# 非平衡热力学的极值定理

Energy dissipation and entropy production extremal principles are ideas developed within non-equilibrium thermodynamics that attempt to predict the likely steady states and dynamical structures that a physical system might show. The search for extremum principles for non-equilibrium thermodynamics follows their successful use in other branches of physics.[1][2][3][4][5][6] According to Kondepudi (2008),[7] and to Grandy (2008),[8] there is no general rule that provides an extremum principle that governs the evolution of a far-from-equilibrium system to a steady state. According to Glansdorff and Prigogine (1971, page 16),[9] irreversible processes usually are not governed by global extremal principles because description of their evolution requires differential equations which are not self-adjoint, but local extremal principles can be used for local solutions. Lebon Jou and Casas-Vásquez (2008)[10] state that "In non-equilibrium ... it is generally not possible to construct thermodynamic potentials depending on the whole set of variables". Šilhavý (1997)[11] offers the opinion that "... the extremum principles of thermodynamics ... do not have any counterpart for [non-equilibrium] steady states (despite many claims in the literature)." It follows that any general extremal principle for a non-equilibrium problem will need to refer in some detail to the constraints that are specific for the structure of the system considered in the problem.

Energy dissipation and entropy production extremal principles are ideas developed within non-equilibrium thermodynamics that attempt to predict the likely steady states and dynamical structures that a physical system might show. The search for extremum principles for non-equilibrium thermodynamics follows their successful use in other branches of physics. According to Kondepudi (2008), and to Grandy (2008), there is no general rule that provides an extremum principle that governs the evolution of a far-from-equilibrium system to a steady state. According to Glansdorff and Prigogine (1971, page 16), state that "In non-equilibrium ... it is generally not possible to construct thermodynamic potentials depending on the whole set of variables". Šilhavý (1997) offers the opinion that "... the extremum principles of thermodynamics ... do not have any counterpart for [non-equilibrium] steady states (despite many claims in the literature)." It follows that any general extremal principle for a non-equilibrium problem will need to refer in some detail to the constraints that are specific for the structure of the system considered in the problem.

## Fluctuations, entropy, 'thermodynamics forces', and reproducible dynamical structure

Apparent 'fluctuations', which appear to arise when initial conditions are inexactly specified, are the drivers of the formation of non-equilibrium dynamical structures. There is no special force of nature involved in the generation of such fluctuations. Exact specification of initial conditions would require statements of the positions and velocities of all particles in the system, obviously not a remotely practical possibility for a macroscopic system. This is the nature of thermodynamic fluctuations. They cannot be predicted in particular by the scientist, but they are determined by the laws of nature and they are the singular causes of the natural development of dynamical structure.[9]

Apparent 'fluctuations', which appear to arise when initial conditions are inexactly specified, are the drivers of the formation of non-equilibrium dynamical structures. There is no special force of nature involved in the generation of such fluctuations. Exact specification of initial conditions would require statements of the positions and velocities of all particles in the system, obviously not a remotely practical possibility for a macroscopic system. This is the nature of thermodynamic fluctuations. They cannot be predicted in particular by the scientist, but they are determined by the laws of nature and they are the singular causes of the natural development of dynamical structure.

It is pointed out[12][13][14][15] by W.T. Grandy Jr that entropy, though it may be defined for a non-equilibrium system, is when strictly considered, only a macroscopic quantity that refers to the whole system, and is not a dynamical variable and in general does not act as a local potential that describes local 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 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.

W.T.格兰迪(W.T. Grandy Jr) 指出，熵虽然可以为非平衡系统定义，但严格考虑时，它只是一个宏观的量，指的是整个系统，并不是一个动态变量，一般情况下，它并不能作为描述局部物理力的局部势。但在特殊情况下，我们可以比喻为热变量的行为就像局部物理力一样。构成经典不可逆热力学的近似就是建立在这种隐喻思维的基础上。

As indicated by the " " marks of Onsager (1931),[1] such a metaphorical but not categorically mechanical force, the thermal "force", $\displaystyle{ X_{th} }$, 'drives' the conduction of heat. For this so-called "thermodynamic force", we can write

$\displaystyle{ X_{th} = - \frac{1}{T} \nabla T }$.

Actually this thermal "thermodynamic force" is a manifestation of the degree of inexact specification of the microscopic initial conditions for the system, expressed in the thermodynamic variable known as temperature, $\displaystyle{ T }$. Temperature is only one example, and all the thermodynamic macroscopic variables constitute inexact specifications of the initial conditions, and have their respective "thermodynamic forces". These inexactitudes of specification are the source of the apparent fluctuations that drive the generation of dynamical structure, of the very precise but still less than perfect reproducibility of non-equilibrium experiments, and of the place of entropy in thermodynamics. If one did not know of such inexactitude of specification, one might find the origin of the fluctuations mysterious. What is meant here by "inexactitude of specification" is not that the mean values of the macroscopic variables are inexactly specified, but that the use of macroscopic variables to describe processes that actually occur by the motions and interactions of microscopic objects such as molecules is necessarily lacking in the molecular detail of the processes, and is thus inexact. There are many microscopic states compatible with a single macroscopic state, but only the latter is specified, and that is specified exactly for the purposes of the theory.

It is reproducibility in repeated observations that identifies dynamical structure in a system. E.T. Jaynes[16][17][18][19] explains how this reproducibility is why entropy is so important in this topic: entropy is a measure of experimental reproducibility. The entropy tells how many times one would have to repeat the experiment in order to expect to see a departure from the usual reproducible result. When the process goes on in a system with less than a 'practically infinite' number (much much less than Avogadro's or Loschmidt's numbers) of molecules, the thermodynamic reproducibility fades, and fluctuations become easier to see.[20][21]

It is reproducibility in repeated observations that identifies dynamical structure in a system. E.T. Jaynes explains how this reproducibility is why entropy is so important in this topic: entropy is a measure of experimental reproducibility. The entropy tells how many times one would have to repeat the experiment in order to expect to see a departure from the usual reproducible result. When the process goes on in a system with less than a 'practically infinite' number (much much less than Avogadro's or Loschmidt's numbers) of molecules, the thermodynamic reproducibility fades, and fluctuations become easier to see.

According to this view of Jaynes, it is a common and mystificatory abuse of language, that one often sees reproducibility of dynamical structure called "order".[8][22] Dewar[22] writes "Jaynes considered reproducibility - rather than disorder - to be the key idea behind the second law of thermodynamics (Jaynes 1963,[23] 1965,[19] 1988,[24] 1989[25])." Grandy (2008)[8] in section 4.3 on page 55 clarifies the distinction between the idea that entropy is related to order (which he considers to be an "unfortunate" "mischaracterization" that needs "debunking"), and the aforementioned idea of Jaynes that entropy is a measure of experimental reproducibility of process (which Grandy regards as correct). According to this view, even the admirable book of Glansdorff and Prigogine (1971)[9] is guilty of this unfortunate abuse of language.

According to this view of Jaynes, it is a common and mystificatory abuse of language, that one often sees reproducibility of dynamical structure called "order". Dewar "Jaynes considered reproducibility - rather than disorder - to be the key idea behind the second law of thermodynamics (Jaynes 1963,1965,1988,1989)." Grandy (2008) in section 4.3 on page 55 clarifies the distinction between the idea that entropy is related to order (which he considers to be an "unfortunate" "mischaracterization" that needs "debunking"), and the aforementioned idea of Jaynes that entropy is a measure of experimental reproducibility of process (which Grandy regards as correct). According to this view, even the admirable book of Glansdorff and Prigogine (1971)is guilty of this unfortunate abuse of language.

## Local thermodynamic equilibrium

Various principles have been proposed by diverse authors for over a century. According to Glansdorff and Prigogine (1971, page 15),[9] in general, these principles apply only to systems that can be described by thermodynamical variables, in which dissipative processes dominate by excluding large deviations from statistical equilibrium. The thermodynamical variables are defined subject to the kinematical requirement of local thermodynamic equilibrium. This means that collisions between molecules are so frequent that chemical and radiative processes do not disrupt the local Maxwell-Boltzmann distribution of molecular velocities.

Various principles have been proposed by diverse authors for over a century. According to Glansdorff and Prigogine (1971, page 15),in general, these principles apply only to systems that can be described by thermodynamical variables, in which dissipative processes dominate by excluding large deviations from statistical equilibrium. The thermodynamical variables are defined subject to the kinematical requirement of local thermodynamic equilibrium. This means that collisions between molecules are so frequent that chemical and radiative processes do not disrupt the local Maxwell-Boltzmann distribution of molecular velocities.

## Linear and non-linear processes

Dissipative structures can depend on the presence of non-linearity in their dynamical régimes. Autocatalytic reactions provide examples of non-linear dynamics, and may lead to the natural evolution of self-organized dissipative structures.

Dissipative structures can depend on the presence of non-linearity in their dynamical régimes. Autocatalytic reactions provide examples of non-linear dynamics, and may lead to the natural evolution of self-organized dissipative structures.

## Continuous and discontinuous motions of fluids

Much of the theory of classical non-equilibrium thermodynamics is concerned with the spatially continuous motion of fluids, but fluids can also move with spatial discontinuities. Helmholtz (1868)[26] wrote about how in a flowing fluid, there can arise a zero fluid pressure, which sees the fluid broken asunder. This arises from the momentum of the fluid flow, showing a different kind of dynamical structure from that of the conduction of heat or electricity. Thus for example: water from a nozzle can form a shower of droplets (Rayleigh 1878,[27] and in section 357 et seq. of Rayleigh (1896/1926)[28]); waves on the surface of the sea break discontinuously when they reach the shore (Thom 1975[29]). Helmholtz pointed out that the sounds of organ pipes must arise from such discontinuity of flow, occasioned by the passage of air past a sharp-edged obstacle; otherwise the oscillatory character of the sound wave would be damped away to nothing. The definition of the rate of entropy production of such a flow is not covered by the usual theory of classical non-equilibrium thermodynamics. There are many other commonly observed discontinuities of fluid flow that also lie beyond the scope of the classical theory of non-equilibrium thermodynamics, such as: bubbles in boiling liquids and in effervescent drinks; also protected towers of deep tropical convection (Riehl, Malkus 1958[30]), also called penetrative convection (Lindzen 1977[31]).

Much of the theory of classical non-equilibrium thermodynamics is concerned with the spatially continuous motion of fluids, but fluids can also move with spatial discontinuities. Helmholtz (1868) wrote about how in a flowing fluid, there can arise a zero fluid pressure, which sees the fluid broken asunder. This arises from the momentum of the fluid flow, showing a different kind of dynamical structure from that of the conduction of heat or electricity. Thus for example: water from a nozzle can form a shower of droplets (Rayleigh 1878, and in section 357 et seq. of Rayleigh (1896/1926)); waves on the surface of the sea break discontinuously when they reach the shore (Thom 1975). Helmholtz pointed out that the sounds of organ pipes must arise from such discontinuity of flow, occasioned by the passage of air past a sharp-edged obstacle; otherwise the oscillatory character of the sound wave would be damped away to nothing. The definition of the rate of entropy production of such a flow is not covered by the usual theory of classical non-equilibrium thermodynamics. There are many other commonly observed discontinuities of fluid flow that also lie beyond the scope of the classical theory of non-equilibrium thermodynamics, such as: bubbles in boiling liquids and in effervescent drinks; also protected towers of deep tropical convection (Riehl, Malkus 1958), also called penetrative convection (Lindzen 1977)

## Historical development

### W. Thomson, Baron Kelvin

William Thomson, later Baron Kelvin, (1852 a, 1852 b) wrote

"II. When heat is created by any unreversible process (such as friction), there is a dissipation of mechanical energy, and a full restoration of it to its primitive condition is impossible.

“二.当热是由任何不可逆的过程(如摩擦)产生的时候，就会有机械能的耗散，完全恢复到原始状态是不可能的。

III. When heat is diffused by conduction, there is a dissipation of mechanical energy, and perfect restoration is impossible.

IV. When radiant heat or light is absorbed, otherwise than in vegetation, or in a chemical reaction, there is a dissipation of mechanical energy, and perfect restoration is impossible."

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)[32] 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

In 1869, Hermann von Helmholtz stated his Helmholtz minimum dissipation theorem,[33] subject to a certain kind of boundary condition, a principle of least viscous dissipation of kinetic energy: "For a steady flow in a viscous liquid, with the speeds of flow on the boundaries of the fluid being given steady, in the limit of small speeds, the currents in the liquid so distribute themselves that the dissipation of kinetic energy by friction is minimum."[34]

In 1869, Hermann von Helmholtz stated his Helmholtz minimum dissipation theorem, subject to a certain kind of boundary condition, a principle of least viscous dissipation of kinetic energy: "For a steady flow in a viscous liquid, with the speeds of flow on the boundaries of the fluid being given steady, in the limit of small speeds, the currents in the liquid so distribute themselves that the dissipation of kinetic energy by friction is minimum."

1869年，赫尔曼•冯•赫尔姆霍兹Hermann von Helmholtz 在一定的边界条件下，陈述了他的Helmholtz最小耗散定理，即动能最小粘性耗散原理："对于粘性液体中的稳定流动，在给定液体边界上的流速稳定的情况下，在速度较小的情况下，液体中的电流如此分布，摩擦力对动能的耗散是最小的。"

In 1878, Helmholtz,[35] 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.

1878年，Helmholtz和Thomson一样，也引用了Carnot和Clausius的观点，写出了电解质溶液中的电流与浓度梯度。这表明了一种非平衡耦合，在电效应和浓度驱动的扩散之间。和上面提到的Thomson（开尔文Kelvin）一样，Helmholtz也发现了一种相互关系，这也是Onsager指出的另一个观点。

### J. W. Strutt, Baron Rayleigh

Rayleigh (1873)[36] (and in Sections 81 and 345 of Rayleigh (1896/1926)[28]) introduced the dissipation function for the description of dissipative processes involving viscosity. More general versions of this function have been used by many subsequent investigators of the nature of dissipative processes and dynamical structures. Rayleigh's dissipation function was conceived of from a mechanical viewpoint, and it did not refer in its definition to temperature, and it needed to be 'generalized' to make a dissipation function suitable for use in non-equilibrium thermodynamics.

Rayleigh (1873)(and in Sections 81 and 345 of Rayleigh (1896/1926)) introduced the dissipation function for the description of dissipative processes involving viscosity. More general versions of this function have been used by many subsequent investigators of the nature of dissipative processes and dynamical structures. Rayleigh's dissipation function was conceived of from a mechanical viewpoint, and it did not refer in its definition to temperature, and it needed to be 'generalized' to make a dissipation function suitable for use in non-equilibrium thermodynamics. Studying jets of water from a nozzle, Rayleigh (1878,1896/1926) noted that when a jet is in a state of conditionally stable dynamical structure, the mode of fluctuation most likely to grow to its full extent and lead to another state of conditionally stable dynamical structure is the one with the fastest growth rate. In other words, a jet can settle into a conditionally stable state, but it is likely to suffer fluctuation so as to pass to another, less unstable, conditionally stable state. He used like reasoning in a study of Bénard convection.These physically lucid considerations of Rayleigh seem to contain the heart of the distinction between the principles of minimum and maximum rates of dissipation of energy and entropy production, which have been developed in the course of physical investigations by later authors.

Rayleigh(1873)(以及Rayleigh(1896/1926)的第81和345节)引入了耗散函数，用于描述涉及粘度的耗散过程。这个函数的更多通用版本已经被许多后来的研究者用来研究耗散过程和动力学结构的性质。Rayleigh的耗散函数是从力学的角度设想的，它的定义中并没有提到温度，它需要被 "泛化"，使耗散函数适合用于非平衡热力学。 Rayleigh(1878,1896/1926)在研究喷头的射流时指出，当射流处于条件稳定的动力结构状态时，最有可能增长到其全部程度并导致另一种条件稳定的动力结构状态的波动模式是增长速度最快的模式。换句话说，一个喷流可以稳定在一个条件稳定的状态，但它很可能遭受波动，从而传递到另一个不那么不稳定的条件稳定状态。他在对贝纳德Bénard对流的研究中使用了类似的推理。Rayleigh的这些物理上的明晰考虑似乎包含了能量耗散和熵产生的最小速率和最大速率原则的区别的核心，这些原则在后来的物理研究过程中得到了发展。

### Korteweg

Korteweg (1883)[37] 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) 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。

### Onsager

Great theoretical progress was made by Onsager in 1931[1][38] and in 1953.[39][40]

Great theoretical progress was made by Onsager in 1931 and in 1953.

Onsager 1931年和1953年在理论上取得了重大进展。

### Prigogine

Further progress was made by Prigogine in 1945[41] and later.[9][42] Prigogine (1947)[41] cites Onsager (1931).[1][38]

Further progress was made by Prigogine in 1945 and later. Prigogine (1947) extended the theory of Onsager.

Prigogine在1945年及其后取得了进一步的进展。Prigogine(1947)扩展了Onsager的理论。

### Casimir

Casimir (1945)extended the theory of Onsager. Casimir（1945）扩展了Onsager理论。

### Ziman

Ziman (1956)[44] 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)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

Hans Ziegler extended the Melan-Prager non-equilibrium theory of materials to the non-isothermal case.

Hans Ziegler 将材料的 Melan-Prager 非平衡理论扩展到了非等温情况。

### Gyarmati

Gyarmati (1967/1970)[2] also gives in Section III 5 a very helpful precis of the subtleties of Casimir (1945)).[43] 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。

### 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).[26] 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),[46] "On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975,[47] 2001[48]). Initially Paltridge (1975)[47] used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978),[49] and in Paltridge (1979)[50]), he used the now current terminology "maximum entropy production" to describe the same thing. This point is clarified in the review by Ozawa, Ohmura, Lorenz, Pujol (2003).[51] Paltridge (1978)[49] cited Busse's (1967)[52] fluid mechanical work concerning an extremum principle. Nicolis and Nicolis (1980) [53] discuss Paltridge's work, and they comment that the behaviour of the entropy production is far from simple and universal. This seems natural in the context of the requirement of some classical theory of non-equilibrium thermodynamics that the threshold of turbulence not be crossed. Paltridge himself nowadays tends to prefer to think in terms of the dissipation function rather than in terms of rate of entropy production.

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).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),"On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975,2001). Initially Paltridge (1975)used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978),and in Paltridge (1979)), he used the now current terminology "maximum entropy production" to describe the same thing. This point is clarified in the review by Ozawa, Ohmura, Lorenz, Pujol (2003).Paltridge (1978)cited Busse's (1967)fluid mechanical work concerning an extremum principle. Nicolis and Nicolis (1980) discuss Paltridge's work, and they comment that the behaviour of the entropy production is far from simple and universal. This seems natural in the context of the requirement of some classical theory of non-equilibrium thermodynamics that the threshold of turbulence not be crossed. Paltridge himself nowadays tends to prefer to think in terms of the dissipation function rather than in terms of rate of entropy production.

## Speculated thermodynamic extremum principles for energy dissipation and entropy production

Jou, Casas-Vazquez, Lebon (1993)[54] note that classical non-equilibrium thermodynamics "has seen an extraordinary expansion since the second world war", and they refer to the Nobel prizes for work in the field awarded to Lars Onsager and Ilya Prigogine. Martyushev and Seleznev (2006)[4] note the importance of entropy in the evolution of natural dynamical structures: "Great contribution has been done in this respect by two scientists, namely Clausius, ... , and Prigogine." Prigogine in his 1977 Nobel Lecture[55] said: "... non-equilibrium may be a source of order. Irreversible processes may lead to a new type of dynamic states of matter which I have called “dissipative structures”." Glansdorff and Prigogine (1971)[9] wrote on page xx: "Such 'symmetry breaking instabilities' are of special interest as they lead to a spontaneous 'self-organization' of the system both from the point of view of its space order and its function."

Analyzing the Rayleigh–Bénard convection cell phenomenon, Chandrasekhar (1961)[56] wrote "Instability occurs at the minimum temperature gradient at which a balance can be maintained between the kinetic energy dissipated by viscosity and the internal energy released by the buoyancy force." With a temperature gradient greater than the minimum, viscosity can dissipate kinetic energy as fast as it is released by convection due to buoyancy, and a steady state with convection is stable. The steady state with convection is often a pattern of macroscopically visible hexagonal cells with convection up or down in the middle or at the 'walls' of each cell, depending on the temperature dependence of the quantities; in the atmosphere under various conditions it seems that either is possible. (Some details are discussed by Lebon, Jou, and Casas-Vásquez (2008)[10] on pages 143–158.) With a temperature gradient less than the minimum, viscosity and heat conduction are so effective that convection cannot keep going.

Glansdorff and Prigogine (1971)[9] on page xv wrote "Dissipative structures have a quite different [from equilibrium structures] status: they are formed and maintained through the effect of exchange of energy and matter in non-equilibrium conditions." They were referring to the dissipation function of Rayleigh (1873)[36] that was used also by Onsager (1931, I,[1] 1931, II[38]). On pages 78–80 of their book[9] Glansdorff and Prigogine (1971) consider the stability of laminar flow that was pioneered by Helmholtz; they concluded that at a stable steady state of sufficiently slow laminar flow, the dissipation function was minimum.

Jou, Casas-Vazquez, Lebon (1993)note that classical non-equilibrium thermodynamics "has seen an extraordinary expansion since the second world war", and they refer to the Nobel prizes for work in the field awarded to Lars Onsager and Ilya Prigogine. Martyushev and Seleznev (2006)note the importance of entropy in the evolution of natural dynamical structures: "Great contribution has been done in this respect by two scientists, namely Clausius, ... , and Prigogine." Prigogine in his 1977 Nobel Lecture said: "... non-equilibrium may be a source of order. Irreversible processes may lead to a new type of dynamic states of matter which I have called “dissipative structures”." Glansdorff and Prigogine (1971) wrote on page xx: "Such 'symmetry breaking instabilities' are of special interest as they lead to a spontaneous 'self-organization' of the system both from the point of view of its space order and its function." Analyzing the Rayleigh–Bénard convection cell phenomenon, Chandrasekhar (1961) wrote "Instability occurs at the minimum temperature gradient at which a balance can be maintained between the kinetic energy dissipated by viscosity and the internal energy released by the buoyancy force." With a temperature gradient greater than the minimum, viscosity can dissipate kinetic energy as fast as it is released by convection due to buoyancy, and a steady state with convection is stable. The steady state with convection is often a pattern of macroscopically visible hexagonal cells with convection up or down in the middle or at the 'walls' of each cell, depending on the temperature dependence of the quantities; in the atmosphere under various conditions it seems that either is possible. (Some details are discussed by Lebon, Jou, and Casas-Vásquez (2008) on pages 143–158.) With a temperature gradient less than the minimum, viscosity and heat conduction are so effective that convection cannot keep going. Glansdorff and Prigogine (1971) on page xv wrote "Dissipative rstructures have a quite different [from equilibrium structures] status: they are formed and maintained through the effect of exchange of energy and matter in non-equilibrium conditions." They were referring to the dissipation function of Rayleigh (1873) that was used also by Onsager (1931, I,1931, II). On pages 78–80 of their book Glansdorff and Prigogine (1971) consider the stability of laminar flow that was pioneered by Helmholtz; they concluded that at a stable steady state of sufficiently slow laminar flow, the dissipation function was minimum. These advances have led to proposals for various extremal principles for the "self-organized" régimes that are possible for systems governed by classical linear and non-linear non-equilibrium thermodynamical laws, with stable stationary régimes being particularly investigated. Convection introduces effects of momentum which appear as non-linearity in the dynamical equations. In the more restricted case of no convective motion, Prigogine wrote of "dissipative structures". Šilhavý (1997) offers the opinion that "... the extremum principles of [equilibrium] thermodynamics ... do not have any counterpart for [non-equilibrium] steady states (despite many claims in the literature)."

Jou、Casas-Vazquez、Lebon(1993)指出，经典非平衡热力学 "自第二次世界大战以来得到了非凡的发展"，他们指的是诺贝尔奖获得者Lars Onsager和Ilya Prigogine。 Martyushev和Seleznev(2006)指出了熵在自然动力结构演化中的重要性。"在这方面，两位科学家，例如Clausius和Prigogine做出了巨大贡献。” Prigogine在1977年的诺贝尔演讲中说："... 非平衡可能是秩序的源泉。不可逆的过程可能导致一种新的物质动态状态，我称之为 "耗散结构" 。Glansdorff和Prigogine(1971)在第xx页上写道："这种'对称性破坏的不稳定性'特别令人关注，因为它们从系统的空间秩序和功能的角度来看，都会导致系统自发的'自组织'。" Chandrasekhar(1961)在分析Rayleigh-Bénard对流池现象时写道："不稳定性发生在最小温度梯度处，在这个温度梯度上，粘度所耗散的动能和浮力所释放的内能之间可以保持平衡。" 在温度梯度大于最小值的情况下，由于浮力的作用，粘度散失的动能与对流释放的动能一样快，有对流的稳态是稳定的。有对流的稳态往往是宏观可见的六边形单元的模式，每个单元的中间或 "壁 "处都有上下对流，这取决于量的温度依赖性；在大气中的各种条件下，似乎两种情况都有可能。一些细节由Lebon、Jou和Casasas-Vásquez（2008）在第143-158页上讨论。在温度梯度小于最小值的情况下，粘度和热传导非常有效，以至于对流无法继续进行。 Glansdorff和Prigogine(1971)在第xv页上写道："耗散结构具有完全不同的[与平衡结构]地位：它们是通过非平衡条件下能量和物质交换的作用形成和维持的"。他们指的是雷利(1873)的耗散函数，Onsager(1931,I,1931,II)也使用了这个函数。在他们的书的78-80页上，Glansdorff和Prigogine(1971)考虑了由Helmholtz开创的层流的稳定性；他们的结论是，在足够慢的层流的稳定稳态下，耗散函数是最小的。 这些进展导致了对经典线性和非线性非平衡热力学定律所控制的系统可能出现的 "自组织 "规则的各种极限原理的提出，其中稳定的静止规则被特别研究。对流引入了动量的影响，在动力学方程中表现为非线性。在没有对流运动的情况下，Prigogine写道 "耗散结构"。Šilhavý(1997年)提出了这样的观点："平衡热力学的极端原则对于非平衡稳定状态没有任何对应的原则(尽管文献中有许多说法)"。

## Prigogine's proposed theorem of minimum entropy production for very slow purely diffusive transfer

Prigogine提出的用于极慢的纯扩散转移的最小熵产生定理。

In 1945 Prigogine (see also Prigogine (1947)) proposed a “Theorem of Minimum Entropy Production” which applies only to the purely diffusive linear regime, with negligible inertial terms, near a stationary thermodynamically non-equilibrium state. Prigogine's proposal is that the rate of entropy production is locally minimum at every point. The proof offered by Prigogine is open to serious criticism. A critical and unsupportive discussion of Prigogine's proposal is offered by Grandy (2008). It has been shown by Barbera that the total whole body entropy production cannot be minimum, but this paper did not consider the pointwise minimum proposal of Prigogine. A proposal closely related to Prigogine's is that the pointwise rate of entropy production should have its maximum value minimized at the steady state. This is compatible, but not identical, with the Prigogine proposal. Moreover, N. W. Tschoegl proposes a proof, perhaps more physically motivated than Prigogine's, that would if valid support the conclusion of Helmholtz and of Prigogine, that under these restricted conditions, the entropy production is at a pointwise minimum.

1945年Prigogine(另见Prigogine(1947))提出了 "最小熵产生定理"，该定理只适用于静止的热力学非平衡状态附近的纯扩散线性体系，惯性项可忽略不计。Prigogine提出的是熵的产生速率在每一点上都是局部最小的。Prigogine提出的证明受到了严重的批评。Grandy(2008)对Prigogine的提议进行了批判性的和不支持的讨论。Barbera已经证明了全身熵产不能最小，但本文没有考虑Prigogine的点最小建议。与Prigogine的提议密切相关的是，熵产生的点率在稳态时应该有其最大值最小化。这与Prigogine的建议是一致的，但不完全相同。此外，N.W.Tschoegl提出了一个证明，也许比Prigogine的证明更有物理动机，如果有效的话，它将支持Helmholtz和Prigogine的结论，即在这些限制条件下，熵的产生是在一个点上最小的。

## Faster transfer with convective circulation: second entropy

In contrast to the case of sufficiently slow transfer with linearity between flux and generalized force with negligible inertial terms, there can be heat transfer that is not very slow. Then there is consequent non-linearity, and heat flow can develop into phases of convective circulation. In these cases, the time rate of entropy production has been shown to be a non-monotonic function of time during the approach to steady state heat convection. This makes these cases different from the near-thermodynamic-equilibrium regime of very-slow-transfer with linearity. Accordingly, the local time rate of entropy production, defined according to the local thermodynamic equilibrium hypothesis, is not an adequate variable for prediction of the time course of far-from-thermodynamic equilibrium processes. The principle of minimum entropy production is not applicable to these cases.

To cover these cases, there is needed at least one further state variable, a non-equilibrium quantity, the so-called second entropy. This appears to be a step towards generalization beyond the classical second law of thermodynamics, to cover non-equilibrium states or processes. The classical law refers only to states of thermodynamic equilibrium, and local thermodynamic equilibrium theory is an approximation that relies upon it. Still it is invoked to deal with phenomena near but not at thermodynamic equilibrium, and has some uses then. But the classical law is inadequate for description of the time course of processes far from thermodynamic equilibrium. For such processes, a more powerful theory is needed, and the second entropy is part of such a theory.

## Speculated principles of maximum entropy production and minimum energy dissipatio

Onsager (1931, I)wrote: "Thus the vector field J of the heat flow is described by the condition that the rate of increase of entropy, less the dissipation function, be a maximum." Careful note needs to be taken of the opposite signs of the rate of entropy production and of the dissipation function, appearing in the left-hand side of Onsager's equation (5.13) on Onsager's page 423.

Onsager(1931，I)写道："因此，热流的矢量场J是由熵的增加率减去耗散函数的条件来描述的，是一个最大值。" 需要仔细注意的是，在Onsager第423页的Onsager方程(5.13)左侧出现的熵产生率和耗散函数的相反符号。

Although largely unnoticed at the time, Ziegler proposed an idea early with his work in the mechanics of plastics in 1961,and later in his book on thermomechanics revised in 1983,and in various papers (e.g., Ziegler (1987),). Ziegler never stated his principle as a universal law but he may have intuited this. He demonstrated his principle using vector space geometry based on an “orthogonality condition” which only worked in systems where the velocities were defined as a single vector or tensor, and thus, as he wrote at p. 347, was “impossible to test by means of macroscopic mechanical models”, and was, as he pointed out, invalid in “compound systems where several elementary processes take place simultaneously”. In relation to the earth's atmospheric energy transport process, according to Tuck (2008), "On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975,2001)." Initially Paltridge (1975)used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978),and in Paltridge (1979),he used the now current terminology "maximum entropy production" to describe the same thing. The logic of Paltridge's earlier work is open to serious criticism.Nicolis and Nicolis (1980) discuss Paltridge's work, and they comment that the behaviour of the entropy production is far from simple and universal. Later work by Paltridge focuses more on the idea of a dissipation function than on the idea of rate of production of entropy. Sawada (1981),also in relation to the Earth's atmospheric energy transport process, postulating a principle of largest amount of entropy increment per unit time, cites work in fluid mechanics by Malkus and Veronis (1958) as having "proven a principle of maximum heat current, which in turn is a maximum entropy production for a given boundary condition", but this inference is not logically valid. Again investigating planetary atmospheric dynamics, Shutts (1981) used an approach to the definition of entropy production, different from Paltridge's, to investigate a more abstract way to check the principle of maximum entropy production, and reported a good fit.

## Prospects

Until recently, prospects for useful extremal principles in this area have seemed clouded. C. Nicolis (1999) concludes that one model of atmospheric dynamics has an attractor which is not a regime of maximum or minimum dissipation; she says this seems to rule out the existence of a global organizing principle, and comments that this is to some extent disappointing; she also points to the difficulty of finding a thermodynamically consistent form of entropy production. Another top expert offers an extensive discussion of the possibilities for principles of extrema of entropy production and of dissipation of energy: Chapter 12 of Grandy (2008) and chemical reactions do not obey extremal principles for the secondary differential of entropy production, hence the development of a general extremal principle seems infeasible.

• Non-equilibrium thermodynamics 非平衡热力学
• Dissipative system 耗散系统
• Self-organization 自组织
• Autocatalytic reactions and order creation 自动催化反应和秩序创造
• Fluctuation theorem涨落定理
• Fluctuation dissipation theorem波动耗散定理

## References

This page was moved from wikipedia:en:Extremal principles in non-equilibrium thermodynamics. Its edit history can be viewed at 非平衡热力学的极值定理/edithistory

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