. 2546:{\displaystyle x_{1},x_{2},...,x_{n}} 2038:) is a function of the collection of 7: 6845:- 1992- book by Xavier de Hemptinne. 6567:, Wiley-Interscience, London, 1971, 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 1947:Extended irreversible thermodynamics 1942:Extended irreversible thermodynamics 1829:extended irreversible thermodynamics 5667:, Oxford University Press, New York 5642:, John Wiley & Sons, New York, 5511:Journal of the Atmospheric Sciences 6635:An introduction to Thermomechanics 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: 6159:Amendola, Giovambattista (2012). 3387:. The middle term in (1) depicts 1921:in the nineteenth century and by 18:Law of Maximum Entropy Production 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): 5112:Sobolev, S. L. (October 2015). 4712:. IOP Publishing, Bristol, UK. 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 4733:Non-equilibrium thermodynamics 4675:Physica D: Nonlinear Phenomena 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: 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). 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 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 4523:transport through membranes 3246:{\displaystyle h_{\alpha }} 2471:Entropy in evolving systems 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 7255: 7239:Branches of thermodynamics 7008:Classical electromagnetism 6593:Gyarmati, I. (1967/1970). 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} 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 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: 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: 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: 210: 91: 77: 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: 6505: 6504: 6500: 6488: 6487: 6483: 6435: 6434: 6430: 6408: 6407: 6403: 6373: 6372: 6368: 6314: 6313: 6309: 6293: 6292: 6288: 6283: 6279: 6241: 6240: 6236: 6215: 6211: 6206: 6202: 6190: 6186: 6177: 6168: 6158: 6157: 6153: 6143: 6142: 6138: 6128: 6127: 6123: 6113: 6112: 6108: 6100: 6096: 6054: 6053: 6049: 5997: 5996: 5992: 5940: 5939: 5935: 5897: 5896: 5892: 5862: 5855: 5838: 5834: 5796: 5795: 5791: 5774: 5770: 5763: 5740: 5739: 5735: 5718: 5711: 5694: 5687: 5674:Wayback Machine 5661: 5654: 5637: 5630: 5621: 5617: 5587: 5586: 5582: 5546: 5545: 5541: 5504: 5503: 5499: 5469: 5468: 5464: 5459: 5455: 5430: 5423: 5406: 5397: 5380: 5371: 5354: 5350: 5333: 5329: 5324: 5320: 5303: 5292: 5263:Physical Review 5256: 5255: 5242: 5216: 5215: 5208: 5173:Acta Materialia 5166: 5165: 5161: 5111: 5110: 5106: 5072: 5071: 5067: 5033: 5032: 5028: 4970: 4969: 4965: 4960: 4949: 4907: 4906: 4899: 4857: 4856: 4849: 4789: 4788: 4781: 4772: 4763: 4756: 4754: 4750: 4743: 4730: 4729: 4725: 4707: 4706: 4702: 4668: 4667: 4663: 4656: 4633: 4632: 4623: 4619: 4614: 4590:Maxwell's demon 4585:Vlasov equation 4535: 4521:/unfolding and 4519:protein folding 4515: 4506: 4500: 4465: 4464: 4437: 4436: 4405: 4401: 4390: 4386: 4371: 4367: 4356: 4352: 4337: 4310: 4309: 4279: 4275: 4264: 4260: 4245: 4222: 4217: 4216: 4194: 4193: 4162: 4158: 4147: 4143: 4131: 4110: 4109: 4077: 4076: 4055: 4050: 4049: 4028: 4023: 4022: 4019: 4013: 3974: 3969: 3968: 3932: 3927: 3926: 3907: 3906: 3887: 3886: 3865: 3860: 3859: 3840: 3839: 3820: 3819: 3795: 3790: 3789: 3787:particle number 3763: 3758: 3757: 3738: 3737: 3715: 3714: 3690: 3685: 3684: 3658: 3653: 3652: 3633: 3632: 3610: 3609: 3584: 3583: 3556: 3538: 3463: 3462: 3456: 3429: 3424: 3423: 3402: 3397: 3396: 3369: 3368: 3343: 3338: 3337: 3316: 3303: 3290: 3285: 3284: 3263: 3255: 3254: 3233: 3228: 3227: 3216: 3187: 3173: 3138: 3124: 3086: 3085: 3061: 3060: 3038: 3037: 3032:was brought by 3018: 2983: 2970: 2957: 2941: 2928: 2905: 2904: 2865: 2864: 2840: 2821: 2808: 2789: 2776: 2771: 2770: 2759: 2675: 2674: 2670: 2657: 2636: 2625: 2621: 2614: 2613: 2605:non-equilibrium 2573: 2560: 2555: 2554: 2533: 2508: 2495: 2490: 2489: 2486: 2473: 2451:Edward A. Milne 2448: 2429: 2416: 2399: 2356: 2342: 2307: 2299: 2298: 2268: 2260: 2259: 2229: 2219: 2182: 2174: 2173: 2135: 2130: 2120: 2105: 2100: 2099: 2075: 2070: 2069: 2048: 2043: 2042: 1959: 1944: 1931: 1887: 1882: 1881: 1860: 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: 1167: 1166: 1165: 1112: 1111: 1110: 1057: 1056: 1055: 1017: 1016: 1015: 1013:Internal energy 1008: 993: 983: 982: 957: 932: 922: 921: 920: 891: 890: 870: 869: 844: 843: 823: 822: 795: 794: 759: 758: 738: 737: 712: 711: 691: 690: 660: 659: 654:Compressibility 624: 623: 603: 602: 577: 576: 556: 555: 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: 15: 12: 11: 5: 7252: 7250: 7242: 7241: 7236: 7226: 7225: 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: 6901: 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: 4620: 4618: 4615: 4613: 4612: 4607: 4602: 4597: 4592: 4587: 4582: 4577: 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: 4370: 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: 3668: 3665: 3661: 3640: 3617: 3594: 3591: 3577: 3576: 3563: 3559: 3555: 3550: 3545: 3541: 3533: 3528: 3525: 3522: 3518: 3514: 3511: 3508: 3503: 3500: 3495: 3492: 3489: 3484: 3481: 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: 24: 14: 13: 10: 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: 6831: 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: 6623: 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: 6519: 6511: 6510: 6502: 6499: 6494: 6493: 6485: 6482: 6477: 6473: 6469: 6465: 6461: 6457: 6452: 6447: 6443: 6439: 6432: 6429: 6424: 6420: 6416: 6412: 6405: 6402: 6397: 6393: 6389: 6385: 6381: 6377: 6370: 6367: 6362: 6358: 6353: 6348: 6343: 6338: 6334: 6330: 6326: 6322: 6318: 6311: 6308: 6302: 6297: 6290: 6287: 6281: 6278: 6273: 6269: 6265: 6261: 6257: 6253: 6249: 6245: 6238: 6235: 6231: 6230:0-387-94299-8 6227: 6223: 6219: 6213: 6210: 6204: 6201: 6197: 6193: 6192:Prigogine, I. 6188: 6185: 6181: 6175: 6173: 6171: 6167: 6162: 6155: 6152: 6147: 6140: 6137: 6132: 6125: 6122: 6117: 6110: 6107: 6103: 6098: 6095: 6090: 6086: 6082: 6078: 6074: 6070: 6066: 6062: 6058: 6051: 6048: 6043: 6039: 6035: 6031: 6027: 6023: 6018: 6013: 6009: 6005: 6001: 5994: 5991: 5986: 5982: 5978: 5974: 5970: 5966: 5961: 5956: 5952: 5948: 5944: 5937: 5934: 5928: 5923: 5919: 5915: 5911: 5907: 5906: 5901: 5894: 5891: 5887: 5883: 5879: 5878:0-306-10895-X 5875: 5871: 5870: 5865: 5864:Zubarev D. N. 5860: 5858: 5854: 5850: 5849:0-471-97394-7 5846: 5842: 5836: 5833: 5828: 5824: 5820: 5816: 5812: 5808: 5804: 5800: 5793: 5790: 5786: 5782: 5778: 5772: 5769: 5764: 5758: 5754: 5753:10.1142/11729 5750: 5746: 5745: 5737: 5734: 5730: 5729:0-387-96501-7 5726: 5722: 5716: 5714: 5710: 5706: 5705:3-528-06343-2 5702: 5698: 5692: 5690: 5686: 5682: 5681:0-19-503437-6 5678: 5675: 5671: 5668: 5666: 5659: 5657: 5653: 5649: 5648:0-471-04600-0 5645: 5641: 5635: 5633: 5629: 5625: 5619: 5616: 5611: 5607: 5603: 5599: 5595: 5591: 5584: 5581: 5575: 5570: 5566: 5562: 5558: 5554: 5550: 5543: 5540: 5533: 5528: 5524: 5520: 5516: 5512: 5508: 5501: 5498: 5493: 5489: 5485: 5481: 5477: 5473: 5466: 5463: 5457: 5454: 5450: 5449:0-387-55874-8 5446: 5442: 5441:3-540-55874-8 5438: 5434: 5428: 5426: 5422: 5418: 5414: 5410: 5404: 5402: 5400: 5396: 5392: 5391:0-471-30280-5 5388: 5384: 5378: 5376: 5374: 5370: 5366: 5365:0-471-04600-0 5362: 5358: 5352: 5349: 5345: 5344:0-444-86503-9 5341: 5337: 5331: 5328: 5322: 5319: 5315: 5314:0-333-21616-4 5311: 5307: 5301: 5299: 5297: 5295: 5291: 5285: 5280: 5276: 5272: 5268: 5264: 5260: 5253: 5251: 5249: 5247: 5245: 5241: 5236: 5232: 5228: 5224: 5220: 5213: 5211: 5207: 5202: 5198: 5194: 5190: 5186: 5182: 5178: 5174: 5170: 5163: 5160: 5155: 5151: 5147: 5143: 5139: 5135: 5131: 5127: 5123: 5119: 5115: 5108: 5105: 5100: 5096: 5092: 5088: 5084: 5080: 5076: 5069: 5066: 5061: 5057: 5053: 5049: 5045: 5041: 5037: 5030: 5027: 5022: 5018: 5014: 5010: 5006: 5002: 4998: 4994: 4990: 4986: 4982: 4978: 4974: 4967: 4964: 4958: 4956: 4954: 4952: 4948: 4943: 4939: 4935: 4931: 4927: 4923: 4919: 4915: 4911: 4904: 4902: 4898: 4893: 4889: 4885: 4881: 4877: 4873: 4869: 4865: 4861: 4854: 4852: 4848: 4843: 4839: 4835: 4831: 4827: 4823: 4819: 4815: 4810: 4805: 4802:(1): 014131. 4801: 4797: 4793: 4786: 4784: 4780: 4776: 4770: 4768: 4766: 4762: 4752: 4749: 4744: 4738: 4734: 4727: 4724: 4719: 4715: 4711: 4704: 4701: 4696: 4692: 4688: 4684: 4680: 4676: 4672: 4665: 4662: 4657: 4651: 4647: 4643: 4639: 4638: 4630: 4628: 4626: 4622: 4616: 4611: 4608: 4606: 4603: 4601: 4598: 4596: 4593: 4591: 4588: 4586: 4583: 4581: 4578: 4576: 4573: 4571: 4568: 4566: 4563: 4561: 4558: 4556: 4553: 4551: 4548: 4546: 4543: 4541: 4538: 4537: 4532: 4530: 4528: 4527:nanoparticles 4524: 4520: 4512: 4510: 4505: 4497: 4495: 4492: 4490: 4486: 4470: 4462: 4458: 4442: 4434: 4410: 4406: 4395: 4391: 4376: 4372: 4361: 4357: 4345: 4342: 4338: 4332: 4329: 4326: 4322: 4318: 4315: 4308: 4307: 4306: 4284: 4280: 4269: 4265: 4253: 4250: 4246: 4240: 4236: 4232: 4227: 4223: 4215: 4214: 4213: 4199: 4191: 4167: 4163: 4152: 4148: 4136: 4132: 4126: 4122: 4118: 4115: 4108: 4107: 4106: 4104: 4085: 4060: 4056: 4033: 4029: 4018: 4010: 4008: 4004: 4001: 3995: 3979: 3975: 3965: 3963: 3957: 3953: 3937: 3933: 3912: 3892: 3870: 3866: 3845: 3825: 3816: 3800: 3796: 3788: 3771: 3768: 3764: 3743: 3736: 3720: 3713: 3695: 3691: 3683: 3666: 3663: 3659: 3638: 3631: 3615: 3608: 3592: 3589: 3582: 3561: 3557: 3553: 3548: 3543: 3539: 3531: 3526: 3523: 3520: 3516: 3512: 3509: 3506: 3501: 3498: 3493: 3490: 3487: 3482: 3479: 3474: 3471: 3468: 3461: 3460: 3459: 3453: 3451: 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: 3174: 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: 2954: 2951: 2946: 2942: 2938: 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: 2802: 2794: 2790: 2786: 2781: 2777: 2763: 2756: 2754: 2740: 2737: 2734: 2731: 2727: 2724: 2721: 2718: 2714: 2710: 2701: 2693: 2689: 2685: 2680: 2676: 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: 2483: 2481: 2477: 2470: 2468: 2465: 2460: 2456: 2452: 2443: 2441: 2435: 2433: 2426: 2424: 2422: 2413: 2411: 2407: 2403: 2396: 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: 1956: 1954: 1952: 1948: 1941: 1939: 1936: 1928: 1926: 1924: 1920: 1916: 1910: 1894: 1891: 1888: 1865: 1861: 1850: 1843: 1841: 1834: 1832: 1830: 1825: 1823: 1819: 1815: 1811: 1810:steady states 1806: 1804: 1800: 1792: 1790: 1786: 1779: 1777: 1775: 1771: 1765: 1762: 1756: 1749: 1744: 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: 1434: 1428: 1425: 1423: 1420: 1419: 1417: 1416: 1411: 1405: 1404: 1397: 1394: 1393: 1386: 1383: 1382: 1375: 1374: 1373: 1372: 1369: 1365: 1360: 1357: 1355: 1352: 1350: 1346: 1342: 1341: 1337: 1334: 1333: 1331: 1330: 1325: 1319: 1316: 1314: 1311: 1309: 1306: 1304: 1301: 1299: 1296: 1294: 1291: 1289: 1286: 1285: 1283: 1282: 1279: 1275: 1270: 1267: 1266: 1262: 1259: 1257: 1254: 1252: 1249: 1248: 1246: 1245: 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: 433: 427: 424: 422: 419: 418: 416: 415: 412: 408: 403: 400: 398: 395: 394: 393: 392: 388: 384: 377: 372: 371: 360: 357: 355: 352: 350: 347: 346: 345: 344: 341: 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: 264: 259: 256: 254: 251: 249: 246: 244: 241: 239: 236: 234: 231: 229: 226: 224: 221: 220: 219: 218: 215: 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: 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 3115:∑ 3111:− 3105:Δ 3034:Prigogine 2997:… 2985:ξ 2972:ξ 2872:ξ 2835:… 2791:τ 2778:τ 2738:… 2690:ξ 2686:− 2677:ξ 2659:τ 2650:− 2627:ξ 2587:… 2575:ξ 2562:ξ 2331:∑ 2294:, whence 2208:∑ 2204:− 2132:∂ 2122:∂ 1892:− 1729:is lost. 1611:Waterston 1561:von Mayer 1516:de Donder 1506:Clapeyron 1486:Boltzmann 1481:Bernoulli 1442:Education 1413:Timelines 1197:− 1142:− 930:Equations 897:∂ 850:∂ 801:α 765:∂ 718:∂ 672:− 666:β 630:∂ 583:∂ 291:Adiabatic 281:Isochoric 267:Processes 228:Ideal gas 111:Classical 58:talk page 6809:(1984). 6807:Mazur P. 6684:(1974): 6633:(1977): 6615:: 1–96. 6563:(1971). 6476:10827167 6468:23520980 6361:16592132 6042:18573684 5985:13096766 5953:(1): 1. 5670:Archived 5154:59366545 5021:31762955 5013:15901339 4842:34412362 4533:See also 3630:pressure 1793:Overview 1663:Category 1601:Thompson 1511:Clausius 1491:Bridgman 1345:Vis viva 1327:Theories 1261:Gas laws 1053:Enthalpy 461:Pressure 276:Isobaric 233:Real gas 121:Chemical 104:Branches 7186:Related 7070:General 7065:Special 6923:Applied 6520:Sources 6384:Bibcode 6329:Bibcode 6252:Bibcode 6069:Bibcode 6022:Bibcode 5965:Bibcode 5914:Bibcode 5807:Bibcode 5598:Bibcode 5561:Bibcode 5519:Bibcode 5480:Bibcode 5271:Bibcode 5181:Bibcode 5126:Bibcode 5079:Calphad 4985:Bibcode 4922:Bibcode 4872:Bibcode 4814:Bibcode 4714:Bibcode 4683:Bibcode 3581:entropy 3058:entropy 2457:of the 2290:is the 2032:entropy 2028:entropy 1814:entropy 1799:Onsager 1738:entropy 1586:Smeaton 1581:Rankine 1571:Onsager 1556:Maxwell 1551:Massieu 1256:Entropy 1251:General 1242:History 1232:Culture 1229:History 453: ( 450:Entropy 387:italics 188:Systems 7097:Atomic 7052:Modern 6902:Major 6817: 6798: 6783: 6768: 6753: 6735: 6717: 6702: 6694: 6664: 6649: 6641: 6586: 6571: 6552: 6537: 6474: 6466: 6359: 6352:387963 6349: 6270: 6228: 6087: 6040: 5983: 5884: 5876: 5847: 5825: 5783: 5759: 5727: 5703: 5679: 5646: 5447: 5439: 5415: 5389: 5363: 5342: 5312: 5199: 5152: 5144: 5097: 5058: 5019: 5011: 5003: 4940: 4890: 4864:Energy 4840: 4832: 4739: 4652: 4190:matrix 3735:volume 3712:energy 2769:where 2459:matter 2305: 2266: 2258:where 2180: 1951:fluxes 1919:Kelvin 1576:Planck 1566:Nernst 1541:Kelvin 1501:Carnot 791: 656: 524: 466:Volume 381:Note: 340:Cycles 169:Second 159:Zeroth 56:. 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