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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.〔Onsager, L. (1931). Reciprocal relations in irreversible processes, I, ''Physical Review'' 37:405-426〕〔Gyarmati, I. (1970). ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', Springer, Berlin.〕〔Ziegler, H., (1983). ''An Introduction to Thermomechanics'', North-Holland, Amsterdam, ISBN 0-444-86503-9〕〔(Martyushev, L.M., Seleznev, V.D. (2006). Maximum entropy production principle in physics, chemistry and biology, ''Physics Reports'' 426: 1-45 )〕〔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: 371-380.〕〔Hillert, M., Agren, J. (2006). Extremum principles for irreversible processes, ''Acta Materialia'' 54: 2063-2066.〕 According to Kondepudi (2008),〔Kondepudi, D. (2008)., ''Introduction to Modern Thermodynamics'', Wiley, Chichester UK, ISBN 978-0-470-01598-8, page 172.〕 and to 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.〕 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),〔 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)〔Lebon, G., Jou, J., Casas-Vásquez (2008). ''Understanding Non-equilibrium Thermodynamics. Foundations, Applications, Frontiers'', Springer, Berlin, ISBN 978-3-540-74251-7.〕 state that "In non-equilibrium ... it is generally not possible to construct thermodynamic potentials depending on the whole set of variables". Šilhavý (1997)〔Šilhavý, M. (1997). ''The Mechanics and Thermodynamics of Continuous Media'', Springer, Berlin, ISBN 3-540-58378-5, page 209.〕 offers the opinion that "... the extremum principles of thermodynamics ... do not have any counterpart for () steady states (despite many claims in the literature)." It follows that any general extremal principle for a non-equilibrium problem will need to refer in some detail to the constraints that are specific for the structure of the system considered in the problem. ==Fluctuations, entropy, 'thermodynamics forces', and reproducible dynamical structure== Apparent 'fluctuations', which appear to arise when initial conditions are inexactly specified, are the drivers of the formation of non-equilibrium dynamical structures. There is no special force of nature involved in the generation of such fluctuations. Exact specification of initial conditions would require statements of the positions and velocities of all particles in the system, obviously not a remotely practical possibility for a macroscopic system. This is the nature of thermodynamic fluctuations. They cannot be predicted in particular by the scientist, but they are determined by the laws of nature and they are the singular causes of the natural development of dynamical structure.〔 It is pointed out〔Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. I: Equations of motion. ''Found. Phys.'' 34: 1-20. See ().〕〔Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. II: The entropy. ''Found. Phys.'' 34: 21-57. See ().〕〔Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. III: Selected applications. ''Found. Phys.'' 34: 771-813. See ().〕〔Grandy 2004 see also ().〕 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),〔 such a metaphorical but not categorically mechanical force, the thermal "force", , '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, . Temperature is only one example, and all the thermodynamic macroscopic variables constitute inexact specifications of the initial conditions, and have their respective "thermodynamic forces". These inexactitudes of specification are the source of the apparent fluctuations that drive the generation of dynamical structure, of the very precise but still less than perfect reproducibility of non-equilibrium experiments, and of the place of entropy in thermodynamics. If one did not know of such inexactitude of specification, one might find the origin of the fluctuations mysterious. What is meant here by "inexactitude of specification" is not that the mean values of the macroscopic variables are inexactly specified, but that the use of macroscopic variables to describe processes that actually occur by the motions and interactions of microscopic objects such as molecules is necessarily lacking in the molecular detail of the processes, and is thus inexact. There are many microscopic states compatible with a single macroscopic state, but only the latter is specified, and that is specified exactly for the purposes of the theory. It is reproducibility in repeated observations that identifies dynamical structure in a system. E.T. Jaynes〔(Jaynes, E.T. (1957). Information theory and statistical mechanics, ''Physical Review'' 106: 620-630. )〕〔(Jaynes, E.T. (1957). Information theory and statistical mechanics. II, ''Physical Review'' 108: 171-190. )〕〔(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. )〕〔(Jaynes, E.T. (1965). Gibbs vs Boltzmann Entropies, ''American Journal of Physics'' 33: 391-398. )〕 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.〔Evans, D.J., Searles, D.J. (2002). The fluctuation theorem, ''Advances in Physics'' 51: 1529-1585〕〔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.〕 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, 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.〕 Dewar〔 writes "Jaynes considered reproducibility - rather than disorder - to be the key idea behind the second law of thermodynamics (Jaynes 1963,〔(Jaynes, E.T. (1963). pp. 181-218 in ''Brandeis Summer Institute 1962, Statistical Physics'', edited by K.W. Ford, Benjamin, New York. )〕 1965,〔 1988,〔(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. )〕 1989〔(Jaynes, E.T. (1989). Clearing up mysteries, the original goal, pp. 1-27 in ''Maximum entropy and Bayesian methods'', Kluwer, Dordrecht. )〕)." 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Extremal principles in non-equilibrium thermodynamics」の詳細全文を読む スポンサード リンク
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