'''能量耗散和熵增极值原理(Energy dissipation and entropy production extremal principles )'''是在非平衡热力学中发展起来的思想,试图预测一个物理系统可能呈现的稳定状态和动态结构。
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'''Energy dissipation and entropy production extremal principles''' are ideas developed within [[non-equilibrium thermodynamics]] that attempt to predict the likely steady states and dynamical structures that a physical system might show. The search for extremum principles for non-equilibrium thermodynamics follows their successful use in other branches of physics.<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.
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对非平衡热力学极值原则的探索是在其成功应用于物理学的其他分支之后进行的。<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">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><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>
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Energy dissipation and entropy production extremal principles are ideas developed within non-equilibrium thermodynamics that attempt to predict the likely steady states and dynamical structures that a physical system might show. The search for extremum principles for non-equilibrium thermodynamics follows their successful use in other branches of physics. 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.
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根据Kondepudi(2008)<ref>Kondepudi, D. (2008)., ''Introduction to Modern Thermodynamics'', Wiley, Chichester UK, {{ISBN|978-0-470-01598-8}}, page 172.</ref>和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>的说法,没有任何普适规则可以提供一个极值原则来管理一个远离平衡的系统向稳定状态的演变。
根据Glansdorff和Prigogine(1971年,第16页)<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>的理论,不可逆过程通常不受全局极值原则的支配,因为描述其演化需要微分方程,而微分方程是不可自交的,但局部极值原则可用于局部解。
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>认为:"......热力学的极值原则......对非平衡稳态没有任何对应原理(尽管文献中有许多说法)。"
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==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"/>
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Apparent 'fluctuations', which appear to arise when initial conditions are inexactly specified, are the drivers of the formation of non-equilibrium dynamical structures. There is no special force of nature involved in the generation of such fluctuations. Exact specification of initial conditions would require statements of the positions and velocities of all particles in the system, obviously not a remotely practical possibility for a macroscopic system. This is the nature of thermodynamic fluctuations. They cannot be predicted in particular by the scientist, but they are determined by the laws of nature and they are the singular causes of the natural development of dynamical structure.
It is pointed out<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 pointed out by W.T. Grandy Jr that entropy, though it may be defined for a non-equilibrium system, is when strictly considered, only a macroscopic quantity that refers to the whole system, and is not a dynamical variable and in general does not act as a local potential that describes local physical forces. Under special circumstances, however, one can metaphorically think as if the thermal variables behaved like local physical forces. The approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking.
W.T. Grandy Jr指出<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>,尽管熵可以被定义为非平衡系统,但当严格考虑时,它只是一个宏观的量,指的是整个系统,而不是一个动态变量,一般来说,它不会像描述局部物理力的局部势那样发挥作用。然而,在特殊情况下,人们可以比喻为热变量的行为就像局部物理力。构成经典不可逆热力学的近似值就是建立在这种隐喻性思维之上的。
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
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
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.
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>
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It is reproducibility in repeated observations that identifies dynamical structure in a system. E.T. Jaynes explains how this reproducibility is why entropy is so important in this topic: entropy is a measure of experimental reproducibility. The entropy tells how many times one would have to repeat the experiment in order to expect to see a departure from the usual reproducible result. When the process goes on in a system with less than a 'practically infinite' number (much much less than Avogadro's or Loschmidt's numbers) of molecules, the thermodynamic reproducibility fades, and fluctuations become easier to see.
According to this view of [[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.
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According to this view of Jaynes, it is a common and mystificatory abuse of language, that one often sees reproducibility of dynamical structure called "order". Dewar "Jaynes considered reproducibility - rather than disorder - to be the key idea behind the second law of thermodynamics (Jaynes 1963,1965,1988,1989)." Grandy (2008) in section 4.3 on page 55 clarifies the distinction between the idea that entropy is related to order (which he considers to be an "unfortunate" "mischaracterization" that needs "debunking"), and the aforementioned idea of Jaynes that entropy is a measure of experimental reproducibility of process (which Grandy regards as correct). According to this view, even the admirable book of Glansdorff and Prigogine (1971)is guilty of this unfortunate abuse of language.
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正是在重复观察中的可重复性,确定了系统中的动态结构。E.T.杰恩斯<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>解释了这种可重复性是为什么熵在这个主题中如此重要:熵是实验可重复性的衡量标准。熵告诉人们必须重复实验多少次,才能期望看到偏离通常的可重复性结果。当这个过程在一个分子数量少于 "实际无限"(比阿伏伽德罗数或洛施密特数少得多)的系统中进行时,热力学可重复性就会消失,波动就会变得更容易看到。<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>
根据杰恩斯的这一观点,人们经常看到动态结构的再现性被称为 "秩序"<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>,这是一种常见的和神秘的语言滥用。杜瓦<ref name="Dewar 2005" />写道:"杰恩斯认为重现性--而不是无序--是热力学第二定律背后的关键思想,杰恩斯(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" />在第55页的4.3节中澄清了熵与秩序有关的观点(他认为这是一个 "不幸的""错误描述",需要 "揭穿")与上述Jaynes的观点之间的区别,即熵是对过程的实验可重复性的衡量(Grandy认为这是正确的)。根据这种观点,即使是令人钦佩的Glansdorff和Prigogine(1971)<ref name="G&P 1971" />的书也犯了这种不幸的滥用语言的错误。
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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.
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==局部热力学平衡==
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Various principles have been proposed by diverse authors for over a century. According to Glansdorff and Prigogine (1971, page 15),in general, these principles apply only to systems that can be described by thermodynamical variables, in which dissipative processes dominate by excluding large deviations from statistical equilibrium. The thermodynamical variables are defined subject to the kinematical requirement of local thermodynamic equilibrium. This means that collisions between molecules are so frequent that chemical and radiative processes do not disrupt the local Maxwell-Boltzmann distribution of molecular velocities.
[[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 structures can depend on the presence of non-linearity in their dynamical régimes. Autocatalytic reactions provide examples of non-linear dynamics, and may lead to the natural evolution of self-organized dissipative structures.
==Continuous and discontinuous motions of fluids==
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经典非平衡热力学的大部分理论都是关于流体的空间连续运动的,但流体的运动也可以有空间不连续的情况。赫尔姆霍兹(1868年)<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>写道,在流动的流体中,可能会出现零流体压力,这时流体会断裂。这产生于流体流动的动量,显示出与热或电的传导不同的动态结构。因此,例如:从喷嘴喷出的水可以形成水滴雨(Rayleigh 1878年,<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>以及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>的第357节及以下内容);海面上的波浪在到达海岸时不连续地破裂(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>。赫尔姆霍兹指出,风琴管的声音必须来自于空气经过尖角障碍物时产生的这种流动的不连续性;否则,声波的振荡特性就会被阻尼得无影无踪了。
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流体的连续和不连续运动
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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>).
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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)
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经典非平衡热力学的通常理论并没有涵盖这种流动的熵产生率的定义。还有许多其他通常观察到的流体流动的不连续性,也超出了经典非平衡热力学理论的范围,例如:沸腾液体和泡腾饮料中的气泡;还有热带深层对流的保护塔(Riehl,Malkus 1958),也叫穿透性对流(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>。
William Thomson, later Baron Kelvin, (1852 a, 1852 b) wrote
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威廉-汤姆森,后来的开尔文男爵,(1852 a,1852 b)写道
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威廉 · 汤姆森,后来的凯尔文男爵(1852年 a,1852年 b)写道
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"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.
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。
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。
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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.当辐射热或光被吸收后,除植被外,或在化学反应中,就会发生机械能的耗散,不可能完全恢复。”
四.当辐射热或光被吸收后,除植被外,或在化学反应中,就会发生机械能的耗散,不可能完全恢复。”
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In 1854, Thomson wrote about the relation between two previously known non-equilibrium effects. In the Peltier effect, an electric current driven by an external electric field across a bimetallic junction will cause heat to be carried across the junction when the temperature gradient is constrained to zero. In the Seebeck effect, a flow of heat driven by a temperature gradient across such a junction will cause an electromotive force across the junction when the electric current is constrained to zero. Thus thermal and electric effects are said to be coupled. Thomson (1854)<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.
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In 1854, Thomson wrote about the relation between two previously known non-equilibrium effects. In the Peltier effect, an electric current driven by an external electric field across a bimetallic junction will cause heat to be carried across the junction when the temperature gradient is constrained to zero. In the Seebeck effect, a flow of heat driven by a temperature gradient across such a junction will cause an electromotive force across the junction when the electric current is constrained to zero. Thus thermal and electric effects are said to be coupled. Thomson (1854)proposed a theoretical argument, partly based on the work of Carnot and Clausius, and in those days partly simply speculative, that the coupling constants of these two effects would be found experimentally to be equal. Experiment later confirmed this proposal. It was later one of the ideas that led Onsager to his results as noted below.
1869年,赫尔曼-冯-亥姆霍兹陈述了他的亥姆霍兹最小耗散定理<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>,受制于某种边界条件,这是一个动能最小粘性耗散的原则:"对于粘性液体中的稳定流动,在流体边界上的流速给定稳定的情况下,在速度较小的情况下,液体中的电流如此分布,使摩擦力造成的动能耗散最小"。<ref>from page 2 of Helmholtz 1869/1871, translated by Wikipedia editor.</ref>
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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>
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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."
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1869年,赫尔曼•冯•赫尔姆霍兹Hermann von Helmholtz 在一定的边界条件下,陈述了他的Helmholtz最小耗散定理,即动能最小粘性耗散原理:"对于粘性液体中的稳定流动,在给定液体边界上的流速稳定的情况下,在速度较小的情况下,液体中的电流如此分布,摩擦力对动能的耗散是最小的。"
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1878年,亥姆霍兹<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>和汤姆森一样,也引用了卡诺和克劳修斯的话,写了关于浓度梯度的电解质溶液中的电流。这表明在电效应和浓度驱动的扩散之间存在着非平衡的耦合。像上面提到的汤姆森(开尔文)一样,亥姆霍兹也发现了一个相互关系,这也是昂萨格注意到的另一个观点。
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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.
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In 1878, Helmholtz, like Thomson also citing Carnot and Clausius, wrote about electric current in an electrolyte solution with a concentration gradient. This shows a non-equilibrium coupling, between electric effects and concentration-driven diffusion. Like Thomson (Kelvin) as noted above, Helmholtz also found a reciprocal relation, and this was another of the ideas noted by Onsager.
===[[John William Strutt, 3rd Baron Rayleigh|J. W. Strutt, Baron Rayleigh]]===
===[[John William Strutt, 3rd Baron Rayleigh|J. W. Strutt, Baron Rayleigh]]===
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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.
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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>(以及在Rayleigh(1896/1926)<ref name="Rayleigh 1896/1926" />的第81节和第345节)引入了耗散函数,用于描述涉及粘性的耗散过程。后来许多研究耗散过程和动力学结构性质的人都使用了这个函数的更一般版本。 Rayleigh的耗散函数是从机械的角度来设想的,它在定义中没有提到温度,它需要被 "泛化 "以使耗散函数适合用于非平衡热力学。雷利(1878,1896/1926)在研究来自喷嘴的水射流时指出,当射流处于条件稳定的动力结构状态时,最有可能增长到其全部程度并导致另一个条件稳定的动力结构状态的波动模式是具有最快的增长速度。换句话说,喷气可以进入一个条件稳定的状态,但它很可能遭受波动,从而进入另一个不太稳定的条件稳定状态。 他在对贝纳德对流的研究中使用了类似的推理。雷利的这些物理上的清晰考虑似乎包含了能量耗散和熵产生的最小和最大速率原则之间的区别的核心,这在后来的作者的物理调查过程中得到了发展。
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Rayleigh (1873)(and in Sections 81 and 345 of Rayleigh (1896/1926)) introduced the dissipation function for the description of dissipative processes involving viscosity. More general versions of this function have been used by many subsequent investigators of the nature of dissipative processes and dynamical structures. Rayleigh's dissipation function was conceived of from a mechanical viewpoint, and it did not refer in its definition to temperature, and it needed to be 'generalized' to make a dissipation function suitable for use in non-equilibrium thermodynamics.
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Studying jets of water from a nozzle, Rayleigh (1878,1896/1926) noted that when a jet is in a state of conditionally stable dynamical structure, the mode of fluctuation most likely to grow to its full extent and lead to another state of conditionally stable dynamical structure is the one with the fastest growth rate. In other words, a jet can settle into a conditionally stable state, but it is likely to suffer fluctuation so as to pass to another, less unstable, conditionally stable state. He used like reasoning in a study of Bénard convection.These physically lucid considerations of Rayleigh seem to contain the heart of the distinction between the principles of minimum and maximum rates of dissipation of energy and entropy production, which have been developed in the course of physical investigations by later authors.
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>给出了一个证明:"在任何简单连接的区域,当沿边界的速度给定时,只要速度的平方和乘积可以忽略,就只存在一个不可压缩的粘性流体的稳定运动方程的解,而且这个解总是稳定的。" 他把这一定理的第一部分归功于亥姆霍兹,他表明这是一个定理的简单结果,即 "如果运动是稳定的,在一个粘性(不可压缩)流体中的电流是如此分布,以至于由于粘性造成的[动]能损失是最小的,前提是流体沿边界的速度是给定的。" 由于被限制在速度的平方和乘积可以被忽略的情况下,这些运动低于湍流的阈值。
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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.
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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.
Onsager 1931年<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>和1953年<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>在理论上取得了重大进展。
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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>
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Great theoretical progress was made by Onsager in 1931 and in 1953.
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Onsager 1931年和1953年在理论上取得了重大进展。
===[[Ilya Prigogine|Prigogine]]===
===[[Ilya Prigogine|Prigogine]]===
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普利戈金
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Prigogine在1945年<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>及其后<ref name="G&P 1971" /><ref name="Prigogine 1947">Prigogine, I. (1947). ''Étude thermodynamique des Phenomènes Irréversibles'', Desoer, Liège.</ref>取得了进一步的进展。Prigogine(1947)<ref name="Prigogine 1945" />扩展了Onsager的理论。<ref name="Onsager 1931 I" /><ref name="Onsager 1931 II" />
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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"/>
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Further progress was made by Prigogine in 1945 and later. Prigogine (1947) extended the theory of Onsager.
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".
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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".
Hans Ziegler 将材料的 Melan-Prager 非平衡理论扩展到了非等温情况<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>。
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[[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>
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Hans Ziegler extended the Melan-Prager non-equilibrium theory of materials to the non-isothermal case.
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.
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Gyarmati (1967/1970)<ref name="Gyarmati 1970"/> also gives in Section III 5 a very helpful precis of the subtleties of Casimir (1945)).<ref name="Casimir 1945"/> He explains that the Onsager reciprocal relations concern variables which are even functions of the velocities of the molecules, and notes that Casimir went on to derive anti-symmetric relations concerning variables which are odd functions of the velocities of the molecules.
Gyarmati (1967/1970) gives a systematic presentation, and extends Onsager's principle of least dissipation of energy, to give a more symmetric form known as Gyarmati's principle. Gyarmati (1967/1970) cites 11 papers or books authored or co-authored by Prigogine. Gyarmati (1967/1970)also gives in Section III 5 a very helpful precis of the subtleties of Casimir (1945)).He explains that the Onsager reciprocal relations concern variables which are even functions of the velocities of the molecules, and notes that Casimir went on to derive anti-symmetric relations concerning variables which are odd functions of the velocities of the molecules.
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.''
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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.
</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 name=":0">{{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 name=":1">{{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.
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</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.
The physics of the earth's atmosphere includes dramatic events like lightning and the effects of volcanic eruptions, with discontinuities of motion such as noted by Helmholtz (1868).Turbulence is prominent in atmospheric convection. Other discontinuities include the formation of raindrops, hailstones, and snowflakes. The usual theory of classical non-equilibrium thermodynamics will need some extension to cover atmospheric physics. According to Tuck (2008),"On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975,2001). Initially Paltridge (1975)used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978),and in Paltridge (1979)), he used the now current terminology "maximum entropy production" to describe the same thing. This point is clarified in the review by Ozawa, Ohmura, Lorenz, Pujol (2003).Paltridge (1978)cited Busse's (1967)fluid mechanical work concerning an extremum principle. Nicolis and Nicolis (1980) discuss Paltridge's work, and they comment that the behaviour of the entropy production is far from simple and universal. This seems natural in the context of the requirement of some classical theory of non-equilibrium thermodynamics that the threshold of turbulence not be crossed. Paltridge himself nowadays tends to prefer to think in terms of the dissipation function rather than in terms of rate of entropy production.
==Speculated thermodynamic extremum principles for energy dissipation and entropy production==
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==用于能量耗散和熵产生的推测热力学极值原理==
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用于能量耗散和熵产生的推测热力学极值原理
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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''."
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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 name=":2">[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''."
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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.
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Analyzing the [[Rayleigh–Bénard convection|Rayleigh–Bénard convection cell phenomenon]], Chandrasekhar (1961)<ref name=":3">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.
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.
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Jou, Casas-Vazquez, Lebon (1993)note that classical non-equilibrium thermodynamics "has seen an extraordinary expansion since the second world war", and they refer to the Nobel prizes for work in the field awarded to Lars Onsager and Ilya Prigogine. Martyushev and Seleznev (2006)note the importance of entropy in the evolution of natural dynamical structures: "Great contribution has been done in this respect by two scientists, namely Clausius, ... , and Prigogine." Prigogine in his 1977 Nobel Lecture said: "... non-equilibrium may be a source of order. Irreversible processes may lead to a new type of dynamic states of matter which I have called “dissipative structures”." Glansdorff and Prigogine (1971) wrote on page xx: "Such 'symmetry breaking instabilities' are of special interest as they lead to a spontaneous 'self-organization' of the system both from the point of view of its space order and its function."
Analyzing the Rayleigh–Bénard convection cell phenomenon, Chandrasekhar (1961) wrote "Instability occurs at the minimum temperature gradient at which a balance can be maintained between the kinetic energy dissipated by viscosity and the internal energy released by the buoyancy force." With a temperature gradient greater than the minimum, viscosity can dissipate kinetic energy as fast as it is released by convection due to buoyancy, and a steady state with convection is stable. The steady state with convection is often a pattern of macroscopically visible hexagonal cells with convection up or down in the middle or at the 'walls' of each cell, depending on the temperature dependence of the quantities; in the atmosphere under various conditions it seems that either is possible. (Some details are discussed by Lebon, Jou, and Casas-Vásquez (2008) on pages 143–158.) With a temperature gradient less than the minimum, viscosity and heat conduction are so effective that convection cannot keep going.
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Glansdorff and Prigogine (1971) on page xv wrote "Dissipative rstructures have a quite different [from equilibrium structures] status: they are formed and maintained through the effect of exchange of energy and matter in non-equilibrium conditions." They were referring to the dissipation function of Rayleigh (1873) that was used also by Onsager (1931, I,1931, II). On pages 78–80 of their book Glansdorff and Prigogine (1971) consider the stability of laminar flow that was pioneered by Helmholtz; they concluded that at a stable steady state of sufficiently slow laminar flow, the dissipation function was minimum.
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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)."
In 1945 Prigogine (see also Prigogine (1947)) proposed a “Theorem of Minimum Entropy Production” which applies only to the purely diffusive linear regime, with negligible inertial terms, near a stationary thermodynamically non-equilibrium state. Prigogine's proposal is that the rate of entropy production is locally minimum at every point. The proof offered by Prigogine is open to serious criticism. A critical and unsupportive discussion of Prigogine's proposal is offered by Grandy (2008). It has been shown by Barbera that the total whole body entropy production cannot be minimum, but this paper did not consider the pointwise minimum proposal of Prigogine. A proposal closely related to Prigogine's is that the pointwise rate of entropy production should have its maximum value minimized at the steady state. This is compatible, but not identical, with the Prigogine proposal. Moreover, N. W. Tschoegl proposes a proof, perhaps more physically motivated than Prigogine's, that would if valid support the conclusion of Helmholtz and of Prigogine, that under these restricted conditions, the entropy production is at a pointwise minimum.
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.
1945年普里戈金(另见普里戈金(1947))提出了一个 "最小熵产生定理",该定理仅适用于在静止的热力学非平衡状态附近,具有可忽略的惯性条款的纯扩散线性制度。普里戈金的建议是,熵增率在每一个点上都是局部最小。普里戈金提供的证明是可以受到严肃批评的。Grandy(2008)对Prigogine的提议进行了批评和不支持的讨论。Barbera已经证明,整个身体的总熵的产生不可能是最小的,但这篇论文没有考虑Prigogine的点状最小提案。与Prigogine的提议密切相关的是,熵增的点状速率应该在稳定状态下具有最小的最大值。这与普里戈金的建议相一致,但不完全相同。此外,N. W. Tschoegl提出了一个证明,也许比普里戈金的证明更有物理动机,如果有效的话,它将支持亥姆霍兹和普里戈金的结论,即在这些限制条件下,熵增处于点状最小。
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== 更快的对流环流转移:第二熵 ==
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To cover these cases, there is needed at least one further state variable, a non-equilibrium quantity, the so-called second entropy. This appears to be a step towards generalization beyond the classical second law of thermodynamics, to cover non-equilibrium states or processes. The classical law refers only to states of thermodynamic equilibrium, and local thermodynamic equilibrium theory is an approximation that relies upon it. Still it is invoked to deal with phenomena near but not at thermodynamic equilibrium, and has some uses then. But the classical law is inadequate for description of the time course of processes far from thermodynamic equilibrium. For such processes, a more powerful theory is needed, and the second entropy is part of such a theory.
== Speculated principles of maximum entropy production and minimum energy dissipatio==
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最大熵的产生和最小能量耗散的推测原理。
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Onsager (1931, I)wrote: "Thus the vector field J of the heat flow is described by the condition that the rate of increase of entropy, less the dissipation function, be a maximum." Careful note needs to be taken of the opposite signs of the rate of entropy production and of the dissipation function, appearing in the left-hand side of Onsager's equation (5.13) on Onsager's page 423.
Although largely unnoticed at the time, Ziegler proposed an idea early with his work in the mechanics of plastics in 1961,and later in his book on thermomechanics revised in 1983,and in various papers (e.g., Ziegler (1987),). Ziegler never stated his principle as a universal law but he may have intuited this. He demonstrated his principle using vector space geometry based on an “orthogonality condition” which only worked in systems where the velocities were defined as a single vector or tensor, and thus, as he wrote at p. 347, was “impossible to test by means of macroscopic mechanical models”, and was, as he pointed out, invalid in “compound systems where several elementary processes take place simultaneously”.
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In relation to the earth's atmospheric energy transport process, according to Tuck (2008), "On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975,2001)." Initially Paltridge (1975)used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978),and in Paltridge (1979),he used the now current terminology "maximum entropy production" to describe the same thing. The logic of Paltridge's earlier work is open to serious criticism.Nicolis and Nicolis (1980) discuss Paltridge's work, and they comment that the behaviour of the entropy production is far from simple and universal. Later work by Paltridge focuses more on the idea of a dissipation function than on the idea of rate of production of entropy.
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Sawada (1981),also in relation to the Earth's atmospheric energy transport process, postulating a principle of largest amount of entropy increment per unit time, cites work in fluid mechanics by Malkus and Veronis (1958) as having "proven a principle of maximum heat current, which in turn is a maximum entropy production for a given boundary condition", but this inference is not logically valid. Again investigating planetary atmospheric dynamics, Shutts (1981) used an approach to the definition of entropy production, different from Paltridge's, to investigate a more abstract way to check the principle of maximum entropy production, and reported a good fit.
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.
As indicated by the " " marks of Onsager (1931),[1] such a metaphorical but not categorically mechanical force, the thermal "force", [math]\displaystyle{ X_{th} }[/math], 'drives' the conduction of heat. For this so-called "thermodynamic force", we can write
The physics of the earth's atmosphere includes dramatic events like lightning and the effects of volcanic eruptions, with discontinuities of motion such as noted by Helmholtz (1868).[26] Turbulence is prominent in atmospheric convection. Other discontinuities include the formation of raindrops, hailstones, and snowflakes. The usual theory of classical non-equilibrium thermodynamics will need some extension to cover atmospheric physics. According to Tuck (2008),[43] "On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975,[44] 2001[45]). Initially Paltridge (1975)[44] used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978),[46] and in Paltridge (1979)[47]), 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).[48] Paltridge (1978)[46] cited Busse's (1967)[49] fluid mechanical work concerning an extremum principle. Nicolis and Nicolis (1980) [50] 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.
Jou, Casas-Vazquez, Lebon (1993)[51] note that classical non-equilibrium thermodynamics "has seen an extraordinary expansion since the second world war", and they refer to the Nobel prizes for work in the field awarded to Lars Onsager and Ilya Prigogine. Martyushev and Seleznev (2006)[4] note the importance of entropy in the evolution of natural dynamical structures: "Great contribution has been done in this respect by two scientists, namely Clausius, ... , and Prigogine." Prigogine in his 1977 Nobel Lecture[52] said: "... non-equilibrium may be a source of order. Irreversible processes may lead to a new type of dynamic states of matter which I have called “dissipative structures”." Glansdorff and Prigogine (1971)[9] wrote on page xx: "Such 'symmetry breaking instabilities' are of special interest as they lead to a spontaneous 'self-organization' of the system both from the point of view of its space order and its function."
Analyzing the Rayleigh–Bénard convection cell phenomenon, Chandrasekhar (1961)[53] wrote "Instability occurs at the minimum temperature gradient at which a balance can be maintained between the kinetic energy dissipated by viscosity and the internal energy released by the buoyancy force." With a temperature gradient greater than the minimum, viscosity can dissipate kinetic energy as fast as it is released by convection due to buoyancy, and a steady state with convection is stable. The steady state with convection is often a pattern of macroscopically visible hexagonal cells with convection up or down in the middle or at the 'walls' of each cell, depending on the temperature dependence of the quantities; in the atmosphere under various conditions it seems that either is possible. (Some details are discussed by Lebon, Jou, and Casas-Vásquez (2008)[10] on pages 143–158.) With a temperature gradient less than the minimum, viscosity and heat conduction are so effective that convection cannot keep going.
Glansdorff and Prigogine (1971)[9] on page xv wrote "Dissipative structures have a quite different [from equilibrium structures] status: they are formed and maintained through the effect of exchange of energy and matter in non-equilibrium conditions." They were referring to the dissipation function of Rayleigh (1873)[34] that was used also by Onsager (1931, I,[1] 1931, II[36]). On pages 78–80 of their book[9] Glansdorff and Prigogine (1971) consider the stability of laminar flow that was pioneered by Helmholtz; they concluded that at a stable steady state of sufficiently slow laminar flow, the dissipation function was minimum.
1945年普里戈金(另见普里戈金(1947))提出了一个 "最小熵产生定理",该定理仅适用于在静止的热力学非平衡状态附近,具有可忽略的惯性条款的纯扩散线性制度。普里戈金的建议是,熵增率在每一个点上都是局部最小。普里戈金提供的证明是可以受到严肃批评的。Grandy(2008)对Prigogine的提议进行了批评和不支持的讨论。Barbera已经证明,整个身体的总熵的产生不可能是最小的,但这篇论文没有考虑Prigogine的点状最小提案。与Prigogine的提议密切相关的是,熵增的点状速率应该在稳定状态下具有最小的最大值。这与普里戈金的建议相一致,但不完全相同。此外,N. W. Tschoegl提出了一个证明,也许比普里戈金的证明更有物理动机,如果有效的话,它将支持亥姆霍兹和普里戈金的结论,即在这些限制条件下,熵增处于点状最小。
↑ 2.02.12.2Gyarmati, 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.
↑Ziegler, H., (1983). An Introduction to Thermomechanics, North-Holland, Amsterdam,
↑Martyushev, I.M.; Nazarova, A.S.; Seleznev, V.D. (2007). "On the problem of the minimum entropy production in the nonequilibrium stationary state". Journal of Physics A: Mathematical and Theoretical. 40 (3): 371–380. Bibcode:2007JPhA...40..371M. doi:10.1088/1751-8113/40/3/002.
↑Hillert, M.; Agren, J. (2006). "Extremum principles for irreversible processes". Acta Materialia. 54 (8): 2063–2066. doi:10.1016/j.actamat.2005.12.033.
↑Kondepudi, D. (2008)., Introduction to Modern Thermodynamics, Wiley, Chichester UK,
↑Wang, G.M., Sevick, E.M., Mittag, E., Searles, D.J., Evans, D.J. (2002) Experimental demonstration of violations of the Second Law of Thermodynamics for small systems and short time scales, Physical Review Letters 89: 050601-1 - 050601-4.
↑ 22.022.1Dewar, 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.
↑ 26.026.126.2Helmholtz, 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.
↑ 28.028.1Strutt, J.W. (Baron Rayleigh) (1896/1926). Section 357 et seq. The Theory of Sound, Macmillan, London, reprinted by Dover, New York, 1945.
↑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,
↑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,
↑Helmholtz, H. (1869/1871). Zur Theorie der stationären Ströme in reibenden Flüssigkeiten, Verhandlungen des naturhistorisch-medizinischen Vereins zu Heidelberg, Band V: 1-7. Reprinted in Helmholtz, H. (1882), Wissenschaftliche Abhandlungen, volume 1, Johann Ambrosius Barth, Leipzig, pages 223-230 [5]
↑from page 2 of Helmholtz 1869/1871, translated by Wikipedia editor.
↑ 39.039.1Prigogine, I (1945). "Modération et transformations irréversibles des systèmes ouverts". Bulletin de la Classe des Sciences., Académie Royale de Belgique. 31: 600–606.
↑Prigogine, I. (1947). Étude thermodynamique des Phenomènes Irréversibles, Desoer, Liège.
↑ 44.044.144.2Paltridge, G.W. (1975). Global dynamics and climate - a system of minimum entropy exchange, Quarterly Journal of the Royal Meteorological Society 101:475-484.[6]
↑ 46.046.146.246.3Paltridge, G.W. (1978). "The steady-state format of global climate". Quarterly Journal of the Royal Meteorological Society. 104 (442): 927–945. doi:10.1256/smsqj.44205.