非平衡热力学的极值定理

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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. 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)

能量耗散和产生熵极值原理是在非平衡态热力学中发展起来的概念，它们试图预测物理系统可能表现出的可能的稳态和动态结构。在物理学的其他分支成功应用极值原理之后，我们对非平衡态热力学的极值原理进行了探索。不可逆过程通常不受全局极值原理的控制，因为对其演化的描述需要微分方程，而微分方程不是自伴的，但局部极值原理可用于局部解。Lebon Jou 和 Casas-Vásquez (2008)


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

It is pointed out such a metaphorical but not categorically mechanical force, the thermal "force", [math]\displaystyle{ X_{th} }[/math], 'drives' the conduction of heat. For this so-called "thermodynamic force", we can write

有人指出，这种隐喻性但并非绝对的机械力---- 热力---- 驱动着热传导。对于这个所谓的“热力学力” ，我们可以写

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]

[math]\displaystyle{ X_{th} = - \frac{1}{T} \nabla T }[/math].

< math > x _ { th } =-frac {1}{ t } nabla t </math > .


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, [math]\displaystyle{ T }[/math]. 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.

实际上，这种热力学“热力学力”是体系的微观初始条件不精确程度的表现，用称为温度的热力学变量来表示。温度只是一个例子，所有的热力学宏观变量构成的初始条件的不精确规格，并有各自的“热力学力”。这些不确定的规范是驱动动力学结构产生的明显波动的来源，是非平衡实验非常精确但仍然不完美的可重复性的来源，也是熵在热力学中的地位的来源。如果一个人不知道这种不准确的说明，他可能会发现波动的起源是神秘的。这里所说的“规格的不确定性”并不是指宏观变量的平均值没有准确地确定，而是指用宏观变量来描述微观物体(如分子)的运动和相互作用实际发生的过程，必然缺乏过程的分子细节，因此是不精确的。有许多微观状态可以与单一的宏观状态相兼容，但只有后者是确定的，而这正是为了理论的目的而确定的。

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 reproducibility in repeated observations that identifies dynamical structure in a system. E.T. Jaynes writes "Jaynes considered reproducibility - rather than disorder - to be the key idea behind the second law of thermodynamics (Jaynes 1963, 1988, 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.

在重复的观察中，确定一个系统的动力学结构是可重复的。外星人。Jaynes 写道: “ Jaynes 认为可再生性而不是无序性是热力学第二定律理论背后的关键思想(Jaynes 1963,1988，在55页的第4.3节澄清了熵与秩序有关的概念(他认为这是一个“不幸的”“错误描述” ，需要“揭穿”)和 Jaynes 前面提到的熵是实验过程可再生性的度量(Grandy 认为是正确的)之间的区别。根据这种观点，即使是令人钦佩的《 Glansdorff 与普里戈金》(1971)也对这种不幸的滥用语言负有责任。

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

[math]\displaystyle{ X_{th} = - \frac{1}{T} \nabla T }[/math].

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, [math]\displaystyle{ T }[/math]. 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.

Various principles have been proposed by diverse authors for over a century. According to Glansdorff and Prigogine (1971, page 15),) 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.

一个多世纪以来，不同的作家提出了各种各样的原则。根据格兰斯多夫和普里高金(1971年，第15页) ，引入耗散函数描述耗散过程涉及粘性。这个函数的更一般的形式已经被许多随后的耗散过程和动力结构性质的研究者所使用。瑞利耗散函数是从机械学的观点出发构想出来的，它在定义中并没有提到温度，它需要被广义化，以使耗散函数适用于非平衡态热力学。


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]

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. cites Onsager (1931).

瑞利(1878,1896/1926)研究了喷嘴内的水射流，发现当喷嘴处于条件稳定动力结构状态时，最有可能发展到充分发展并导致另一种条件稳定动力结构状态的涨落模式是增长速度最快的。换句话说，一个射流可以稳定到一个条件稳定的状态，但是它很可能遭受波动，从而转移到另一个条件稳定的不太稳定的状态。他在贝纳德对流研究中使用了类似推理。昂萨格(1931)。


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.


Casimir (1945) cites 11 papers or books authored or co-authored by Prigogine.

卡西米尔(1945)引用了普里戈金的11篇论文或书籍。

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.

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)也在第三部分5给出了一个非常有用的精确的卡西米尔(1945)。他解释说，昂萨格互反关系涉及的变量是分子速度的偶数函数，并指出，卡西米尔继续推导反对称关系的变量是分子速度的奇数函数。


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.

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), used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978), cited Busse's (1967) 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 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."

地球大气物理学包括诸如闪电和火山爆发的影响等戏剧性事件，以及诸如亥姆霍兹(1868)所指出的运动中断。在变形(气象学) ，湍流是突出的。其他不连续面包括雨滴、冰雹和雪花的形成。通常的经典非平衡态热力学理论将需要一些扩展来涵盖大气物理学。据塔克(2008年) ，使用术语“最小熵交换” ，但在此之后，例如在 Paltridge (1978年) ，引用 Busse (1967年)的注意到熵在自然动力结构的进化中的重要性: “两个科学家在这方面做出了巨大贡献，即克劳修斯，... ，和 Prigogine。”普里戈金在他1977年的诺贝尔演讲中在第 xx 页写道: “这种‘对称性破缺不稳定性’特别有意义，因为从空间秩序及其功能的角度来看，它们导致了系统自发的‘自我组织’。”


Continuous and discontinuous motions of fluids

Analyzing the Rayleigh–Bénard convection cell phenomenon, Chandrasekhar (1961) on pages 143–158.) With a temperature gradient less than the minimum, viscosity and heat conduction are so effective that convection cannot keep going.

分析瑞利-贝纳德对流单元现象，钱德拉塞卡(1961) ，第143-158页由于温度梯度低于最小值，粘性和热传导非常有效，以至于对流无法继续。

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]).


Glansdorff and Prigogine (1971) 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) 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.

格兰斯多夫和普里戈金(1971)在第十五页写道: “耗散结构有一个完全不同的[平衡结构]地位: 他们是通过在非平衡条件下能量和物质的交换效应形成和维持的。”他们指的是瑞利(1873)的耗散功能，也被昂萨格(1931，i，1931，II)使用。在他们的书格兰斯多夫和普里戈金(1971)78-80页考虑层流的稳定性，由亥姆霍兹开创; 他们的结论是，在足够慢的层流稳定状态下，耗散功能是最小的。

Historical development

W. Thomson, Baron Kelvin

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)."

这些进展导致了关于”自组织”网络的各种极值原理的建议，这些原理对于受经典线性和非线性非平衡热力学定律支配的系统是可能的，特别是对稳定平稳网络的研究。对流引入了动量效应，在动力学方程中表现为非线性。在没有对流运动的更为严格的情况下，普里戈金写到了“耗散结构”。Ilhavý (1997)认为“ ... 热力学[平衡态]的极值原理... 对于[非平衡态]稳态没有任何相应的对应物(尽管文献中有许多说法)。”


William Thomson, later Baron Kelvin, (1852 a,[32] 1852 b[33]) wrote


In 1945 Prigogine (see also Prigogine (1947) 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. 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.

1945年，Prigogine (又见 Prigogine (1947) Barbera 证明了全身产生熵不可能是最小的，但本文没有考虑 Prigogine 的逐点最小提议。他写道: “因此，热流的矢量场 j 被描述为熵增率减去耗散函数后为最大值的条件。”需要仔细注意的是产生熵和耗散函数的相反符号，出现在昂萨格方程的左手边(5.13)的423页。

"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.


Although largely unnoticed at the time, Ziegler proposed an idea early with his work in the mechanics of plastics in 1961, and in various papers (e.g., Ziegler (1987), 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”.

虽然当时基本上没有被注意到，但是齐格勒在1961年塑料力学的早期工作中提出了一个想法，并且在各种论文中(例如，齐格勒(1987) ，第347页，“不可能用宏观力学模型来测试” ，并且，正如他指出的，在“几个基本过程同时发生的复合系统”中是无效的。

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


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 used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978), and in Paltridge (1979), 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.

关于地球的大气能量传输过程，根据 Tuck (2008年) ，“在宏观层面上，这种方法是由一位气象学家开创的(Paltridge 1975年，2001年使用了术语“最小熵交换” ，但在那之后，例如在 Paltridge (1978年) ，在 Paltridge (1979年) ，Nicolis 和 Nicolis (1980年)讨论了 Paltridge 的工作，他们评论说产生熵的行为远非简单和普遍。帕特里奇后来的著作更多地集中在耗散函数的概念上，而不是熵产生率的概念上。

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."


Sawada (1981), 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). It has been shown that heat convection does not obey extremal principles for entropy production 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.

泽田教授(1981)非常谨慎，在很多情况下难以定义内部产生熵，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)。已经证明，热对流不服从产生熵的极值原理，化学反应不服从产生熵的二次微分的极值原理，因此发展一般的极值原理似乎是不可行的。

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)[34] 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.


Helmholtz

In 1869, Hermann von Helmholtz stated his Helmholtz minimum dissipation theorem,[35] 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."[36]


In 1878, Helmholtz,[37] 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.


J. W. Strutt, Baron Rayleigh

Rayleigh (1873)[38] (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.


Studying jets of water from a nozzle, Rayleigh (1878,[27] 1896/1926[28]) 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.[39] 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.


Korteweg

Korteweg (1883)[40] 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.

Category:Non-equilibrium thermodynamics

类别: 非平衡态热力学


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  1. ↑ 1.0 1.1 Onsager, L (1931). "Reciprocal relations in irreversible processes, I". Physical Review. 37 (4): 405–426. Bibcode:1931PhRv...37..405O. doi:10.1103/physrev.37.405.
  2. ↑ 引用错误：无效<ref>标签；未给name属性为Gyarmati 1970的引用提供文字
  3. ↑ Ziegler, H., (1983). An Introduction to Thermomechanics, North-Holland, Amsterdam,
  4. ↑ Martyushev, L.M.; Seleznev, V.D. (2006). "Maximum entropy production principle in physics, chemistry and biology" (PDF). Physics Reports. 426 (1): 1–45. Bibcode:2006PhR...426....1M. doi:10.1016/j.physrep.2005.12.001. Archived from the original (PDF) on 2011-03-02. Retrieved 2009-10-10.
  5. ↑ Martyushev, I.M.; Nazarova, A.S.; Seleznev, V.D. (2007). "On the problem of the minimum entropy production in the nonequilibrium stationary state". Journal of Physics A: Mathematical and Theoretical. 40 (3): 371–380. Bibcode:2007JPhA...40..371M. doi:10.1088/1751-8113/40/3/002.
  6. ↑ Hillert, M.; Agren, J. (2006). "Extremum principles for irreversible processes". Acta Materialia. 54 (8): 2063–2066. doi:10.1016/j.actamat.2005.12.033.
  7. ↑ Kondepudi, D. (2008)., Introduction to Modern Thermodynamics, Wiley, Chichester UK,
    , page 172.
  8. ↑ 8.0 8.1 8.2 Grandy, W.T., Jr (2008). Entropy and the Time Evolution of Macroscopic Systems, Oxford University Press, Oxford,
    .
  9. ↑ 9.0 9.1 9.2 9.3 Glansdorff, P., Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience, London.
  10. ↑ Lebon, G., Jou, J., Casas-Vásquez (2008). Understanding Non-equilibrium Thermodynamics. Foundations, Applications, Frontiers, Springer, Berlin,
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  11. ↑ Šilhavý, M. (1997). The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin,
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  15. ↑ Grandy 2004 see also [4].
  16. ↑ Jaynes, E.T. (1957). "Information theory and statistical mechanics" (PDF). Physical Review. 106 (4): 620–630. Bibcode:1957PhRv..106..620J. doi:10.1103/physrev.106.620.
  17. ↑ Jaynes, E.T. (1957). "Information theory and statistical mechanics. II" (PDF). Physical Review. 108 (2): 171–190. Bibcode:1957PhRv..108..171J. doi:10.1103/physrev.108.171.
  18. ↑ Jaynes, E.T. (1985). Macroscopic prediction, in Complex Systems - Operational Approaches in Neurobiology, edited by H. Haken, Springer-Verlag, Berlin, pp. 254-269
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  19. ↑ 19.0 19.1 Jaynes, E.T. (1965). "Gibbs vs Boltzmann Entropies" (PDF). American Journal of Physics. 33 (5): 391–398. Bibcode:1965AmJPh..33..391J. doi:10.1119/1.1971557.
  20. ↑ Evans, D.J.; Searles, D.J. (2002). "The fluctuation theorem". Advances in Physics. 51 (7): 1529–1585. Bibcode:2002AdPhy..51.1529E. doi:10.1080/00018730210155133. S2CID 10308868.
  21. ↑ Wang, G.M., Sevick, E.M., Mittag, E., Searles, D.J., Evans, D.J. (2002) Experimental demonstration of violations of the Second Law of Thermodynamics for small systems and short time scales, Physical Review Letters 89: 050601-1 - 050601-4.
  22. ↑ 22.0 22.1 Dewar, R.C. (2005). Maximum entropy production and non-equilibrium statistical mechanics, pp. 41-55 in Non-equilibrium Thermodynamics and the Production of Entropy, edited by A. Kleidon, R.D. Lorenz, Springer, Berlin.
    .
  23. ↑ Jaynes, E.T. (1963). pp. 181-218 in Brandeis Summer Institute 1962, Statistical Physics, edited by K.W. Ford, Benjamin, New York.
  24. ↑ Jaynes, E.T. (1988). The evolution of Carnot's Principle, pp. 267-282 in Maximum-entropy and Bayesian methods in science and engineering, edited by G.J. Erickson, C.R. Smith, Kluwer, Dordrecht, volume 1
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  25. ↑ Jaynes, E.T. (1989). Clearing up mysteries, the original goal, pp. 1-27 in Maximum entropy and Bayesian methods, Kluwer, Dordrecht.
  26. ↑ Helmholtz, H. (1868). On discontinuous movements of fluids, Philosophical Magazine series 4, vol. 36: 337-346, translated by F. Guthrie from Monatsbericht der koeniglich preussischen Akademie der Wissenschaften zu Berlin April 1868, page 215 et seq.
  27. ↑ 27.0 27.1 Strutt, J.W. (1878). "On the instability of jets". Proceedings of the London Mathematical Society. 10: 4–13. doi:10.1112/plms/s1-10.1.4.
  28. ↑ 28.0 28.1 28.2 Strutt, J.W. (Baron Rayleigh) (1896/1926). Section 357 et seq. The Theory of Sound, Macmillan, London, reprinted by Dover, New York, 1945.
  29. ↑ Thom, R. (1975). Structural Stability and Morphogenesis: An outline of a general theory of models, translated from the French by D.H. Fowler, W.A. Benjamin, Reading Ma,
  30. ↑ Riehl, H.; Malkus, J.S. (1958). "On the heat balance in the equatorial trough zone". Geophysica. 6: 503–538.
  31. ↑ Lindzen, R.S. (1977). Some aspects of convection in meteorology, pp. 128-141 in Problems of Stellar Convection, volume 71 of Lecture Notes in Physics, Springer, Berlin,
    .
  32. ↑ Thomson, William (1852 a). "On a Universal Tendency in Nature to the Dissipation of Mechanical Energy" Proceedings of the Royal Society of Edinburgh for April 19, 1852 [This version from Mathematical and Physical Papers, vol. i, art. 59, pp. 511.]
  33. ↑ Thomson, W (1852). "b). On a universal tendency in nature to the dissipation of mechanical energy". Philosophical Magazine. 4: 304–306.
  34. ↑ Thomson, W. (1854). On a mechanical theory of thermo-electric currents, Proceedings of the Royal Society of Edinburgh pp. 91-98.
  35. ↑ Helmholtz, H. (1869/1871). Zur Theorie der stationären Ströme in reibenden Flüssigkeiten, Verhandlungen des naturhistorisch-medizinischen Vereins zu Heidelberg, Band V: 1-7. Reprinted in Helmholtz, H. (1882), Wissenschaftliche Abhandlungen, volume 1, Johann Ambrosius Barth, Leipzig, pages 223-230 [5]
  36. ↑ from page 2 of Helmholtz 1869/1871, translated by Wikipedia editor.
  37. ↑ Helmholtz, H (1878). "Ueber galvanische Ströme, verursacht durch Concentrationsunterschiede; Folgeren aus der mechanischen Wärmetheorie, Wiedermann's". Annalen der Physik und Chemie. 3 (2): 201–216. doi:10.1002/andp.18782390204.
  38. ↑ Strutt, J.W. (1873). "Some theorems relating to vibrations". Proceedings of the London Mathematical Society. 4: 357–368. doi:10.1112/plms/s1-4.1.357.
  39. ↑ Strutt, J.W. (Baron Rayleigh) (1916). On convection currents in a horizontal layer of fluids, when the higher temperature is on the under side, The London, Edinburgh, and Dublin Philosophical Magazine series 6, volume 32: 529-546.
  40. ↑ Korteweg, D.J. (1883). "On a general theorem of the stability of the motion of a viscous fluid". The London, Edinburgh and Dublin Philosophical Journal of Science. 16 (98): 112–118. doi:10.1080/14786448308627405.
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