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{{short description|Branch of thermodynamics}}

{{short description|Branch of thermodynamics}}

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. Some systems and processes are, however, in a useful sense, near enough to thermodynamic equilibrium to allow description with useful accuracy by currently known non-equilibrium thermodynamics. Nevertheless, many natural systems and processes will always remain far beyond the scope of non-equilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of free energy is lost.<ref>{{cite journal |last1=Bodenschatz |first1=Eberhard |last2=Cannell |first2=David S. |last3=de Bruyn |first3=John R. |last4=Ecke |first4=Robert |last5=Hu |first5=Yu-Chou |last6=Lerman |first6=Kristina |last7=Ahlers |first7=Guenter |title=Experiments on three systems with non-variational aspects |journal=Physica D: Nonlinear Phenomena |date=December 1992 |volume=61 |issue=1–4 |pages=77–93 |doi=10.1016/0167-2789(92)90150-L}}</ref>

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. Some systems and processes are, however, in a useful sense, near enough to thermodynamic equilibrium to allow description with useful accuracy by currently known non-equilibrium thermodynamics. Nevertheless, many natural systems and processes will always remain far beyond the scope of non-equilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of free energy is lost.<ref>{{cite journal |last1=Bodenschatz |first1=Eberhard |last2=Cannell |first2=David S. |last3=de Bruyn |first3=John R. |last4=Ecke |first4=Robert |last5=Hu |first5=Yu-Chou |last6=Lerman |first6=Kristina |last7=Ahlers |first7=Guenter |title=Experiments on three systems with non-variational aspects |journal=Physica D: Nonlinear Phenomena |date=December 1992 |volume=61 |issue=1–4 |pages=77–93 |doi=10.1016/0167-2789(92)90150-L}}</ref>

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. Some systems and processes are, however, in a useful sense, near enough to thermodynamic equilibrium to allow description with useful accuracy by currently known non-equilibrium thermodynamics. Nevertheless, many natural systems and processes will always remain far beyond the scope of non-equilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of free energy is lost. Equilibrium thermodynamics does, however, for theoretical development, use the idealized concept of the "quasi-static process". A quasi-static process is a conceptual (timeless and physically impossible) smooth mathematical passage along a continuous path of states of thermodynamic equilibrium. In reality, these requirements are very demanding, and it may be difficult or practically, or even theoretically, impossible to satisfy them. This is part of why non-equilibrium thermodynamics is a work in progress.

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. Some systems and processes are, however, in a useful sense, near enough to thermodynamic equilibrium to allow description with useful accuracy by currently known non-equilibrium thermodynamics. Nevertheless, many natural systems and processes will always remain far beyond the scope of non-equilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of free energy is lost.

The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by [[equilibrium thermodynamics]]. 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 or impossibility, in general, in defining [[entropy]] at an instant of time in macroscopic terms for systems not in thermodynamic equilibrium; it can be done, to useful approximation, only in carefully chosen special cases, namely those that are throughout in local thermodynamic equilibrium.<ref name="Grandy 2008">Grandy, W.T., Jr (2008).</ref><ref name="Lebon Jou Casas-Vázquez 2008">Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, e-{{ISBN|978-3-540-74252-4}}.</ref>

The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by [[equilibrium thermodynamics]]. 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 or impossibility, in general, in defining [[entropy]] at an instant of time in macroscopic terms for systems not in thermodynamic equilibrium; it can be done, to useful approximation, only in carefully chosen special cases, namely those that are throughout in local thermodynamic equilibrium.<ref name="Grandy 2008">Grandy, W.T., Jr (2008).</ref><ref name="Lebon Jou Casas-Vázquez 2008">Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, e-{{ISBN|978-3-540-74252-4}}.</ref>

==Scope==

==Scope==

===Difference between equilibrium and non-equilibrium thermodynamics===

===Difference between equilibrium and non-equilibrium thermodynamics===

One problem of interest is the thermodynamic study of non-equilibrium steady states, in which entropy production and some flows are non-zero, but there is no time variation of physical variables.

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.

 

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.

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.

Equilibrium thermodynamics restricts its considerations to processes that have initial and final states of thermodynamic equilibrium; the time-courses of processes are deliberately ignored. Consequently, equilibrium thermodynamics allows processes that pass through states far from thermodynamic equilibrium, that cannot be described even by the variables admitted for non-equilibrium thermodynamics,<ref name="EY 5">[[Elliott H. Lieb|Lieb, E.H.]], [[Jakob Yngvason|Yngvason, J.]] (1999), p.&nbsp;5.</ref> such as time rates of change of temperature and pressure.<ref>Gyarmati, I. (1967/1970), pp.&nbsp;8–12.</ref> For example, in equilibrium thermodynamics, a process is allowed to include even a violent explosion that cannot be described by non-equilibrium thermodynamics.<ref name="EY 5"/> Equilibrium thermodynamics does, however, for theoretical development, use the idealized concept of the "quasi-static process". A quasi-static process is a conceptual (timeless and physically impossible) smooth mathematical passage along a continuous path of states of thermodynamic equilibrium.<ref>[[Herbert Callen|Callen, H.B.]] (1960/1985), §&nbsp;4–2.</ref> It is an exercise in differential geometry rather than a process that could occur in actuality.

Equilibrium thermodynamics restricts its considerations to processes that have initial and final states of thermodynamic equilibrium; the time-courses of processes are deliberately ignored. Consequently, equilibrium thermodynamics allows processes that pass through states far from thermodynamic equilibrium, that cannot be described even by the variables admitted for non-equilibrium thermodynamics,<ref name="EY 5">[[Elliott H. Lieb|Lieb, E.H.]], [[Jakob Yngvason|Yngvason, J.]] (1999), p.&nbsp;5.</ref> such as time rates of change of temperature and pressure.<ref>Gyarmati, I. (1967/1970), pp.&nbsp;8–12.</ref> For example, in equilibrium thermodynamics, a process is allowed to include even a violent explosion that cannot be described by non-equilibrium thermodynamics.<ref name="EY 5"/> Equilibrium thermodynamics does, however, for theoretical development, use the idealized concept of the "quasi-static process". A quasi-static process is a conceptual (timeless and physically impossible) smooth mathematical passage along a continuous path of states of thermodynamic equilibrium.<ref>[[Herbert Callen|Callen, H.B.]] (1960/1985), §&nbsp;4–2.</ref> It is an exercise in differential geometry rather than a process that could occur in actuality.

Non-equilibrium thermodynamics, on the other hand, attempting to describe continuous time-courses, needs its [[State function|state variables]] to have a very close connection with those of equilibrium thermodynamics.<ref>Glansdorff, P., Prigogine, I. (1971), Ch. {{math|II}},§&nbsp;2.</ref> This profoundly restricts the scope of non-equilibrium thermodynamics, and places heavy demands on its conceptual framework.

Non-equilibrium thermodynamics, on the other hand, attempting to describe continuous time-courses, needs its [[State function|state variables]] to have a very close connection with those of equilibrium thermodynamics.<ref>Glansdorff, P., Prigogine, I. (1971), Ch. {{math|II}},§&nbsp;2.</ref> This profoundly restricts the scope of non-equilibrium thermodynamics, and places heavy demands on its conceptual framework.

The suitable relationship that defines non-equilibrium thermodynamic state variables is as follows. On occasions when the system happens to be in states that are sufficiently close to thermodynamic 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 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. 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. On occasions when the system is sufficiently close to thermodynamic equilibrium, intensive non-equilibrium 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.<ref name="Gyarmati 1970"/> In reality, these requirements are very demanding, and it may be difficult or practically, or even theoretically, impossible to satisfy them. This is part of why non-equilibrium thermodynamics is a work in progress.

The suitable relationship that defines non-equilibrium thermodynamic state variables is as follows. On occasions when the system happens to be in states that are sufficiently close to thermodynamic 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 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. 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. On occasions when the system is sufficiently close to thermodynamic equilibrium, intensive non-equilibrium 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.<ref name="Gyarmati 1970"/> In reality, these requirements are very demanding, and it may be difficult or practically, or even theoretically, impossible to satisfy them. This is part of why non-equilibrium thermodynamics is a work in progress.

Non-equilibrium thermodynamics is a work in progress, not an established edifice. This article is an attempt to sketch some approaches to it and some concepts important for it.

Non-equilibrium thermodynamics is a work in progress, not an established edifice. This article is an attempt to sketch some approaches to it and some concepts important for it.

Some concepts of particular importance for non-equilibrium thermodynamics include time rate of dissipation of energy (Rayleigh 1873,<ref name="Rayleigh 1873">{{Cite journal | last1 = Strutt | first1 = J. W. | doi = 10.1112/plms/s1-4.1.357 | title = Some General Theorems relating to Vibrations | journal = Proceedings of the London Mathematical Society | year = 1871 | volume = s1-4 | pages = 357–368 | url = https://zenodo.org/record/1447754 }}</ref> [[Lars Onsager|Onsager]] 1931,<ref name="Onsager 1931 I">{{Cite journal | doi = 10.1103/PhysRev.37.405 | last1 = Onsager | first1 = L. | year = 1931 | title = Reciprocal relations in irreversible processes, I | url = | journal = Physical Review | volume = 37 | issue = 4| pages = 405–426 |bibcode = 1931PhRv...37..405O | doi-access = free }}</ref> also<ref name="Gyarmati 1970">Gyarmati, I. (1967/1970).</ref><ref name="Lavenda 1978">Lavenda, B.H. (1978). ''Thermodynamics of Irreversible Processes'', Macmillan, London, {{ISBN|0-333-21616-4}}.</ref>), time rate of entropy production (Onsager 1931),<ref name="Onsager 1931 I"/> thermodynamic fields,<ref>Gyarmati, I. (1967/1970), pages 4-14.</ref><ref name="Ziegler 1983">Ziegler, H., (1983). ''An Introduction to Thermomechanics'', North-Holland, Amsterdam, {{ISBN|0-444-86503-9}}.</ref><ref name="Balescu">Balescu, R. (1975). ''Equilibrium and Non-equilibrium Statistical Mechanics'', Wiley-Interscience, New York, {{ISBN|0-471-04600-0}}, Section 3.2, pages 64-72.</ref> [[dissipative structure]],<ref name="G&P 1971"/> and non-linear dynamical structure.<ref name="Lavenda 1978"/>

Some concepts of particular importance for non-equilibrium thermodynamics include time rate of dissipation of energy (Rayleigh 1873,<ref name="Rayleigh 1873">{{Cite journal | last1 = Strutt | first1 = J. W. | doi = 10.1112/plms/s1-4.1.357 | title = Some General Theorems relating to Vibrations | journal = Proceedings of the London Mathematical Society | year = 1871 | volume = s1-4 | pages = 357–368 | url = https://zenodo.org/record/1447754 }}</ref> [[Lars Onsager|Onsager]] 1931,<ref name="Onsager 1931 I">{{Cite journal | doi = 10.1103/PhysRev.37.405 | last1 = Onsager | first1 = L. | year = 1931 | title = Reciprocal relations in irreversible processes, I | url = | journal = Physical Review | volume = 37 | issue = 4| pages = 405–426 |bibcode = 1931PhRv...37..405O | doi-access = free }}</ref> also<ref name="Gyarmati 1970">Gyarmati, I. (1967/1970).</ref><ref name="Lavenda 1978">Lavenda, B.H. (1978). ''Thermodynamics of Irreversible Processes'', Macmillan, London, {{ISBN|0-333-21616-4}}.</ref>), time rate of entropy production (Onsager 1931),<ref name="Onsager 1931 I"/> thermodynamic fields,<ref>Gyarmati, I. (1967/1970), pages 4-14.</ref><ref name="Ziegler 1983">Ziegler, H., (1983). ''An Introduction to Thermomechanics'', North-Holland, Amsterdam, {{ISBN|0-444-86503-9}}.</ref><ref name="Balescu">Balescu, R. (1975). ''Equilibrium and Non-equilibrium Statistical Mechanics'', Wiley-Interscience, New York, {{ISBN|0-471-04600-0}}, Section 3.2, pages 64-72.</ref> [[dissipative structure]],<ref name="G&P 1971"/> and non-linear dynamical structure.<ref name="Lavenda 1978"/>

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 Onsager reciprocal relations.

The terms 'classical irreversible thermodynamics' 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.

One problem of interest is the thermodynamic study of non-equilibrium [[steady state]]s, in which [[entropy]] production and some [[flux|flows]] are non-zero, but there is no [[Time-variant system|time variation]] of physical variables.

One problem of interest is the thermodynamic study of non-equilibrium [[steady state]]s, in which [[entropy]] production and some [[flux|flows]] are non-zero, but there is no [[Time-variant system|time variation]] of physical variables.

Establishing the relation between such forces and flux densities is a problem in statistical mechanics. Flux densities (<math>J_i</math>) 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.

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 remote from one another. In the classical irreversible thermodynamic approach, there is allowed very small spatial variation, from very small volume element to adjacent very small 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 means that spatial structure cannot contribute as it properly should to the global entropy assessment for the system. This approach assumes spatial and temporal continuity and even differentiability of locally defined intensive variables such as temperature and internal energy density. All of these are very stringent demands. Consequently, this approach can deal with only a very limited range of phenomena. This approach is nevertheless valuable because it can deal well with some macroscopically observable phenomena.

建立这种力量和通量密度之间的关系在统计力学是一个问题。通量密度(< math > j _ i </math >)可能是耦合的。本文在昂萨格互反关系的基础上，考虑了稳定的近稳态热动力学非平衡态，它在力和流量密度方面具有线性动力学性质。

同时假设局部熵密度与平衡状态下其他局部密集型变量的函数相同，这被称为局部热力学平衡假设(参见 Keizer (1987))。辐射之所以被忽略，是因为它是能量在区域之间的转移，而区域之间可以相互远离。在经典的不可逆热力学方法中，允许有非常小的空间变化，从非常小的体积元到相邻的非常小的体积元，但是假定系统的总体熵可以通过简单的局部熵密度的空间积分得到，这意味着空间结构不能对系统的总体熵评价作出贡献，因为它应该对系统的总体熵评价作出贡献。这种方法假设空间和时间的连续性，甚至可微的局部定义的强度变量，如温度和内部能量密度。所有这些都是非常严格的要求。因此，这种方法只能处理非常有限范围的现象。然而，这种方法是有价值的，因为它可以很好地处理一些宏观上可观察到的现象。

One initial approach to non-equilibrium thermodynamics is sometimes called 'classical irreversible thermodynamics'.<ref name="Lebon Jou Casas-Vázquez 2008"/> There are other approaches to non-equilibrium thermodynamics, for example [[extended irreversible thermodynamics]],<ref name="Lebon Jou Casas-Vázquez 2008"/><ref name="JCVL 1993"/> and generalized thermodynamics,<ref>Eu, B.C. (2002).</ref> but they are hardly touched on in the present article.

One initial approach to non-equilibrium thermodynamics is sometimes called 'classical irreversible thermodynamics'.<ref name="Lebon Jou Casas-Vázquez 2008"/> There are other approaches to non-equilibrium thermodynamics, for example [[extended irreversible thermodynamics]],<ref name="Lebon Jou Casas-Vázquez 2008"/><ref name="JCVL 1993"/> and generalized thermodynamics,<ref>Eu, B.C. (2002).</ref> but they are hardly touched on in the present article.

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 Ilya Prigogine 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.

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 thermoelectric phenomena known as the Seebeck and the Peltier effects, considered by Kelvin in the nineteenth century and by Lars Onsager 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.

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

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

According to Wildt<ref name="Wildt 1972">{{Cite journal |last=Wildt |first=R. |year=1972 |title=Thermodynamics of the gray atmosphere. IV. Entropy transfer and production |journal=Astrophysical Journal |volume=174 |issue= |pages=69–77 |doi=10.1086/151469 |bibcode=1972ApJ...174...69W}}</ref> (see also Essex<ref name="Essex 1984a">{{Cite journal |last=Essex |first=C. |year=1984a |title=Radiation and the irreversible thermodynamics of climate |journal=Journal of the Atmospheric Sciences |volume=41 |issue=12 |pages=1985–1991 |doi=10.1175/1520-0469(1984)041<1985:RATITO>2.0.CO;2 |bibcode = 1984JAtS...41.1985E |doi-access=free }}.</ref><ref name="Essex 1984b">{{Cite journal |last=Essex |first=C. |year=1984b |title=Minimum entropy production in the steady state and radiative transfer |journal=Astrophysical Journal |volume=285 |issue= |pages=279–293 |doi=10.1086/162504 |bibcode=1984ApJ...285..279E}}</ref><ref name="Essex 1984c">{{Cite journal |last=Essex |first=C. |year=1984c |title=Radiation and the violation of bilinearity in the irreversible thermodynamics of irreversible processes |journal=Planetary and Space Science |volume=32 |pages=1035–1043 |doi=10.1016/0032-0633(84)90060-6 |bibcode = 1984P&SS...32.1035E |issue=8 }}</ref>), 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 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.

According to Wildt<ref name="Wildt 1972">{{Cite journal |last=Wildt |first=R. |year=1972 |title=Thermodynamics of the gray atmosphere. IV. Entropy transfer and production |journal=Astrophysical Journal |volume=174 |issue= |pages=69–77 |doi=10.1086/151469 |bibcode=1972ApJ...174...69W}}</ref> (see also Essex<ref name="Essex 1984a">{{Cite journal |last=Essex |first=C. |year=1984a |title=Radiation and the irreversible thermodynamics of climate |journal=Journal of the Atmospheric Sciences |volume=41 |issue=12 |pages=1985–1991 |doi=10.1175/1520-0469(1984)041<1985:RATITO>2.0.CO;2 |bibcode = 1984JAtS...41.1985E |doi-access=free }}.</ref><ref name="Essex 1984b">{{Cite journal |last=Essex |first=C. |year=1984b |title=Minimum entropy production in the steady state and radiative transfer |journal=Astrophysical Journal |volume=285 |issue= |pages=279–293 |doi=10.1086/162504 |bibcode=1984ApJ...285..279E}}</ref><ref name="Essex 1984c">{{Cite journal |last=Essex |first=C. |year=1984c |title=Radiation and the violation of bilinearity in the irreversible thermodynamics of irreversible processes |journal=Planetary and Space Science |volume=32 |pages=1035–1043 |doi=10.1016/0032-0633(84)90060-6 |bibcode = 1984P&SS...32.1035E |issue=8 }}</ref>), 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 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.

===Local equilibrium thermodynamics===

===Local equilibrium thermodynamics===

Following Section III of Rayleigh (1873), Onsager (1931, I) showed that in the regime where both the flows (<math>J_i</math>) are small and the thermodynamic forces (<math>F_i</math>) vary slowly, the rate of creation of entropy <math>(\sigma)</math> is linearly related to the flows:

Extended irreversible thermodynamics 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 fluxes of mass, momentum and energy and eventually higher order fluxes.

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<ref name="Gyarmati 1970"/><ref name="Lavenda 1978"/><ref name="G&P 1971">Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.</ref><ref name="JCVL 1993">Jou, D., Casas-Vázquez, J., Lebon, G. (1993). ''Extended Irreversible Thermodynamics'', Springer, Berlin, {{ISBN|3-540-55874-8}}, {{ISBN|0-387-55874-8}}.</ref><ref name="De Groot Mazur 1962">De Groot, S.R., Mazur, P. (1962). ''Non-equilibrium Thermodynamics'', North-Holland, Amsterdam.</ref><ref name="Balescu 1975">Balescu, R. (1975). ''Equilibrium and Non-equilibrium Statistical Mechanics'', John Wiley & Sons, New York, {{ISBN|0-471-04600-0}}.</ref><ref name="Mihalas Mihalas 1984">[http://www.filestube.com/9c5b2744807c2c3d03e9/details.html Mihalas, D., Weibel-Mihalas, B. (1984). ''Foundations of Radiation Hydrodynamics'', Oxford University Press, New York] {{ISBN|0-19-503437-6}}.</ref><ref name="Schloegl 1989">Schloegl, F. (1989). ''Probability and Heat: Fundamentals of Thermostatistics'', Freidr. Vieweg & Sohn, Braunschweig, {{ISBN|3-528-06343-2}}.</ref> (see also Keizer (1987)<ref name="Keizer 1987">Keizer, J. (1987). ''Statistical Thermodynamics of Nonequilibrium Processes'', Springer-Verlag, New York, {{ISBN|0-387-96501-7}}.</ref>). Radiation is ignored because it is transfer of energy between regions, which can be remote from one another. In the classical irreversible thermodynamic approach, there is allowed very small spatial variation, from very small volume element to adjacent very small 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 means that spatial structure cannot contribute as it properly should to the global entropy assessment for the system. This approach assumes spatial and temporal continuity and even differentiability of locally defined intensive variables such as temperature and internal energy density. All of these are very stringent demands. Consequently, this approach can deal with only a very limited range of phenomena. This approach is nevertheless valuable because it can deal well with some macroscopically observable phenomena.{{examples|date=February 2015}}

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<ref name="Gyarmati 1970"/><ref name="Lavenda 1978"/><ref name="G&P 1971">Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.</ref><ref name="JCVL 1993">Jou, D., Casas-Vázquez, J., Lebon, G. (1993). ''Extended Irreversible Thermodynamics'', Springer, Berlin, {{ISBN|3-540-55874-8}}, {{ISBN|0-387-55874-8}}.</ref><ref name="De Groot Mazur 1962">De Groot, S.R., Mazur, P. (1962). ''Non-equilibrium Thermodynamics'', North-Holland, Amsterdam.</ref><ref name="Balescu 1975">Balescu, R. (1975). ''Equilibrium and Non-equilibrium Statistical Mechanics'', John Wiley & Sons, New York, {{ISBN|0-471-04600-0}}.</ref><ref name="Mihalas Mihalas 1984">[http://www.filestube.com/9c5b2744807c2c3d03e9/details.html Mihalas, D., Weibel-Mihalas, B. (1984). ''Foundations of Radiation Hydrodynamics'', Oxford University Press, New York] {{ISBN|0-19-503437-6}}.</ref><ref name="Schloegl 1989">Schloegl, F. (1989). ''Probability and Heat: Fundamentals of Thermostatistics'', Freidr. Vieweg & Sohn, Braunschweig, {{ISBN|3-528-06343-2}}.</ref> (see also Keizer (1987)<ref name="Keizer 1987">Keizer, J. (1987). ''Statistical Thermodynamics of Nonequilibrium Processes'', Springer-Verlag, New York, {{ISBN|0-387-96501-7}}.</ref>). Radiation is ignored because it is transfer of energy between regions, which can be remote from one another. In the classical irreversible thermodynamic approach, there is allowed very small spatial variation, from very small volume element to adjacent very small 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 means that spatial structure cannot contribute as it properly should to the global entropy assessment for the system. This approach assumes spatial and temporal continuity and even differentiability of locally defined intensive variables such as temperature and internal energy density. All of these are very stringent demands. Consequently, this approach can deal with only a very limited range of phenomena. This approach is nevertheless valuable because it can deal well with some macroscopically observable phenomena.{{examples|date=February 2015}}

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 [[thermoelectric effect|thermoelectric phenomena]] known as the Seebeck and the Peltier effects, considered by [[William Thomson, 1st Baron Kelvin|Kelvin]] in the nineteenth century and by [[Lars Onsager]] in the twentieth.<ref name="De Groot Mazur 1962"/><ref>Kondepudi, D. (2008). ''Introduction to Modern Thermodynamics'', Wiley, Chichester UK, {{ISBN|978-0-470-01598-8}}, pages 333-338.</ref> These effects occur at metal junctions, which were originally effectively treated as two-dimensional surfaces, with no spatial volume, and no spatial variation.

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 [[thermoelectric effect|thermoelectric phenomena]] known as the Seebeck and the Peltier effects, considered by [[William Thomson, 1st Baron Kelvin|Kelvin]] in the nineteenth century and by [[Lars Onsager]] in the twentieth.<ref name="De Groot Mazur 1962"/><ref>Kondepudi, D. (2008). ''Introduction to Modern Thermodynamics'', Wiley, Chichester UK, {{ISBN|978-0-470-01598-8}}, pages 333-338.</ref> These effects occur at metal junctions, which were originally effectively treated as two-dimensional surfaces, with no spatial volume, and no spatial variation.

and the flows are related to the gradient of the forces, parametrized by a matrix of coefficients conventionally denoted <math>L</math>:

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 Couette flow, a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls. Laser 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, 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 complex fluids, turbulent systems and glasses are other examples of non-equilibrium systems.

====Local equilibrium thermodynamics with materials with "memory"====

====Local equilibrium thermodynamics with materials with "memory"====

A further extension of local equilibrium thermodynamics is to allow that materials may have "memory", so that their [[constitutive equation]]s 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.<ref>{{cite journal | last1 = Coleman | first1 = B.D. | last2 = Noll | first2 = W. | year = 1963 | title = The thermodynamics of elastic materials with heat conduction and viscosity | url = | journal = Arch. Ration. Mach. Analysis | volume = 13 | issue = 1| pages = 167–178 | doi=10.1007/bf01262690| bibcode = 1963ArRMA..13..167C | s2cid = 189793830 }}</ref>

A further extension of local equilibrium thermodynamics is to allow that materials may have "memory", so that their [[constitutive equation]]s 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.<ref>{{cite journal | last1 = Coleman | first1 = B.D. | last2 = Noll | first2 = W. | year = 1963 | title = The thermodynamics of elastic materials with heat conduction and viscosity | url = | journal = Arch. Ration. Mach. Analysis | volume = 13 | issue = 1| pages = 167–178 | doi=10.1007/bf01262690| bibcode = 1963ArRMA..13..167C | s2cid = 189793830 }}</ref>

<math>J_i = \sum_{j} L_{ij} \frac{\partial F_j}{\partial x_j} </math>

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 extensive quantities. 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 Helmholtz free energy (A = U - TS), a Legendre transformation of the energy. If, next to fluctuations of the energy, the macroscopic dimensions (volume) of the system are left fluctuating, we use the Gibbs free energy (G = U + PV - TS), where the system's properties are determined both by the temperature and by the pressure.

===Extended irreversible thermodynamics===

===Extended irreversible thermodynamics===

from which it follows that:

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 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 entropy in equilibrium thermodynamics. That is why in such cases a more generalized Legendre transformation should be considered. This is the extended Massieu potential.

'''Extended irreversible thermodynamics''' 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 [[flux]]es of mass, momentum and energy and eventually higher order fluxes.

'''Extended irreversible thermodynamics''' 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 [[flux]]es 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.

The formalism is well-suited for describing high-frequency processes and small-length scales materials.

==Basic concepts==

==Basic concepts==

The second law of thermodynamics requires that the matrix <math>L</math> be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that the matrix <math>L</math> is symmetric. This fact is called the Onsager reciprocal relations.

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 [[Couette flow]], a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls. [[Laser]] 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, 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 [[complex fluids]], turbulent systems and glasses are other examples of non-equilibrium systems.

The generalization of the above equations for the rate of creation of entropy was given by Pokrovskii.

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 [[extensive quantity|extensive quantiti]]es. 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 [[Helmholtz free energy]] (''A'' = ''U'' - ''TS''), a [[Legendre transformation]] of the energy. If, next to fluctuations of the energy, the macroscopic dimensions (volume) of the system are left fluctuating, we use the [[Gibbs free energy]] (''G'' = ''U'' + ''PV'' - ''TS''), where the system's properties are determined both by the temperature and by the pressure.

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 [[extensive quantity|extensive quantiti]]es. 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 [[Helmholtz free energy]] (''A'' = ''U'' - ''TS''), a [[Legendre transformation]] of the energy. If, next to fluctuations of the energy, the macroscopic dimensions (volume) of the system are left fluctuating, we use the [[Gibbs free energy]] (''G'' = ''U'' + ''PV'' - ''TS''), where the system's properties are determined both by the temperature and by the pressure.

[ math > k { rm b } m = s-sum _ i (i _ i e _ i) ,

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 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 [[entropy]] in equilibrium thermodynamics. That is why in such cases a more generalized Legendre transformation should be considered. This is the extended Massieu potential.

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 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 [[entropy]] 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 [[entropy]] (''S'') is a function of the collection of [[extensive quantity|extensive quantiti]]es <math>E_i</math>. Each extensive quantity has a conjugate intensive variable <math>I_i</math> (a restricted definition of intensive variable is used here by comparison to the definition given in this link) so that:

By definition, the [[entropy]] (''S'') is a function of the collection of [[extensive quantity|extensive quantiti]]es <math>E_i</math>. Each extensive quantity has a conjugate intensive variable <math>I_i</math> (a restricted definition of intensive variable is used here by comparison to the definition given in this link) so that:

Until recently, prospects for useful extremal principles in this area have seemed clouded. Nicolis (1999) 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).

<math>\ k_{\rm B} \, dM = \sum_i (E_i \, dI_i).</math>

直到最近，这个领域中有用的极端原理的前景似乎还很模糊。尼古利斯(1999)非常谨慎，在许多情况下难以定义内产生熵，他发现有时为了预测一个过程的进程，一个叫做能量耗散率的极值可能比产生熵的极值更有用; 这个量出现在昂萨格尔1931年创立的这个主题中。其他作家也认为，一般的全球极端原则的前景是模糊的。这些作家包括格兰斯多夫和普里戈金(1971年)、莱邦、乔和卡萨斯-瓦斯奎斯(2008年) ，以及伊尔哈维(1997年)。

<math>\ k_{\rm B} \, dM = \sum_i (E_i \, dI_i).</math>

:<math> I_i = \frac{\partial{S}}{\partial{E_i}}.</math>

:<math> I_i = \frac{\partial{S}}{\partial{E_i}}.</math>

We then define the extended [[Massieu function]] as follows:

We then define the extended [[Massieu function]] as follows:

Non-equilibrium thermodynamics has been successfully applied to describe biological processes such as protein folding/unfolding and transport through membranes.

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 thermodynamic forces driving fluxes of extensive properties through the system.

:<math>\ k_{\rm B} M = S - \sum_i( I_i E_i),</math>

:<math>\ k_{\rm B} M = S - \sum_i( I_i E_i),</math>

Also, ideas from non-equilibrium thermodynamics and the informatic theory of entropy have been adapted to describe general economic systems.

where <math>\ k_{\rm B}</math> is [[Boltzmann's constant]], whence

The independent variables are the intensities.

The independent variables are the intensities.

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 thermodynamic forces driving fluxes of extensive properties through the system.

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 thermodynamic forces driving fluxes of extensive properties through the system.

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.

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.

==Stationary states, fluctuations, and stability==

==Stationary states, fluctuations, and stability==

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 significant level of probability. Fluctuations about stable stationary states are extremely small except near critical points (Kondepudi and Prigogine 1998, page 323).<ref>Kondepudi, D., Prigogine, I, (1998). ''Modern Thermodynamics. From Heat Engines to Dissipative Structures'', Wiley, Chichester, 1998, {{ISBN|0-471-97394-7}}.</ref> 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.

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 significant level of probability. Fluctuations about stable stationary states are extremely small except near critical points (Kondepudi and Prigogine 1998, page 323).<ref>Kondepudi, D., Prigogine, I, (1998). ''Modern Thermodynamics. From Heat Engines to Dissipative Structures'', Wiley, Chichester, 1998, {{ISBN|0-471-97394-7}}.</ref> 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.

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.

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.

==Local thermodynamic equilibrium==

==Local thermodynamic equilibrium==

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 ''local thermodynamic equilibrium''.

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 ''local thermodynamic equilibrium''.

''Local thermodynamic equilibrium of matter''<ref name="Gyarmati 1970"/><ref name="G&P 1971"/><ref name="Balescu 1975"/><ref name="Mihalas Mihalas 1984"/><ref name="Schloegl 1989"/> (see also Keizer (1987)<ref name="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.

''Local thermodynamic equilibrium of matter''<ref name="Gyarmati 1970"/><ref name="G&P 1971"/><ref name="Balescu 1975"/><ref name="Mihalas Mihalas 1984"/><ref name="Schloegl 1989"/> (see also Keizer (1987)<ref name="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.

One can think here of two 'relaxation times' separated by order of magnitude.<ref name="Zubarev 1971/1974">[[Dmitry Zubarev|Zubarev D. N.]],(1974). ''[https://books.google.com/books?id=SQy3AAAAIAAJ&hl=ru&source=gbs_ViewAPI Nonequilibrium Statistical Thermodynamics]'', translated from the Russian by P.J. Shepherd, New York, Consultants Bureau. {{ISBN|0-306-10895-X}}; {{ISBN|978-0-306-10895-2}}.</ref> 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 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<ref name="Zubarev 1971/1974"/> and other approaches have to be proposed, see for instance [[Extended irreversible thermodynamics]]. 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&nbsp;km where sound propagates, but not above 100&nbsp;km, where, because of the paucity of intermolecular collisions, sound does not propagate.

One can think here of two 'relaxation times' separated by order of magnitude.<ref name="Zubarev 1971/1974">[[Dmitry Zubarev|Zubarev D. N.]],(1974). ''[https://books.google.com/books?id=SQy3AAAAIAAJ&hl=ru&source=gbs_ViewAPI Nonequilibrium Statistical Thermodynamics]'', translated from the Russian by P.J. Shepherd, New York, Consultants Bureau. {{ISBN|0-306-10895-X}}; {{ISBN|978-0-306-10895-2}}.</ref> 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 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<ref name="Zubarev 1971/1974"/> and other approaches have to be proposed, see for instance [[Extended irreversible thermodynamics]]. 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&nbsp;km where sound propagates, but not above 100&nbsp;km, where, because of the paucity of intermolecular collisions, sound does not propagate.

===Milne's definition in terms of radiative equilibrium===

===Milne's definition in terms of radiative equilibrium===

[[Edward Arthur Milne|Edward A. Milne]], thinking about stars, gave a definition of 'local thermodynamic equilibrium' in terms of the [[thermal radiation]] of the [[matter]] in each small local 'cell'.<ref name="Milne 1928">{{cite journal | last1= Milne |first1= E.A. |year=1928 | title= The effect of collisions on monochromatic radiative equilibrium |journal=[[Monthly Notices of the Royal Astronomical Society]] | volume= 88|issue= 6 |pages=493–502|bibcode=1928MNRAS..88..493M | doi = 10.1093/mnras/88.6.493 |doi-access= free }}</ref> 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 [[temperature]] 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 exceed the rates of creation and annihilation of photons.

[[Edward Arthur Milne|Edward A. Milne]], thinking about stars, gave a definition of 'local thermodynamic equilibrium' in terms of the [[thermal radiation]] of the [[matter]] in each small local 'cell'.<ref name="Milne 1928">{{cite journal | last1= Milne |first1= E.A. |year=1928 | title= The effect of collisions on monochromatic radiative equilibrium |journal=[[Monthly Notices of the Royal Astronomical Society]] | volume= 88|issue= 6 |pages=493–502|bibcode=1928MNRAS..88..493M | doi = 10.1093/mnras/88.6.493 |doi-access= free }}</ref> 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 [[temperature]] 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 exceed the rates of creation and annihilation of photons.

==Entropy in evolving systems==

==Entropy in evolving systems==

It is pointed out by W.T. Grandy Jr,<ref>{{cite journal | doi = 10.1023/B:FOOP.0000012007.06843.ed | title = Time Evolution in Macroscopic Systems. I. Equations of Motion | year = 2004 | last1 = Grandy | first1 = W.T., Jr. | journal = Foundations of Physics | volume = 34 | issue = 1 | page = 1 |url=http://physics.uwyo.edu/~tgrandy/evolve.html |arxiv = cond-mat/0303290 |bibcode = 2004FoPh...34....1G }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/entropy.html | doi=10.1023/B:FOOP.0000012008.36856.c1 | title=Time Evolution in Macroscopic Systems. II. The Entropy | year=2004 | last1=Grandy | first1=W.T., Jr. | journal=Foundations of Physics | volume=34 | issue=1 | page=21 |arxiv = cond-mat/0303291 |bibcode = 2004FoPh...34...21G | s2cid=18573684 }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/applications.html | doi = 10.1023/B:FOOP.0000022187.45866.81 | title=Time Evolution in Macroscopic Systems. III: Selected Applications | year=2004 | last1=Grandy | first1=W. T., Jr | journal=Foundations of Physics | volume=34 | issue=5 | page=771 |bibcode = 2004FoPh...34..771G | s2cid = 119406182 }}</ref><ref>Grandy 2004 see also [http://physics.uwyo.edu/~tgrandy/Statistical_Mechanics.html].</ref> 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 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.

It is pointed out by W.T. Grandy Jr,<ref>{{cite journal | doi = 10.1023/B:FOOP.0000012007.06843.ed | title = Time Evolution in Macroscopic Systems. I. Equations of Motion | year = 2004 | last1 = Grandy | first1 = W.T., Jr. | journal = Foundations of Physics | volume = 34 | issue = 1 | page = 1 |url=http://physics.uwyo.edu/~tgrandy/evolve.html |arxiv = cond-mat/0303290 |bibcode = 2004FoPh...34....1G }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/entropy.html | doi=10.1023/B:FOOP.0000012008.36856.c1 | title=Time Evolution in Macroscopic Systems. II. The Entropy | year=2004 | last1=Grandy | first1=W.T., Jr. | journal=Foundations of Physics | volume=34 | issue=1 | page=21 |arxiv = cond-mat/0303291 |bibcode = 2004FoPh...34...21G | s2cid=18573684 }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/applications.html | doi = 10.1023/B:FOOP.0000022187.45866.81 | title=Time Evolution in Macroscopic Systems. III: Selected Applications | year=2004 | last1=Grandy | first1=W. T., Jr | journal=Foundations of Physics | volume=34 | issue=5 | page=771 |bibcode = 2004FoPh...34..771G | s2cid = 119406182 }}</ref><ref>Grandy 2004 see also [http://physics.uwyo.edu/~tgrandy/Statistical_Mechanics.html].</ref> 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 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.

This point of view shares many points in common with the concept and the use of entropy in continuum thermomechanics,<ref>{{cite book| title = Rational Thermodynamics | year = 1984 | last1 = Truesdell | first1 = Clifford | publisher = Springer | edition = 2 }}</ref><ref>{{cite book| title = Continuum Thermomechanics | year = 2002 | last1 = Maugin | first1 = Gérard A. | publisher = Kluwer }}</ref><ref>{{cite book| title = The Mechanics and Thermodynamics of Continua | year = 2010 | last1 = Gurtin | first1 = Morton E. | publisher = Cambridge University Press }}</ref><ref>{{cite book| title = Thermodynamics of Materials with Memory: Theory and Applications | year = 2012| last1 = Amendola | first1 = Giovambattista | publisher = Springer }}</ref> which evolved completely independently of statistical mechanics and maximum-entropy principles.

This point of view shares many points in common with the concept and the use of entropy in continuum thermomechanics,<ref>{{cite book| title = Rational Thermodynamics | year = 1984 | last1 = Truesdell | first1 = Clifford | publisher = Springer | edition = 2 }}</ref><ref>{{cite book| title = Continuum Thermomechanics | year = 2002 | last1 = Maugin | first1 = Gérard A. | publisher = Kluwer }}</ref><ref>{{cite book| title = The Mechanics and Thermodynamics of Continua | year = 2010 | last1 = Gurtin | first1 = Morton E. | publisher = Cambridge University Press }}</ref><ref>{{cite book| title = Thermodynamics of Materials with Memory: Theory and Applications | year = 2012| last1 = Amendola | first1 = Giovambattista | publisher = Springer }}</ref> which evolved completely independently of statistical mechanics and maximum-entropy principles.

===Entropy in non-equilibrium===

===Entropy in non-equilibrium===

To describe deviation of the thermodynamic system from equilibrium, in addition to constitutive variables <math>x_1, x_2, ..., x_n</math> that are used to fix the equilibrium state, as was described above, a set of variables <math>\xi_1, \xi_2,\ldots</math> that are called ''internal variables'' have been introduced. The equilibrium state is considered to be stable and the main property of the internal variables, as measures of [[non-equilibrium]] of the system, is their trending to disappear; the local law of disappearing can be written as relaxation equation for each internal variable

To describe deviation of the thermodynamic system from equilibrium, in addition to constitutive variables <math>x_1, x_2, ..., x_n</math> that are used to fix the equilibrium state, as was described above, a set of variables <math>\xi_1, \xi_2,\ldots</math> that are called ''internal variables'' have been introduced. The equilibrium state is considered to be stable and the main property of the internal variables, as measures of [[non-equilibrium]] of the system, is their trending to disappear; the local law of disappearing can be written as relaxation equation for each internal variable

{{NumBlk|:|<math>

{{NumBlk|:|<math>

\frac{d\xi_i}{dt} = - \frac{1}{\tau_i} \, \left(\xi_i - \xi_i^{(0)} \right),\quad i =1,\,2,\ldots ,  

\frac{d\xi_i}{dt} = - \frac{1}{\tau_i} \, \left(\xi_i - \xi_i^{(0)} \right),\quad i =1,\,2,\ldots ,  

</math>|{{EquationRef|1}}}}

</math>|{{EquationRef|1}}}}

where <math> \tau_i= \tau_i(T, x_1, x_2, \ldots, x_n)</math> is a relaxation time of a corresponding variables. It is convenient to consider the initial value <math> \xi_i^0</math> 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.<ref name="dx.doi.org">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.  https://dx.doi.org/10.1155/2013/906136.</ref>

where <math> \tau_i= \tau_i(T, x_1, x_2, \ldots, x_n)</math> is a relaxation time of a corresponding variables. It is convenient to consider the initial value <math> \xi_i^0</math> 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.<ref name="dx.doi.org">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.  https://dx.doi.org/10.1155/2013/906136.</ref>

Entropy of the system in non-equilibrium is a function of the total set of variables  

Entropy of the system in non-equilibrium is a function of the total set of variables  

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</math>|{{EquationRef|1}}}}

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The essential contribution to the thermodynamics of the [[non-equilibrium systems]] was brought by [[Prigogine]], 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,<ref>[[Ilya Prigogine|Prigogine, I.]] (1955/1961/1967). ''Introduction to Thermodynamics of Irreversible Processes''. 3rd edition, Wiley Interscience, New York.</ref> Prigogine has specified three contributions to the variation of entropy of the considered system at the given volume and constant temperature <math> T</math> . The increment of [[entropy]] <math> S</math> can be calculated according to the formula  

The essential contribution to the thermodynamics of the [[non-equilibrium systems]] was brought by [[Prigogine]], 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,<ref>[[Ilya Prigogine|Prigogine, I.]] (1955/1961/1967). ''Introduction to Thermodynamics of Irreversible Processes''. 3rd edition, Wiley Interscience, New York.</ref> Prigogine has specified three contributions to the variation of entropy of the considered system at the given volume and constant temperature <math> T</math> . The increment of [[entropy]] <math> S</math> can be calculated according to the formula  

{{NumBlk|:|<math>

{{NumBlk|:|<math>

T\,dS = \Delta Q - \sum_{j} \, \Xi_{j} \,\Delta \xi_j  + \sum_{\alpha =1}^k\, \mu_\alpha \, \Delta N_\alpha.

T\,dS = \Delta Q - \sum_{j} \, \Xi_{j} \,\Delta \xi_j  + \sum_{\alpha =1}^k\, \mu_\alpha \, \Delta N_\alpha.

Category:Concepts in physics

Category:Concepts in physics

分类: 物理概念

分类: 物理概念

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.

* [[Self-organizing criticality]]

Category:Branches of thermodynamics

Category:Branches of thermodynamics