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定义非平衡热力学状态变量的关系如下：在系统恰好处于很接近热力学平衡状态的情况下，非平衡状态变量可以通过与测量热力学状态变量相同的技术，或通过相应的时空导数，包括物质和能量通量，以足够的精度在本地进行测量。通常，非平衡热力学系统在空间和时间上都是非均匀的，但是它们的非均匀性仍然具有足够的平滑度，以保证非平衡状态变量的时空导数适当存在。另外由于空间的不均匀性，必须将非平衡状态变量（对应于广义热力学状态变量）定义为相应的广义平衡状态变量的空间密度。在系统足够接近热力学平衡的情况下，密集的非平衡状态变量（例如温度和压力）与平衡状态变量紧密对应。测量时探头必须足够小，并且响应速度要足够快，以捕获相关的不均匀性。此外，要求非平衡状态变量在数学上彼此函数相关，其方式应类似于平衡热力学状态变量之间的对应关系。实际上，这些要求非常苛刻，可能很难实现，或实际上，甚至在理论上无法满足它们。这就是非平衡热力学的研究一直处在探索中的部分原因。

定义非平衡热力学状态变量的关系如下：在系统恰好处于很接近热力学平衡状态的情况下，非平衡状态变量可以通过与测量热力学状态变量相同的技术，或通过相应的时空导数，包括物质和能量通量，以足够的精度在本地进行测量。通常，非平衡热力学系统在空间和时间上都是非均匀的，但是它们的非均匀性仍然具有足够的平滑度，以保证非平衡状态变量的时空导数适当存在。另外由于空间的不均匀性，必须将非平衡状态变量（对应于广义热力学状态变量）定义为相应的广义平衡状态变量的空间密度。在系统足够接近热力学平衡的情况下，密集的非平衡状态变量（例如温度和压力）与平衡状态变量紧密对应。测量时探头必须足够小，并且响应速度要足够快，以捕获相关的不均匀性。此外，要求非平衡状态变量在数学上彼此函数相关，其方式应类似于平衡热力学状态变量之间的对应关系。实际上，这些要求非常苛刻，可能很难实现，或实际上，甚至在理论上无法满足它们。这就是非平衡热力学的研究一直处在探索中的部分原因。

==Overview==

== Overview 概述 ==

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"/>

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.

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.

=== Quasi-radiationless non-equilibrium thermodynamics of matter in laboratory conditions 实验室条件下物质的准无辐射非平衡热力学 ===

 

===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 局部平衡热力学 ===

 

 

===Local equilibrium thermodynamics===

The terms 'classical irreversible thermodynamics'<ref name="Lebon Jou Casas-Vázquez 2008"/> 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 system effectively homogeneous, or well-mixed, or without an effective spatial structure, and without kinetic energy of bulk flow or of diffusive flux. Even within the thought-frame of classical irreversible thermodynamics, care<ref name="Lavenda 1978"/> is needed in choosing the independent variables<ref>Prigogine, I., Defay, R. (1950/1954). ''Chemical Thermodynamics'', Longmans, Green & Co, London, page 1.</ref> 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.

The terms 'classical irreversible thermodynamics'<ref name="Lebon Jou Casas-Vázquez 2008"/> 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 system effectively homogeneous, or well-mixed, or without an effective spatial structure, and without kinetic energy of bulk flow or of diffusive flux. Even within the thought-frame of classical irreversible thermodynamics, care<ref name="Lavenda 1978"/> is needed in choosing the independent variables<ref>Prigogine, I., Defay, R. (1950/1954). ''Chemical Thermodynamics'', Longmans, Green & Co, London, page 1.</ref> 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.

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.

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>

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.

局部平衡热力学的进一步扩展是允许材料具有“记忆”，因此它们的'''<font color="#ff8000"> 本构方程Constitutive equations</font>'''不仅取决于局部平衡变量的当前值，而且还取决于过去的值。因此，与无记忆材料的依时性局部平衡热力学相比，时间对图像的影响更深，不过通量不是状态的独立变量。

===Extended irreversible thermodynamics===

'''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==