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In 1854, Thomson wrote about the relation between two previously known non-equilibrium effects. In the Peltier effect, an electric current driven by an external electric field across a bimetallic junction will cause heat to be carried across the junction when the temperature gradient is constrained to zero. In the Seebeck effect, a flow of heat driven by a temperature gradient across such a junction will cause an electromotive force across the junction when the electric current is constrained to zero. Thus thermal and electric effects are said to be coupled. Thomson (1854)proposed a theoretical argument, partly based on the work of Carnot and Clausius, and in those days partly simply speculative, that the coupling constants of these two effects would be found experimentally to be equal. Experiment later confirmed this proposal. It was later one of the ideas that led Onsager to his results as noted below.

In 1854, Thomson wrote about the relation between two previously known non-equilibrium effects. In the Peltier effect, an electric current driven by an external electric field across a bimetallic junction will cause heat to be carried across the junction when the temperature gradient is constrained to zero. In the Seebeck effect, a flow of heat driven by a temperature gradient across such a junction will cause an electromotive force across the junction when the electric current is constrained to zero. Thus thermal and electric effects are said to be coupled. Thomson (1854)proposed a theoretical argument, partly based on the work of Carnot and Clausius, and in those days partly simply speculative, that the coupling constants of these two effects would be found experimentally to be equal. Experiment later confirmed this proposal. It was later one of the ideas that led Onsager to his results as noted below.

1854年，汤姆森Thomson写下了两个以前已知的非平衡效应之间的关系。在佩尔蒂埃效应中，当温度梯度被限制为零时，由外部电场驱动的电流穿过双金属交界处将导致热流穿过该交界处。在塞贝克效应中，当电流被约束为零时，由温度梯度驱动的热流穿过这样的结，将导致电动力穿过结。因此热效应和电效应可以说是耦合的。Thomson(1854)提出了一个理论上的论点，部分是基于卡诺和克劳修斯的工作，在当时部分只是推测，这两种效应的耦合常数在实验中会发现是相等的。实验后来证实了这个提议。这也是后来导致昂萨格得出他的结果的想法之一，如下所述。

1854年，汤姆森Thomson写下了两个以前已知的非平衡效应之间的关系。在佩尔蒂埃效应中，当温度梯度被限制为零时，由外部电场驱动的电流穿过双金属交界处将导致热流穿过该交界处。在塞贝克效应中，当电流被约束为零时，由温度梯度驱动的热流穿过这样的结，将引起结上产生电动势。因此热效应和电效应可以说是耦合的。Thomson(1854)提出了一个理论上的论点，部分是基于卡诺Carnot和克劳修斯Clausius的工作，在当时部分只是推测，这两种效应的耦合常数在实验中会发现是相等的。实验后来证实了这个论点。这也是后来导致Onsager得出他的结果的想法之一，如下所述。

===[[Helmholtz]]===

===[[Helmholtz]]===

In 1878, Helmholtz, like Thomson also citing Carnot and Clausius, wrote about electric current in an electrolyte solution with a concentration gradient. This shows a non-equilibrium coupling, between electric effects and concentration-driven diffusion. Like Thomson (Kelvin) as noted above, Helmholtz also found a reciprocal relation, and this was another of the ideas noted by Onsager.

In 1878, Helmholtz, like Thomson also citing Carnot and Clausius, wrote about electric current in an electrolyte solution with a concentration gradient. This shows a non-equilibrium coupling, between electric effects and concentration-driven diffusion. Like Thomson (Kelvin) as noted above, Helmholtz also found a reciprocal relation, and this was another of the ideas noted by Onsager.

===[[John William Strutt, 3rd Baron Rayleigh|J. W. Strutt, Baron Rayleigh]]===

===[[John William Strutt, 3rd Baron Rayleigh|J. W. Strutt, Baron Rayleigh]]===

===[[Diederik Korteweg|Korteweg]]===

===[[Diederik Korteweg|Korteweg]]===

Korteweg (1883)<ref name="Korteweg 1883">{{cite journal | last1 = Korteweg | first1 = D.J. | year = 1883 | title = On a general theorem of the stability of the motion of a viscous fluid | url = https://zenodo.org/record/1431165| journal = The London, Edinburgh and Dublin Philosophical Journal of Science | volume = 16 | issue = 98| pages = 112–118 | doi=10.1080/14786448308627405}}</ref> gave a proof "that in any simply connected region, when the velocities along the boundaries are given, there exists, as far as the squares and products of the velocities may be neglected, only one solution of the equations for the steady motion of an incompressible viscous fluid, and that this solution is always stable." He attributed the first part of this theorem to Helmholtz, who had shown that it is a simple consequence of a theorem that "if the motion be steady, the currents in a viscous [incompressible] fluid are so distributed that the loss of [kinetic] energy due to viscosity is a minimum, on the supposition that the velocities along boundaries of the fluid are given." Because of the restriction to cases in which the squares and products of the velocities can be neglected, these motions are below the threshold for turbulence.

Korteweg (1883)<ref name="Korteweg 1883">{{cite journal | last1 = Korteweg | first1 = D.J. | year = 1883 | title = On a general theorem of the stability of the motion of a viscous fluid | url = https://zenodo.org/record/1431165| journal = The London, Edinburgh and Dublin Philosophical Journal of Science | volume = 16 | issue = 98| pages = 112–118 | doi=10.1080/14786448308627405}}</ref> gave a proof "that in any simply connected region, when the velocities along the boundaries are given, there exists, as far as the squares and products of the velocities may be neglected, only one solution of the equations for the steady motion of an incompressible viscous fluid, and that this solution is always stable." He attributed the first part of this theorem to Helmholtz, who had shown that it is a simple consequence of a theorem that "if the motion be steady, the currents in a viscous [incompressible] fluid are so distributed that the loss of [kinetic] energy due to viscosity is a minimum, on the supposition that the velocities along boundaries of the fluid are given." Because of the restriction to cases in which the squares and products of the velocities can be neglected, these motions are below the threshold for turbulence.

Korteweg (1883)证明了“在任何简单连接的区域，当沿边界的速度给定时，只要速度的平方和乘积可以忽略，就存在一个不可压缩粘性流体稳定运动方程的唯一解，而且这个解总是稳定的"。他把这个定理的第一部分归功于Helmholtz，他曾证明，这是一个定理的简单结果，即 "如果运动是稳定的，那么在给定沿流体边界的速度的前提下，粘性[不可压缩]流体中的电流是如此分布，以至于粘性引起的[动能]损失是最小的"。"由于限制在速度的平方和乘积可以忽略的情况下，这些运动低于湍流的阈值。

Korteweg (1883)证明了“在任何简单连接的区域，当沿边界的速度给定时，只要速度的平方和乘积可以忽略，就存在一个不可压缩粘性流体稳定运动方程的唯一解，而且这个解总是稳定的"。他把这个定理的第一部分归功于Helmholtz，他曾证明，这是一个定理的简单结果，即 "如果运动是稳定的，那么在给定沿流体边界的速度的前提下，粘性[不可压缩]流体中的电流是如此分布，以至于粘性引起的[动能]损失是最小的"。"由于限制在速度的平方和乘积可以忽略的情况下，这些运动低于湍流的阈值。

===[[Lars Onsager|Onsager]]===

===[[Lars Onsager|Onsager]]===

Great theoretical progress was made by Onsager in 1931<ref name="Onsager 1931 I"/><ref name="Onsager 1931 II">{{cite journal | last1 = Onsager | first1 = L | year = 1931 | title = Reciprocal relations in irreversible processes. II | url = | journal = Physical Review | volume = 38 | issue = 12| pages = 2265–2279 | doi=10.1103/physrev.38.2265| bibcode = 1931PhRv...38.2265O| doi-access = free }}</ref> and in 1953.<ref name="Onsager Machlup 1953">{{cite journal | last1 = Onsager | first1 = L. | last2 = Machlup | first2 = S. | year = 1953 | title = Fluctuations and Irreversible Processes | url = | journal = Physical Review | volume = 91 | issue = 6| pages = 1505–1512 | doi=10.1103/physrev.91.1505| bibcode = 1953PhRv...91.1505O }}</ref><ref name="Machlup Onsager 1953">{{cite journal | last1 = Machlup | first1 = S. | last2 = Onsager | first2 = L. | year = 1953 | title = Fluctuations and Irreversible Processes. II. Systems with kinetic energy | url = | journal = Physical Review | volume = 91 | issue = 6| pages = 1512–1515 | doi=10.1103/physrev.91.1512| bibcode = 1953PhRv...91.1512M }}</ref>

Great theoretical progress was made by Onsager in 1931<ref name="Onsager 1931 I"/><ref name="Onsager 1931 II">{{cite journal | last1 = Onsager | first1 = L | year = 1931 | title = Reciprocal relations in irreversible processes. II | url = | journal = Physical Review | volume = 38 | issue = 12| pages = 2265–2279 | doi=10.1103/physrev.38.2265| bibcode = 1931PhRv...38.2265O| doi-access = free }}</ref> and in 1953.<ref name="Onsager Machlup 1953">{{cite journal | last1 = Onsager | first1 = L. | last2 = Machlup | first2 = S. | year = 1953 | title = Fluctuations and Irreversible Processes | url = | journal = Physical Review | volume = 91 | issue = 6| pages = 1505–1512 | doi=10.1103/physrev.91.1505| bibcode = 1953PhRv...91.1505O }}</ref><ref name="Machlup Onsager 1953">{{cite journal | last1 = Machlup | first1 = S. | last2 = Onsager | first2 = L. | year = 1953 | title = Fluctuations and Irreversible Processes. II. Systems with kinetic energy | url = | journal = Physical Review | volume = 91 | issue = 6| pages = 1512–1515 | doi=10.1103/physrev.91.1512| bibcode = 1953PhRv...91.1512M }}</ref>

===[[Ilya Prigogine|Prigogine]]===

===[[Ilya Prigogine|Prigogine]]===

Further progress was made by Prigogine in 1945<ref name="Prigogine 1945">{{cite journal | last1 = Prigogine | first1 = I | year = 1945 | title = Modération et transformations irréversibles des systèmes ouverts | url = | journal = Bulletin de la Classe des Sciences., Académie Royale de Belgique | volume = 31 | issue = | pages = 600–606 }}</ref> and later.<ref name="G&P 1971"/><ref name="Prigogine 1947">Prigogine, I. (1947). ''Étude thermodynamique des Phenomènes Irréversibles'', Desoer, Liège.</ref> Prigogine (1947)<ref name="Prigogine 1945"/> cites Onsager (1931).<ref name="Onsager 1931 I"/><ref name="Onsager 1931 II"/>

Further progress was made by Prigogine in 1945<ref name="Prigogine 1945">{{cite journal | last1 = Prigogine | first1 = I | year = 1945 | title = Modération et transformations irréversibles des systèmes ouverts | url = | journal = Bulletin de la Classe des Sciences., Académie Royale de Belgique | volume = 31 | issue = | pages = 600–606 }}</ref> and later.<ref name="G&P 1971"/><ref name="Prigogine 1947">Prigogine, I. (1947). ''Étude thermodynamique des Phenomènes Irréversibles'', Desoer, Liège.</ref> Prigogine (1947)<ref name="Prigogine 1945"/> cites Onsager (1931).<ref name="Onsager 1931 I"/><ref name="Onsager 1931 II"/>

===[[Hendrik Casimir|Casimir]]===

===[[Hendrik Casimir|Casimir]]===

 

Casimir (1945)<ref name="Casimir 1945">{{cite journal | last1 = Casimir | first1 = H.B.G. | s2cid = 53386496 | year = 1945 | title = On Onsager's principle of microscopic reversibility | journal = Reviews of Modern Physics | volume = 17 | issue = 2–3| pages = 343–350 | doi=10.1103/revmodphys.17.343| bibcode = 1945RvMP...17..343C}}</ref> extended the theory of Onsager.

Casimir (1945)<ref name="Casimir 1945">{{cite journal | last1 = Casimir | first1 = H.B.G. | s2cid = 53386496 | year = 1945 | title = On Onsager's principle of microscopic reversibility | journal = Reviews of Modern Physics | volume = 17 | issue = 2–3| pages = 343–350 | doi=10.1103/revmodphys.17.343| bibcode = 1945RvMP...17..343C}}</ref> extended the theory of Onsager.

===[[John Ziman|Ziman]]===

===[[John Ziman|Ziman]]===

Ziman (1956)<ref name="Ziman 1956">{{cite journal | last1 = Ziman | first1 = J.M. | year = 1956 | title = The general variational principle of transport theory | url = | journal = Canadian Journal of Physics | volume = 34 | issue = 12A| pages = 1256–1273 | doi=10.1139/p56-139| bibcode = 1956CaJPh..34.1256Z}}</ref> gave very readable account. He proposed the following as a general principle of the thermodynamics of irreversible processes: "''Consider all distributions of currents such that the intrinsic entropy production equals the extrinsic entropy production for the given set of forces. Then, of all current distributions satisfying this condition, the steady state distribution makes the entropy production a maximum.''" He commented that this was a known general principle, discovered by Onsager, but was "not quoted in any of the books on the subject". He notes the difference between this principle and "Prigogine's theorem, which states, crudely speaking, that if not all the forces acting on a system are fixed the free forces will take such values as to make the entropy production a minimum." Prigogine was present when this paper was read and he is reported by the journal editor to have given "notice that he doubted the validity of part of Ziman's thermodynamic interpretation".

Ziman (1956)<ref name="Ziman 1956">{{cite journal | last1 = Ziman | first1 = J.M. | year = 1956 | title = The general variational principle of transport theory | url = | journal = Canadian Journal of Physics | volume = 34 | issue = 12A| pages = 1256–1273 | doi=10.1139/p56-139| bibcode = 1956CaJPh..34.1256Z}}</ref> gave very readable account. He proposed the following as a general principle of the thermodynamics of irreversible processes: "''Consider all distributions of currents such that the intrinsic entropy production equals the extrinsic entropy production for the given set of forces. Then, of all current distributions satisfying this condition, the steady state distribution makes the entropy production a maximum.''" He commented that this was a known general principle, discovered by Onsager, but was "not quoted in any of the books on the subject". He notes the difference between this principle and "Prigogine's theorem, which states, crudely speaking, that if not all the forces acting on a system are fixed the free forces will take such values as to make the entropy production a minimum." Prigogine was present when this paper was read and he is reported by the journal editor to have given "notice that he doubted the validity of part of Ziman's thermodynamic interpretation".

Ziman (1956) gave very readable account. He proposed the following as a general principle of the thermodynamics of irreversible processes: "Consider all distributions of currents such that the intrinsic entropy production equals the extrinsic entropy production for the given set of forces. Then, of all current distributions satisfying this condition, the steady state distribution makes the entropy production a maximum." He commented that this was a known general principle, discovered by Onsager, but was "not quoted in any of the books on the subject". He notes the difference between this principle and "Prigogine's theorem, which states, crudely speaking, that if not all the forces acting on a system are fixed the free forces will take such values as to make the entropy production a minimum." Prigogine was present when this paper was read and he is reported by the journal editor to have given "notice that he doubted the validity of part of Ziman's thermodynamic interpretation".

Ziman (1956) gave very readable account. He proposed the following as a general principle of the thermodynamics of irreversible processes: "Consider all distributions of currents such that the intrinsic entropy production equals the extrinsic entropy production for the given set of forces. Then, of all current distributions satisfying this condition, the steady state distribution makes the entropy production a maximum." He commented that this was a known general principle, discovered by Onsager, but was "not quoted in any of the books on the subject". He notes the difference between this principle and "Prigogine's theorem, which states, crudely speaking, that if not all the forces acting on a system are fixed the free forces will take such values as to make the entropy production a minimum." Prigogine was present when this paper was read and he is reported by the journal editor to have given "notice that he doubted the validity of part of Ziman's thermodynamic interpretation".

齐曼Ziman(1956)给出了非常易读的描述。他提出了以下内容作为不可逆过程热力学的一般原则。"考虑所有的电流分布 使得在给定的力的作用下 内在熵的产生等于外在熵的产生"。那么，在所有满足这一条件的电流分布中，稳态分布使熵产量达到最大值。" 他评论说，这是一个众所周知的一般原理，是由Onsager发现的，但 "在任何有关的书籍中都没有被引用"。他注意到这一原理与 "Prigogine定理 "的区别，Prigogine定理粗略地说，如果不是所有作用在一个系统上的力都是固定的，那么自由力就会取这样的值，使熵的产生成为最小值"。"宣读这篇论文时，Prigogine在场，据期刊编辑报道说，他曾发通知说，他怀疑Ziman的热力学解释的有效性"。

Ziman(1956)给出了非常易读的描述。他提出了以下内容作为不可逆过程热力学的一般原则。"考虑所有的电流分布 使得在给定的力的作用下 内在熵的产生等于外在熵的产生"。那么，在所有满足这一条件的电流分布中，稳态分布使熵产量达到最大值。" 他评论说，这是一个众所周知的一般原理，是由Onsager发现的，但 "在任何有关的书籍中都没有被引用"。他注意到这一原理与 "Prigogine定理 "的区别，Prigogine定理粗略地说，如果不是所有作用在一个系统上的力都是固定的，那么自由力就会取这样的值，使熵的产生成为最小值"。"宣读这篇论文时，Prigogine在场，据期刊编辑报道说，他曾发通知说，他怀疑Ziman的热力学解释的有效性"。

===Ziegler===

===Ziegler===

 

[[Hans Ziegler (physicist)|Hans Ziegler]] extended the Melan-Prager non-equilibrium theory of materials to the non-isothermal case.<ref>T. Inoue (2002). Metallo-Thermo-Mechanics–Application to Quenching. ''In'' G. Totten, M. Howes, and T. Inoue (eds.), Handbook of Residual Stress. pp. 296-311, ASM International, Ohio.</ref>

[[Hans Ziegler (physicist)|Hans Ziegler]] extended the Melan-Prager non-equilibrium theory of materials to the non-isothermal case.<ref>T. Inoue (2002). Metallo-Thermo-Mechanics–Application to Quenching. ''In'' G. Totten, M. Howes, and T. Inoue (eds.), Handbook of Residual Stress. pp. 296-311, ASM International, Ohio.</ref>

===Gyarmati===

===Gyarmati===

Gyarmati (1967/1970)<ref name="Gyarmati 1970">Gyarmati, I. (1970). ''Non-equilibrium Thermodynamics: Field Theory and Variational Principles'', Springer, Berlin; translated, by E. Gyarmati and W.F. Heinz, from the original 1967 Hungarian ''Nemegyensulyi Termodinamika'', Muszaki Konyvkiado, Budapest.</ref> gives a systematic presentation, and extends Onsager's principle of least dissipation of energy, to give a more symmetric form known as Gyarmati's principle. Gyarmati (1967/1970)<ref name="Gyarmati 1970"/> cites 11 papers or books authored or co-authored by Prigogine.

Gyarmati (1967/1970)<ref name="Gyarmati 1970"/> also gives in Section III 5 a very helpful precis of the subtleties of Casimir (1945)).<ref name="Casimir 1945"/> He explains that the Onsager reciprocal relations concern variables which are even functions of the velocities of the molecules, and notes that Casimir went on to derive anti-symmetric relations concerning variables which are odd functions of the velocities of the molecules.

Gyarmati (1967/1970) gives a systematic presentation, and extends Onsager's principle of least dissipation of energy, to give a more symmetric form known as Gyarmati's principle. Gyarmati (1967/1970) cites 11 papers or books authored or co-authored by Prigogine. Gyarmati (1967/1970)also gives in Section III 5 a very helpful precis of the subtleties of Casimir (1945)).He explains that the Onsager reciprocal relations concern variables which are even functions of the velocities of the molecules, and notes that Casimir went on to derive anti-symmetric relations concerning variables which are odd functions of the velocities of the molecules.

Gyarmati (1967/1970) gives a systematic presentation, and extends Onsager's principle of least dissipation of energy, to give a more symmetric form known as Gyarmati's principle. Gyarmati (1967/1970) cites 11 papers or books authored or co-authored by Prigogine. Gyarmati (1967/1970)also gives in Section III 5 a very helpful precis of the subtleties of Casimir (1945)).He explains that the Onsager reciprocal relations concern variables which are even functions of the velocities of the molecules, and notes that Casimir went on to derive anti-symmetric relations concerning variables which are odd functions of the velocities of the molecules.

Gyarmati(1967/1970)给出了一个系统的表述，并扩展了Onsager的能量最小耗散原理，给出了一个更对称的形式，称为Gyarmati原理。Gyarmati(1967/1970)引用了11篇Prigogine撰写或合著的论文或书籍。Gyarmati (1967/1970)还在第三节5中对Casimir (1945)的微妙之处给出了一个非常有用的预言.他解释说，Onsager往复关系涉及的变量是分子速度的偶数函数，并指出Casimir继续推导出反对称关系，涉及的变量是分子速度的奇数函数。

Gyarmati(1967/1970)给出了一个系统的表述，并扩展了Onsager的能量最小耗散原理，给出了一个更对称的形式，称为Gyarmati原理。Gyarmati(1967/1970)引用了11篇Prigogine撰写或合著的论文或书籍。Gyarmati (1967/1970)还在第三节5中对Casimir (1945)的微妙之处给出了一个非常有用的预言.他解释说，Onsager往复关系涉及的变量是分子速度的偶数函数，并指出Casimir继续推导出反对称关系，涉及的变量是分子速度的奇数函数。

===Paltridge===

===Paltridge===

The physics of the earth's atmosphere includes dramatic events like lightning and the effects of volcanic eruptions, with discontinuities of motion such as noted by Helmholtz (1868).<ref name="Helmholtz 1868"/>  Turbulence is prominent in atmospheric convection. Other discontinuities include the formation of raindrops, hailstones, and snowflakes. The usual theory of classical non-equilibrium thermodynamics will need some extension to cover atmospheric physics. According to Tuck (2008),<ref name="Tuck, Adrian F. 2008 page 33">Tuck, Adrian F. (2008) ''Atmospheric Turbulence: a molecular dynamics perspective'', Oxford University Press. {{ISBN|978-0-19-923653-4}}. See page 33.</ref> "On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975,<ref name="Paltridge 1975">Paltridge, G.W. (1975). Global dynamics and climate - a system of minimum entropy exchange, ''Quarterly Journal of the Royal Meteorological Society 101:475-484.

The physics of the earth's atmosphere includes dramatic events like lightning and the effects of volcanic eruptions, with discontinuities of motion such as noted by Helmholtz (1868).<ref name="Helmholtz 1868"/>  Turbulence is prominent in atmospheric convection. Other discontinuities include the formation of raindrops, hailstones, and snowflakes. The usual theory of classical non-equilibrium thermodynamics will need some extension to cover atmospheric physics. According to Tuck (2008),<ref name="Tuck, Adrian F. 2008 page 33">Tuck, Adrian F. (2008) ''Atmospheric Turbulence: a molecular dynamics perspective'', Oxford University Press. {{ISBN|978-0-19-923653-4}}. See page 33.</ref> "On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975,<ref name="Paltridge 1975">Paltridge, G.W. (1975). Global dynamics and climate - a system of minimum entropy exchange, ''Quarterly Journal of the Royal Meteorological Society 101:475-484.