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第1行： − 此词条暂由彩云小译翻译，翻译字数共1436，未经人工整理和审校，带来阅读不便，请见谅。+ 第5行：
第5行： − + − 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)+ − − It is pointed out such a metaphorical but not categorically mechanical force, the thermal "force", <math>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 − <math>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>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<ref>Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. I: Equations of motion. ''Found. Phys.'' '''34''': 1-20. See [http://physics.uwyo.edu/~tgrandy/evolve.html].</ref><ref>Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. II: The entropy. ''Found. Phys.'' '''34''': 21-57. See [http://physics.uwyo.edu/~tgrandy/entropy.html].</ref><ref>Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. III: Selected applications. ''Found. Phys.'' '''34''': 771-813. See [http://physics.uwyo.edu/~tgrandy/applications.html].</ref><ref>Grandy 2004 see also [http://physics.uwyo.edu/~tgrandy/Statistical_Mechanics.html].</ref> 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),<ref name="Onsager 1931 I"/> such a metaphorical but not categorically mechanical force, the thermal "force", <math>X_{th}</math>, 'drives' the conduction of heat. For this so-called "thermodynamic force", we can write+ 第43行：
第39行： − 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。 + + − − 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)。 第59行：
第51行： + + − 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。 第69行：
第61行： − 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==+ − 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’。” + + − 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)<ref name="Helmholtz 1868">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.</ref> 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,<ref name="Rayleigh 1878">{{cite journal | last1 = Strutt | first1 = J.W. | year = 1878 | title = On the instability of jets | url = https://zenodo.org/record/2095384| journal = Proceedings of the London Mathematical Society | volume = 10 | issue = | pages = 4–13 | doi=10.1112/plms/s1-10.1.4}}</ref> and in section 357 et seq. of Rayleigh (1896/1926)<ref name="Rayleigh 1896/1926">Strutt, J.W. (Baron Rayleigh) (1896/1926). Section 357 et seq. ''The Theory of Sound'', Macmillan, London, reprinted by Dover, New York, 1945.</ref>); waves on the surface of the sea break discontinuously when they reach the shore (Thom 1975<ref name="Thom 1975">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, {{ISBN|0-8053-9279-3}}</ref>). 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<ref name="Riehl Malkus 1958">{{cite journal | last1 = Riehl | first1 = H. | last2 = Malkus | first2 = J.S. | year = 1958 | title = On the heat balance in the equatorial trough zone | url = | journal = Geophysica | volume = 6 | issue = | pages = 503–538 }}</ref>), also called penetrative convection (Lindzen 1977<ref name="Lindzen 1977">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, {{ISBN|978-3-540-08532-4}}.</ref>).+ − 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==+ − 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)。” + + − − 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。 + + − 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。 + + − 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。 第137行：
第125行： − 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。+ 第146行：
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Moved page from wikipedia:en:Extremal principles in non-equilibrium thermodynamics (history)
此词条暂由彩云小译翻译，翻译字数共2606，未经人工整理和审校，带来阅读不便，请见谅。
{{Essay|date=December 2010}}
{{Essay|date=December 2010}}
'''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.<ref name="Onsager 1931 I">{{cite journal | last1 = Onsager | first1 = L | year = 1931 | title = Reciprocal relations in irreversible processes, I | url = | journal = Physical Review | volume = 37 | issue = 4| pages = 405–426 | doi=10.1103/physrev.37.405| bibcode = 1931PhRv...37..405O| doi-access = free }}</ref><ref name="Gyarmati 1970"/><ref name="Ziegler 1983">Ziegler, H., (1983). ''An Introduction to Thermomechanics'', North-Holland, Amsterdam, {{ISBN|0-444-86503-9}}</ref><ref name="M&S 2006">{{cite journal | last1 = Martyushev | first1 = L.M. | last2 = Seleznev | first2 = V.D. | year = 2006 | title = Maximum entropy production principle in physics, chemistry and biology | url = http://www.rsbs.anu.edu.au/ResearchGroups/EBG/profiles/Roderick_Dewar/Martyushev%20and%20Seleznev%202006%20Phys%20Rep.pdf | journal = Physics Reports | volume = 426 | issue = 1 | pages = 1–45 | doi = 10.1016/j.physrep.2005.12.001 | bibcode = 2006PhR...426....1M | access-date = 2009-10-10 | archive-url = https://web.archive.org/web/20110302120848/http://www.rsbs.anu.edu.au/ResearchGroups/EBG/profiles/Roderick_Dewar/Martyushev%20and%20Seleznev%202006%20Phys%20Rep.pdf | archive-date = 2011-03-02 | url-status = dead }}</ref><ref name="MNS 2007">{{cite journal | last1 = Martyushev | first1 = I.M. | last2 = Nazarova | first2 = A.S. | last3 = Seleznev | first3 = V.D. | year = 2007 | title = On the problem of the minimum entropy production in the nonequilibrium stationary state | url = | journal = Journal of Physics A: Mathematical and Theoretical | volume = 40 | issue = 3| pages = 371–380 | doi=10.1088/1751-8113/40/3/002| bibcode = 2007JPhA...40..371M }}</ref><ref name="Hillert Agren 2006">{{cite journal | last1 = Hillert | first1 = M. | last2 = Agren | first2 = J. | year = 2006 | title = Extremum principles for irreversible processes | url = | journal = Acta Materialia | volume = 54 | issue = 8| pages = 2063–2066 | doi=10.1016/j.actamat.2005.12.033}}</ref> According to Kondepudi (2008),<ref>Kondepudi, D. (2008)., ''Introduction to Modern Thermodynamics'', Wiley, Chichester UK, {{ISBN|978-0-470-01598-8}}, page 172.</ref> and to Grandy (2008),<ref name="Grandy 2008">Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}.</ref> 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),<ref name="G&P 1971"/> 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)<ref name="Lebon Jou Casas-Vázquez 2008">Lebon, G., Jou, J., Casas-Vásquez (2008). ''Understanding Non-equilibrium Thermodynamics. Foundations, Applications, Frontiers'', Springer, Berlin, {{ISBN|978-3-540-74251-7}}.</ref> state that "In non-equilibrium ... it is generally not possible to construct thermodynamic potentials depending on the whole set of variables". Šilhavý (1997)<ref name="Continuous Media page 209">Šilhavý, M. (1997). ''The Mechanics and Thermodynamics of Continuous Media'', Springer, Berlin, {{ISBN|3-540-58378-5}}, page 209.</ref> 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.<ref name="Onsager 1931 I">{{cite journal | last1 = Onsager | first1 = L | year = 1931 | title = Reciprocal relations in irreversible processes, I | url = | journal = Physical Review | volume = 37 | issue = 4| pages = 405–426 | doi=10.1103/physrev.37.405| bibcode = 1931PhRv...37..405O| doi-access = free }}</ref><ref name="Gyarmati 1970"/><ref name="Ziegler 1983">Ziegler, H., (1983). ''An Introduction to Thermomechanics'', North-Holland, Amsterdam, {{ISBN|0-444-86503-9}}</ref><ref name="M&S 2006">{{cite journal | last1 = Martyushev | first1 = L.M. | last2 = Seleznev | first2 = V.D. | year = 2006 | title = Maximum entropy production principle in physics, chemistry and biology | url = http://www.rsbs.anu.edu.au/ResearchGroups/EBG/profiles/Roderick_Dewar/Martyushev%20and%20Seleznev%202006%20Phys%20Rep.pdf | journal = Physics Reports | volume = 426 | issue = 1 | pages = 1–45 | doi = 10.1016/j.physrep.2005.12.001 | bibcode = 2006PhR...426....1M | access-date = 2009-10-10 | archive-url = https://web.archive.org/web/20110302120848/http://www.rsbs.anu.edu.au/ResearchGroups/EBG/profiles/Roderick_Dewar/Martyushev%20and%20Seleznev%202006%20Phys%20Rep.pdf | archive-date = 2011-03-02 | url-status = dead }}</ref><ref name="MNS 2007">{{cite journal | last1 = Martyushev | first1 = I.M. | last2 = Nazarova | first2 = A.S. | last3 = Seleznev | first3 = V.D. | year = 2007 | title = On the problem of the minimum entropy production in the nonequilibrium stationary state | url = | journal = Journal of Physics A: Mathematical and Theoretical | volume = 40 | issue = 3| pages = 371–380 | doi=10.1088/1751-8113/40/3/002| bibcode = 2007JPhA...40..371M }}</ref><ref name="Hillert Agren 2006">{{cite journal | last1 = Hillert | first1 = M. | last2 = Agren | first2 = J. | year = 2006 | title = Extremum principles for irreversible processes | url = | journal = Acta Materialia | volume = 54 | issue = 8| pages = 2063–2066 | doi=10.1016/j.actamat.2005.12.033}}</ref> According to Kondepudi (2008),<ref>Kondepudi, D. (2008)., ''Introduction to Modern Thermodynamics'', Wiley, Chichester UK, {{ISBN|978-0-470-01598-8}}, page 172.</ref> and to Grandy (2008),<ref name="Grandy 2008">Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}.</ref> 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),<ref name="G&P 1971"/> 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)<ref name="Lebon Jou Casas-Vázquez 2008">Lebon, G., Jou, J., Casas-Vásquez (2008). ''Understanding Non-equilibrium Thermodynamics. Foundations, Applications, Frontiers'', Springer, Berlin, {{ISBN|978-3-540-74251-7}}.</ref> state that "In non-equilibrium ... it is generally not possible to construct thermodynamic potentials depending on the whole set of variables". Šilhavý (1997)<ref name="Continuous Media page 209">Šilhavý, M. (1997). ''The Mechanics and Thermodynamics of Continuous Media'', Springer, Berlin, {{ISBN|3-540-58378-5}}, page 209.</ref> 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)
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.
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。根据 Kondepudi (2008年)和 Grandy (2008年)的说法，没有一般规则可以提供一个极值原理来管理一个远离平衡的系统向稳定状态的演化。根据格兰斯多夫和普里戈金(1971年，第16页) ，指出“在非平衡状态下... ... 通常不可能根据整套变量来构造热力学势”。Ilhavý (1997)认为“ ... 热力学的极值原理... 对于[非平衡]稳态没有任何对应物(尽管文献中有许多说法)。”因此，任何非平衡问题的一般极值原理都需要比较详细地提到问题中所考虑的系统结构的具体约束条件。
==Fluctuations, entropy, 'thermodynamics forces', and reproducible dynamical structure==
==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.<ref name="G&P 1971"/>
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.<ref name="G&P 1971"/>
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. 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.
当初始条件没有确切规定时出现的明显的“涨落”是形成非平衡动力学结构的驱动力。这种波动的产生并不涉及特殊的自然力量。准确地说明初始条件需要描述系统中所有粒子的位置和速度，显然对于宏观系统来说这是不切实际的。这就是热力学涨落的本质。它们是由自然规律决定的，是动力结构自然发展的单一原因。作者: w.t。虽然熵可以定义为一个非平衡系统，但严格考虑时，熵只是一个指向整个系统的宏观量，而不是一个动力学变量，一般不作为描述局部物理力的局部势。然而，在特殊情况下，人们可以隐喻地认为，热变量表现得像局部物理力量。构成经典不可逆热力学的近似是建立在这种隐喻思维之上的。
It is pointed out<ref>Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. I: Equations of motion. ''Found. Phys.'' '''34''': 1-20. See [http://physics.uwyo.edu/~tgrandy/evolve.html].</ref><ref>Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. II: The entropy. ''Found. Phys.'' '''34''': 21-57. See [http://physics.uwyo.edu/~tgrandy/entropy.html].</ref><ref>Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. III: Selected applications. ''Found. Phys.'' '''34''': 771-813. See [http://physics.uwyo.edu/~tgrandy/applications.html].</ref><ref>Grandy 2004 see also [http://physics.uwyo.edu/~tgrandy/Statistical_Mechanics.html].</ref> 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.
As indicated by the " " marks of Onsager (1931), 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.
正如昂萨格(1931)的“”标记所指出的，解释了为什么这种重现性是熵在这个主题中如此重要的原因: 熵是实验重现性的一种度量。熵告诉我们需要重复实验多少次，才能期望看到偏离通常可重复的结果。当这个过程在一个分子数量少于“几乎无穷”的系统中进行时(远远少于阿伏加德罗或洛施密特的数量) ，热力学的可重复性就会消失，波动就更容易被看到。
As indicated by the " " marks of Onsager (1931),<ref name="Onsager 1931 I"/> such a metaphorical but not categorically mechanical force, the thermal "force", <math>X_{th}</math>, 'drives' the conduction of heat. For this so-called "thermodynamic force", we can write
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 1965, 1989)." Grandy (2008) 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.
根据杰尼斯的这一观点，这是语言的一种普遍而神秘的滥用，人们经常看到动力结构的复制性，这种复制性被称为“秩序”。杜瓦1965,1989)。”Grandy (2008)一般来说，这些原理只适用于可以用热力学变量描述的系统，在这些系统中，耗散过程通过排除统计平衡的大偏差而占主导地位。热力学变量的定义取决于局部热力学平衡的运动学要求。这意味着分子之间的碰撞是如此频繁，以至于化学过程和辐射过程不会破坏局部的麦克斯韦-玻尔兹曼分子速度分布。
:::::<math>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>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.
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>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.
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.
耗散结构可以依赖于动力系统中非线性的存在。自催化反应提供了非线性动力学的例子，可能导致自组织耗散结构的自然演化。
It is reproducibility in repeated observations that identifies dynamical structure in a system. [[Edwin Thompson Jaynes|E.T. Jaynes]]<ref name="Jaynes 1957 I">{{cite journal | last1 = Jaynes | first1 = E.T. | year = 1957 | title = Information theory and statistical mechanics | url = http://bayes.wustl.edu/etj/articles/theory.1.pdf | journal = Physical Review | volume = 106 | issue = 4| pages = 620–630 | doi=10.1103/physrev.106.620| bibcode = 1957PhRv..106..620J }}</ref><ref name="Jaynes 1957 II">{{cite journal | last1 = Jaynes | first1 = E.T. | year = 1957 | title = Information theory and statistical mechanics. II | url = http://bayes.wustl.edu/etj/articles/theory.2.pdf | journal = Physical Review | volume = 108 | issue = 2| pages = 171–190 | doi=10.1103/physrev.108.171| bibcode = 1957PhRv..108..171J }}</ref><ref name="Jaynes 1985">[http://bayes.wustl.edu/etj/articles/macroscopic.prediction.pdf Jaynes, E.T. (1985). Macroscopic prediction, in ''Complex Systems - Operational Approaches in Neurobiology'', edited by H. Haken, Springer-Verlag, Berlin, pp. 254-269] {{ISBN|3-540-15923-1}}.</ref><ref name="Jaynes 1965">{{cite journal | last1 = Jaynes | first1 = E.T. | year = 1965 | title = Gibbs vs Boltzmann Entropies | url = http://bayes.wustl.edu/etj/articles/gibbs.vs.boltzmann.pdf | journal = American Journal of Physics | volume = 33 | issue = 5| pages = 391–398 | doi=10.1119/1.1971557| bibcode = 1965AmJPh..33..391J }}</ref> 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.<ref name="Evans Searles 2002">{{cite journal | last1 = Evans | first1 = D.J. | last2 = Searles | first2 = D.J. | s2cid = 10308868 | year = 2002 | title = The fluctuation theorem | journal = Advances in Physics | volume = 51 | issue = 7| pages = 1529–1585 | doi=10.1080/00018730210155133| bibcode = 2002AdPhy..51.1529E }}</ref><ref name="WSMSE 2002">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.</ref>
It is reproducibility in repeated observations that identifies dynamical structure in a system. [[Edwin Thompson Jaynes|E.T. Jaynes]]<ref name="Jaynes 1957 I">{{cite journal | last1 = Jaynes | first1 = E.T. | year = 1957 | title = Information theory and statistical mechanics | url = http://bayes.wustl.edu/etj/articles/theory.1.pdf | journal = Physical Review | volume = 106 | issue = 4| pages = 620–630 | doi=10.1103/physrev.106.620| bibcode = 1957PhRv..106..620J }}</ref><ref name="Jaynes 1957 II">{{cite journal | last1 = Jaynes | first1 = E.T. | year = 1957 | title = Information theory and statistical mechanics. II | url = http://bayes.wustl.edu/etj/articles/theory.2.pdf | journal = Physical Review | volume = 108 | issue = 2| pages = 171–190 | doi=10.1103/physrev.108.171| bibcode = 1957PhRv..108..171J }}</ref><ref name="Jaynes 1985">[http://bayes.wustl.edu/etj/articles/macroscopic.prediction.pdf Jaynes, E.T. (1985). Macroscopic prediction, in ''Complex Systems - Operational Approaches in Neurobiology'', edited by H. Haken, Springer-Verlag, Berlin, pp. 254-269] {{ISBN|3-540-15923-1}}.</ref><ref name="Jaynes 1965">{{cite journal | last1 = Jaynes | first1 = E.T. | year = 1965 | title = Gibbs vs Boltzmann Entropies | url = http://bayes.wustl.edu/etj/articles/gibbs.vs.boltzmann.pdf | journal = American Journal of Physics | volume = 33 | issue = 5| pages = 391–398 | doi=10.1119/1.1971557| bibcode = 1965AmJPh..33..391J }}</ref> 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.<ref name="Evans Searles 2002">{{cite journal | last1 = Evans | first1 = D.J. | last2 = Searles | first2 = D.J. | s2cid = 10308868 | year = 2002 | title = The fluctuation theorem | journal = Advances in Physics | volume = 51 | issue = 7| pages = 1529–1585 | doi=10.1080/00018730210155133| bibcode = 2002AdPhy..51.1529E }}</ref><ref name="WSMSE 2002">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.</ref>
According to this view of [[Edwin Thompson Jaynes|Jaynes]], it is a common and mystificatory abuse of language, that one often sees reproducibility of dynamical structure called "order".<ref name="Grandy 2008"/><ref name="Dewar 2005">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. {{ISBN|3-540-22495-5}}.</ref> Dewar<ref name="Dewar 2005"/> writes "Jaynes considered reproducibility - rather than disorder - to be the key idea behind the second law of thermodynamics (Jaynes 1963,<ref name="Jaynes 1963">[http://bayes.wustl.edu/etj/articles/brandeis.pdf Jaynes, E.T. (1963). pp. 181-218 in ''Brandeis Summer Institute 1962, Statistical Physics'', edited by K.W. Ford, Benjamin, New York.]</ref> 1965,<ref name="Jaynes 1965"/> 1988,<ref name="Jaynes 1988">[http://bayes.wustl.edu/etj/articles/ccarnot.pdf 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] {{ISBN|90-277-2793-7}}.</ref> 1989<ref name="Jaynes 1989">[http://bayes.wustl.edu/etj/articles/cmystery.pdf Jaynes, E.T. (1989). Clearing up mysteries, the original goal, pp. 1-27 in ''Maximum entropy and Bayesian methods'', Kluwer, Dordrecht.]</ref>)." Grandy (2008)<ref name="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 [[Edwin Thompson Jaynes|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)<ref name="G&P 1971"/> is guilty of this unfortunate abuse of language.
According to this view of [[Edwin Thompson Jaynes|Jaynes]], it is a common and mystificatory abuse of language, that one often sees reproducibility of dynamical structure called "order".<ref name="Grandy 2008"/><ref name="Dewar 2005">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. {{ISBN|3-540-22495-5}}.</ref> Dewar<ref name="Dewar 2005"/> writes "Jaynes considered reproducibility - rather than disorder - to be the key idea behind the second law of thermodynamics (Jaynes 1963,<ref name="Jaynes 1963">[http://bayes.wustl.edu/etj/articles/brandeis.pdf Jaynes, E.T. (1963). pp. 181-218 in ''Brandeis Summer Institute 1962, Statistical Physics'', edited by K.W. Ford, Benjamin, New York.]</ref> 1965,<ref name="Jaynes 1965"/> 1988,<ref name="Jaynes 1988">[http://bayes.wustl.edu/etj/articles/ccarnot.pdf 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] {{ISBN|90-277-2793-7}}.</ref> 1989<ref name="Jaynes 1989">[http://bayes.wustl.edu/etj/articles/cmystery.pdf Jaynes, E.T. (1989). Clearing up mysteries, the original goal, pp. 1-27 in ''Maximum entropy and Bayesian methods'', Kluwer, Dordrecht.]</ref>)." Grandy (2008)<ref name="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 [[Edwin Thompson Jaynes|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)<ref name="G&P 1971"/> is guilty of this unfortunate abuse of language.
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).
经典的非平衡态热力学理论大多与流体的空间连续运动有关，但是流体也可以随着空间不连续而运动。亥姆霍兹(1868)写道，在流动的流体中，如何产生零流体压力，使流体分裂。这源于流体流动的动量，表现出一种不同于热传导或电的动力结构。例如: 来自喷嘴的水可以形成水滴淋浴(瑞利1878，在第357节及以下。瑞利(1896/1926) ; 海面上的波浪到达海岸时不连续地破碎(Thom 1975)。赫姆霍兹指出，管风琴的声音一定是由于空气通过尖锐的障碍物而引起的流动的不连续性，否则声波的振荡特性将会被阻尼而消失。这种流动的产生熵的定义并没有被经典的非平衡态热力学理论所涵盖。还有许多其他常见的流体流动的不连续性也超出了经典的非平衡态热力学理论的范围，例如: 沸腾的液体和冒泡的饮料中的气泡; 还有保护热带深对流的塔(Riehl，Malkus 1958) ，也称为穿透性对流(Lindzen 1977)。
==Local thermodynamic equilibrium==
==Local thermodynamic equilibrium==
Various principles have been proposed by diverse authors for over a century. According to Glansdorff and Prigogine (1971, page 15),<ref name="G&P 1971">Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability and Fluctuations'', Wiley-Interscience, London. {{ISBN|0-471-30280-5}}</ref> in general, these principles apply only to systems that can be described by thermodynamical variables, in which [[Dissipative system|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),<ref name="G&P 1971">Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability and Fluctuations'', Wiley-Interscience, London. {{ISBN|0-471-30280-5}}</ref> in general, these principles apply only to systems that can be described by thermodynamical variables, in which [[Dissipative system|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==
William Thomson, later Baron Kelvin, (1852 a, 1852 b) wrote
威廉 · 汤姆森，后来的凯尔文男爵(1852年 a，1852年 b)写道
[[Dissipative system|Dissipative structures]] can depend on the presence of non-linearity in their dynamical régimes. [[Autocatalytic reactions and order creation|Autocatalytic reactions]] provide examples of non-linear dynamics, and may lead to the natural evolution of [[self-organization|self-organized]] dissipative structures.
[[Dissipative system|Dissipative structures]] can depend on the presence of non-linearity in their dynamical régimes. [[Autocatalytic reactions and order creation|Autocatalytic reactions]] provide examples of non-linear dynamics, and may lead to the natural evolution of [[self-organization|self-organized]] dissipative structures.
"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。
==Continuous and discontinuous motions of fluids==
==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)<ref name="Helmholtz 1868">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.</ref> 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,<ref name="Rayleigh 1878">{{cite journal | last1 = Strutt | first1 = J.W. | year = 1878 | title = On the instability of jets | url = https://zenodo.org/record/2095384| journal = Proceedings of the London Mathematical Society | volume = 10 | issue = | pages = 4–13 | doi=10.1112/plms/s1-10.1.4}}</ref> and in section 357 et seq. of Rayleigh (1896/1926)<ref name="Rayleigh 1896/1926">Strutt, J.W. (Baron Rayleigh) (1896/1926). Section 357 et seq. ''The Theory of Sound'', Macmillan, London, reprinted by Dover, New York, 1945.</ref>); waves on the surface of the sea break discontinuously when they reach the shore (Thom 1975<ref name="Thom 1975">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, {{ISBN|0-8053-9279-3}}</ref>). 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<ref name="Riehl Malkus 1958">{{cite journal | last1 = Riehl | first1 = H. | last2 = Malkus | first2 = J.S. | year = 1958 | title = On the heat balance in the equatorial trough zone | url = | journal = Geophysica | volume = 6 | issue = | pages = 503–538 }}</ref>), also called penetrative convection (Lindzen 1977<ref name="Lindzen 1977">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, {{ISBN|978-3-540-08532-4}}.</ref>).
III. When heat is diffused by conduction, there is a dissipation of mechanical energy, and perfect restoration is impossible.
三。当热通过传导扩散时，就会发生机械能的耗散，不可能完全恢复。
==Historical development==
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."
四。当辐射热或光被植物或化学反应以外的其他方式吸收时，就会发生机械能的耗散，不可能完全恢复。”
===[[William Thomson, 1st Baron Kelvin|W. Thomson, Baron Kelvin]]===
===[[William Thomson, 1st Baron Kelvin|W. Thomson, Baron Kelvin]]===
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，汤姆森写了两个之前已知的非平衡效应之间的关系。在佩尔蒂埃效应中，当温度梯度极限为零时，由外部电场驱动的电流通过双金属结，将导致热量通过结传递。在塞贝克效应中，当电流约束为零时，由温度梯度驱动的热流通过这样的结时，将导致电压穿过结。因此，热效应和电效应是耦合的。汤姆森(1854)提出了一个理论论点，部分基于卡诺和克劳修斯的工作，在那些日子里，部分只是推测，这两个效应的耦合常数将被实验发现是相等的。实验后来证实了这个建议。这是后来的想法之一，导致昂萨格的结果，如下所述。
William Thomson, later Baron Kelvin, (1852 a,<ref name="Kelvin 1852a">Thomson, William (1852 a). "[http://zapatopi.net/kelvin/papers/on_a_universal_tendency.html 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.]</ref> 1852 b<ref name="Kelvin 1852b">{{cite journal | last1 = Thomson | first1 = W | year = 1852 | title = b). On a universal tendency in nature to the dissipation of mechanical energy | url = | journal = Philosophical Magazine | volume = 4 | issue = | pages = 304–306 }}</ref>) wrote
William Thomson, later Baron Kelvin, (1852 a,<ref name="Kelvin 1852a">Thomson, William (1852 a). "[http://zapatopi.net/kelvin/papers/on_a_universal_tendency.html 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.]</ref> 1852 b<ref name="Kelvin 1852b">{{cite journal | last1 = Thomson | first1 = W | year = 1852 | title = b). On a universal tendency in nature to the dissipation of mechanical energy | url = | journal = Philosophical Magazine | volume = 4 | issue = | pages = 304–306 }}</ref>) 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.
"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.
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."
在1869年，赫尔曼·冯·亥姆霍兹提出了他的亥姆霍兹最小耗散定理，在一定的边界条件下，一个最小粘性耗散动能的原理: “对于一个在粘性液体中的稳定流动，流体边界上的速度保持稳定，在小速度的极限下，液体中的流动使得摩擦产生的动能耗散最小。”
III. When heat is diffused by ''conduction'', there is a ''dissipation'' of mechanical energy, and perfect ''restoration'' is impossible.
III. When heat is diffused by ''conduction'', there is a ''dissipation'' of mechanical energy, and perfect ''restoration'' is impossible.
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.
在1878年，亥姆霍兹和汤姆森一样，也引用了卡诺和克劳修斯的理论，写了关于在具有浓度梯度的电解质溶液中的电流。这表明了电效应和浓度驱动扩散之间的非平衡耦合。像上面提到的汤姆森(开尔文)一样，亥姆霍兹也发现了一种互惠关系，这是昂萨格指出的另一个想法。
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."
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."
Rayleigh (1873) (and in Sections 81 and 345 of Rayleigh (1896/1926) 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.
瑞利(1873)(和瑞利(1896/1926)的第81和345节)这些物理上清晰的瑞利考虑似乎包含了区分能量和产生熵最低和最高耗散率原则的核心，这些原则是后来的作者在物理研究过程中发展起来的。
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)<ref name="Kelvin 1854">Thomson, W. (1854). On a mechanical theory of thermo-electric currents, ''Proceedings of the Royal Society of Edinburgh'' pp. 91-98.</ref> 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 [[Lars Onsager|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)<ref name="Kelvin 1854">Thomson, W. (1854). On a mechanical theory of thermo-electric currents, ''Proceedings of the Royal Society of Edinburgh'' pp. 91-98.</ref> 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 [[Lars Onsager|Onsager]] to his results as noted below.
===[[Hermann von Helmholtz|Helmholtz]]===
===[[Hermann von Helmholtz|Helmholtz]]===
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“在给定边界速度的任何单连通区域中，只要可以忽略速度的平方和乘积，就只有不可压缩粘性流体定常运动方程的一个解，而且这个解始终是稳定的。”他把这个定理的第一部分归功于亥姆霍兹，他证明了这是一个定理的简单推论，即“如果运动是稳定的，粘性[不可压缩]流体中的流动是如此分布，以至于在假定沿流体边界的速度是给定的情况下，由粘性引起的[动能]损失是最小的。”由于限制在可以忽略速度平方和乘积的情况下，这些运动低于湍流的阈值。
In 1869, [[Hermann von Helmholtz]] stated his [[Helmholtz minimum dissipation theorem]],<ref>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 [http://echo.mpiwg-berlin.mpg.de/ECHOdocuViewfull?url=/mpiwg/online/permanent/einstein_exhibition/sources/QWH2FNX8/index.meta&start=231&viewMode=images&pn=237&mode=texttool]</ref> 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."<ref>from page 2 of Helmholtz 1869/1871, translated by Wikipedia editor.</ref>
In 1869, [[Hermann von Helmholtz]] stated his [[Helmholtz minimum dissipation theorem]],<ref>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 [http://echo.mpiwg-berlin.mpg.de/ECHOdocuViewfull?url=/mpiwg/online/permanent/einstein_exhibition/sources/QWH2FNX8/index.meta&start=231&viewMode=images&pn=237&mode=texttool]</ref> 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."<ref>from page 2 of Helmholtz 1869/1871, translated by Wikipedia editor.</ref>
In 1878, Helmholtz,<ref name="Helmholtz 1978">{{cite journal | last1 = Helmholtz | first1 = H | year = 1878 | title = Ueber galvanische Ströme, verursacht durch Concentrationsunterschiede; Folgeren aus der mechanischen Wärmetheorie, Wiedermann's | url = https://zenodo.org/record/2343658| journal = Annalen der Physik und Chemie | volume = 3 | issue = 2| pages = 201–216 | doi = 10.1002/andp.18782390204 }}</ref> 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,<ref name="Helmholtz 1978">{{cite journal | last1 = Helmholtz | first1 = H | year = 1878 | title = Ueber galvanische Ströme, verursacht durch Concentrationsunterschiede; Folgeren aus der mechanischen Wärmetheorie, Wiedermann's | url = https://zenodo.org/record/2343658| journal = Annalen der Physik und Chemie | volume = 3 | issue = 2| pages = 201–216 | doi = 10.1002/andp.18782390204 }}</ref> 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.
Great theoretical progress was made by Onsager in 1931 and in 1953.
昂萨格1931年和1953年在理论上取得了重大进展。
Rayleigh (1873)<ref name="Rayleigh 1873">{{cite journal | last1 = Strutt | first1 = J.W. | year = 1873 | title = Some theorems relating to vibrations | url = https://zenodo.org/record/1447754| journal = Proceedings of the London Mathematical Society | volume = 4 | issue = | pages = 357–368 | doi=10.1112/plms/s1-4.1.357}}</ref> (and in Sections 81 and 345 of Rayleigh (1896/1926)<ref name="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.
Rayleigh (1873)<ref name="Rayleigh 1873">{{cite journal | last1 = Strutt | first1 = J.W. | year = 1873 | title = Some theorems relating to vibrations | url = https://zenodo.org/record/1447754| journal = Proceedings of the London Mathematical Society | volume = 4 | issue = | pages = 357–368 | doi=10.1112/plms/s1-4.1.357}}</ref> (and in Sections 81 and 345 of Rayleigh (1896/1926)<ref name="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.
Further progress was made by Prigogine in 1945 and later. Prigogine (1947) extended the theory of Onsager.
普里戈金在1945年及其后取得了进一步的进展。普里戈金(1947)扩展了昂萨格的理论。
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(1956)给出了非常可读的帐户。他提出以下作为不可逆过程热力学的一般原理: “考虑所有的电流分布，使得给定的一组力的内产生熵等于外产生熵。然后，在所有满足这个条件的电流分布中，稳态分布使产生熵最大。”他评论说，这是昂萨格发现的一个众所周知的一般原则，但是”关于这个问题的任何书籍都没有引用”。他指出了这个原理和“普里戈金定理”的区别。普里戈金定理粗略地指出，如果作用在一个系统上的所有作用力都是固定的，那么自由作用力就会取这样的值，从而使产生熵极小化普里戈金在读这篇论文时在场，据期刊编辑报道，他曾发出“通知，他怀疑齐曼热力学解释的一部分的有效性”。
===[[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.
Hans Ziegler extended the Melan-Prager non-equilibrium theory of materials to the non-isothermal case.
Hans Ziegler 将材料的 Melan-Prager 非平衡理论推广到非等温情况。
===[[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>
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) "On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975, 2001). Initially Paltridge (1975) 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) 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.
Gyarmati (1967/1970)对能量最小耗散原理进行了系统的介绍，并对昂萨格的最小耗散原理进行了扩展，给出了一种更为对称的形式—— Gyarmati 原理。Gyarmati (1967/1970)“在宏观层面上，这种方法是由一位气象学家开创的(Paltridge 1975,2001)。最初，帕特里奇(1975年)和 Paltridge (1979年) ，他用现在流行的术语“最高产生熵”来描述同样的事情。这一点在小泽、大村、洛伦兹、普约尔(2003)的评论中得到了澄清。关于极值原理的 Paltridge (流体机械功。尼古利斯和 Nicolis (1980)讨论了帕特里奇的工作，他们评论说产生熵的行为远非简单和普遍。在某些经典的非平衡态热力学理论要求湍流门槛不能跨越的背景下，这似乎是很自然的。如今，帕特里奇本人倾向于从耗散功能的角度思考，而不是从产生熵的角度思考。
===[[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"/>
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) 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 "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)) 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). 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.
Jou，Casas-Vazquez，Lebon (1993)指出，古典非平衡态热力学“自二战以来有了非凡的发展” ，他们指的是诺贝尔奖授予 Lars Onsager 和 Ilya Prigogine 的领域工作。Martyushev 和 Seleznev (2006)说: “ ... 不平衡可能是秩序的源泉。不可逆过程可能导致一种新的物质动力学状态，我称之为“耗散结构”格兰斯多夫和普里戈金(1971)写道: “不稳定发生在最低温度梯度，在这个时候粘滞耗散的动能和浮力释放的内能之间可以保持平衡。”当粘度温度梯度大于最小值时，粘度可以像由于浮力而通过对流释放的那样快速地耗散动能，并且具有对流的稳定状态是稳定的。对流的稳定状态通常是宏观上可见的六边形单元格式，对流在每个单元格的中间或壁上升或下降，这取决于数量的温度依赖性; 在大气中，在各种条件下似乎都是可能的。(Lebon，Jou，和 Casas-Vásquez (2008)讨论了一些细节)提出了一个“最小产生熵定理” ，它只适用于纯扩散线性区域，具有可忽略的惯性项，接近稳定的热动力学非平衡态。普里戈金的提议是，产生熵的速率在每一点都是局部极小值。普里戈金提供的证据受到了严肃的批评。对于普里戈金的建议，戈兰迪(Grandy，2008)提供了一个批判性的和不支持的讨论。与 Prigogine 的建议密切相关的一个建议是，点态产生熵的最大值应该在稳态时最小化。这是兼容的，但不完全相同，与普里戈金的建议。此外，n. w. Tschoegl 提出了一个证据，可能比 Prigogine 的更有身体动机，如果有效支持 Helmholtz 和 Prigogine 的结论，在这些限制条件下，产生熵是在一个点态最小值。
===[[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.
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.
与惯性项可忽略不计的通量和广义力之间具有线性关系的足够缓慢的传热情况相反，可能存在传热不是很缓慢的情况。由此产生的非线性效应使热流发展为对流环流阶段。在这些情况下，在接近稳态热对流的过程中，产生熵的时间速率被证明是时间的非单调函数。这使得这些情况不同于线性极慢转移的近热力学平衡区域。因此，产生熵的局部时间速率，根据局部热力学平衡假设定义，不是一个适当的变量来预测远离热力学平衡过程的时间进程。最低产生熵原则不适用于这些情况。
===[[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".
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.
为了涵盖这些情况，还需要至少一个更进一步的状态变量，一个非平衡量，即所谓的第二熵。这似乎是超越经典热力学第二定律的一个推广步骤，涵盖了非平衡态或过程。古典法则只涉及热力学平衡的状态，而当地的热力学平衡理论是依赖于它的近似值。尽管如此，它还是被用来处理接近但不是在热力学平衡的现象，并且在那时还有一些用途。但是，古典法律不足以描述远离热力学平衡的过程的时间进程。对于这样的过程，需要一个更强有力的理论，而第二个熵就是这样一个理论的一部分。
===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>
Onsager (1931, I) and later in his book on thermomechanics revised in 1983,). 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)." Initially Paltridge (1975) 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. 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.
昂萨格(1931年，我)和后来在他的书热力学修订在1983年,)。齐格勒从来没有把他的原理说成是一个普遍规律，但他可能凭直觉知道这一点。他用基于“正交性条件”的向量空间几何论证了他的原理，该条件只适用于速度被定义为单个向量或张量的系统，因此，正如他所写的那样)最初 Paltridge (1975)用现在流行的术语“最大产生熵”来描述同样的事情。帕特里奇早期作品的逻辑受到严肃的批评。同样与地球的大气能量输送过程有关，假定了单位时间内熵增量最大的原则，引用了 Malkus 和 Veronis (1958)在流体力学中的工作“证明了最大热流原则，而最大热流原则又是给定边界条件下的最大产生熵” ，但这个推论在逻辑上是站不住脚的。再次调查行星大气动力学，Shutts (1981)使用了一种不同于 Paltridge 的方法来定义产生熵，以调查一种更抽象的方法来检查最大熵原理的产量，并报告了一个很好的契合。
===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.
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.
直到最近，这个领域中有用的极端原理的前景似乎还很模糊。她说，这似乎排除了存在全球组织原则的可能性，并评论说，这在某种程度上令人失望; 她还指出，很难找到一个热力学上一致的产生熵。另一位顶级专家对产生熵极值原理和能量耗散原理的可能性进行了广泛的讨论: Grandy 的第12章(2008年)和化学反应不遵守产生熵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。
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.
===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.
[http://onlinelibrary.wiley.com/doi/10.1002/qj.49710142906/full]
</ref> 2001<ref>{{cite journal | last1 = Paltridge | first1 = G.W. | year = 2001 | title = A physical basis for a maximum of thermodynamic dissipation of the climate system | url = http://www3.interscience.wiley.com/journal/114028007/abstract | archive-url = https://archive.today/20121018022835/http://www3.interscience.wiley.com/journal/114028007/abstract | url-status = dead | archive-date = 2012-10-18 | journal = Quarterly Journal of the Royal Meteorological Society | volume = 127 | issue = 572| pages = 305–313 | doi=10.1256/smsqj.57202}}</ref>). Initially Paltridge (1975)<ref name="Paltridge 1975"/> used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978),<ref name="Paltridge 1978">{{cite journal | last1 = Paltridge | first1 = G.W. | year = 1978 | title = The steady-state format of global climate | url = | journal = Quarterly Journal of the Royal Meteorological Society | volume = 104 | issue = 442| pages = 927–945 | doi=10.1256/smsqj.44205}}</ref> and in Paltridge (1979)<ref>{{cite journal | last1 = Paltridge | first1 = G.W. | year = 1979 | title = Climate and thermodynamic systems of maximum dissipation | journal = Nature | volume = 279 | issue = 5714| pages = 630–631 | bibcode=1979Natur.279..630P | doi=10.1038/279630a0| s2cid = 4262395 }}</ref>), 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).<ref name="OOLP 2003">{{cite journal | last1 = Ozawa | first1 = H. | last2 = Ohmura | first2 = A. | last3 = Lorenz | first3 = R.D. | last4 = Pujol | first4 = T. | year = 2003 | title = The Second Law of Thermodynamics and the Global Climate System: A Review of the Maximum Entropy Production Principle | url = http://homepage.mac.com/bradmarston/Papers/Ozawa%20etal%20(2003).pdf | journal = Reviews of Geophysics | volume = 41 | issue = 4| pages = 1–24 | doi=10.1029/2002rg000113 | bibcode=2003RvGeo..41.1018O| hdl = 10256/8489 }}</ref> Paltridge (1978)<ref name="Paltridge 1978"/> cited Busse's (1967)<ref>{{cite journal | last1 = Busse | first1 = F.H. | year = 1967 | title = The stability of finite amplitude cellular convection and its relation to an extremum principle | url = | journal = Journal of Fluid Mechanics | volume = 30 | issue = 4| pages = 625–649 | doi=10.1017/s0022112067001661| bibcode = 1967JFM....30..625B}}</ref> fluid mechanical work concerning an extremum principle. Nicolis and Nicolis (1980) <ref name="Nicolis 1980">{{cite journal | last1 = Nicolis | first1 = G. | last2 = Nicolis | first2 = C. | year = 1980 | title = On the entropy balance of the earth-atmosphere system | url = | journal = Quarterly Journal of the Royal Meteorological Society | volume = 106 | issue = 450| pages = 691–706| doi = 10.1002/qj.49710645003 | bibcode = 1980QJRMS.106..691N }}</ref> 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)<ref name="JCVL 1993">Jou, D., Casas-Vázquez, J., Lebon, G. (1993). ''Extended Irreversible Thermodynamics'', Springer, Berlin, {{ISBN|3-540-55874-8}}, {{ISBN|0-387-55874-8}}.</ref> 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)<ref name="M&S 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 [[Rudolf Clausius|Clausius]], ... , and [[Ilya Prigogine|Prigogine]]." Prigogine in his 1977 Nobel Lecture<ref>[http://nobelprize.org/nobel_prizes/chemistry/laureates/1977/prigogine-lecture.pdf Prigogine, I. (1977). Time, Structure and Fluctuations, Nobel Lecture.]</ref> 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)<ref name="G&P 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|Rayleigh–Bénard convection cell phenomenon]], Chandrasekhar (1961)<ref>Chandrasekhar, S. (1961). ''Hydrodynamic and Hydromagnetic Stability'', Clarendon Press, Oxford.</ref> 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)<ref name="Lebon Jou Casas-Vázquez 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)<ref name="G&P 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)<ref name="Rayleigh 1873"/> that was used also by Onsager (1931, I,<ref name="Onsager 1931 I"/> 1931, II<ref name="Onsager 1931 II"/>). On pages 78–80 of their book<ref name="G&P 1971"/> 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.
Category:Non-equilibrium thermodynamics
Category:Non-equilibrium thermodynamics