Statistical mechanical theory for steady state systems. VI. Variational principles

Abstract

Several variational principles that have been proposed for nonequilibrium systems are analyzed. These include the principle of minimum rate of entropy production due to Prigogine [Introduction to Thermodynamics of Irreversible Processes (Interscience, New York, 1967)], the principle of maximum rate of entropy production, which is common on the internet and in the natural sciences, two principles of minimum dissipation due to Onsager [Phys. Rev. 37, 405 (1931)] and to Onsager and Machlup [Phys. Rev. 91, 1505 (1953)], and the principle of maximum second entropy due to Attard [J. Chem.. Phys. 122, 154101 (2005); Phys. Chem. Chem. Phys. 8, 3585 (2006)]. The approaches of Onsager and Attard are argued to be the only viable theories. These two are related, although their physical interpretation and mathematical approximations differ. A numerical comparison with computer simulation results indicates that Attard's expression is the only accurate theory. The implications for the Langevin and other stochastic differential equations are discussed.

Statistical mechanical theory for nonequilibrium systems. X. Nonequilibrium phase transitions

“…Several detailed criticisms of these approaches have been made, [9][10][11][12] including the fact that since the rate of first entropy production is a bilinear function of the forces and fluxes, its extreme values must lie at Ϯϱ. 2,13,14 The second entropy in fluctuation approximation turns out to be a quadratic function that is maximized in the optimum nonequilibrium state, and consequently it is larger in the stable phase than in the unstable phase. Its maximum value in any one phase is shown below to be numerically equal to one quarter of the rate of first entropy production, which implies that the rate of first entropy production is greater in the stable phase than in the unstable phase.…”

Entropy, Closures and Subgrid Modeling

“…Applications of entropy production methods to subgrid modeling have been made by Kazantsev et al (1998) [155], Polyakov (2001) [156] and Holloway (2004) [157]. Critiques of entropy production principles have been presented by Barbera (1999) [158], Attard (2006) [159], Bruers (2007) [160] and Grinstein and Linsker (2007) [161]. More detailed reviews of the literature are presented by Martyushev and Seleznev (2006) [162] and by Dewar et al (2008) [163] in this issue.…”

Statistical mechanical theory for steady state systems. VII. Nonlinear theory

“…[1][2][3][4][5][6] A self-contained review of the theory up to the fifth paper has been given. [1][2][3][4][5][6] A self-contained review of the theory up to the fifth paper has been given.…”

“…7 Briefly, the theory is based on the second entropy, which is the entropy associated with the transition between macrostates. 5 Throughout the flux is treated as a constrained variable whose steady state value maximizes the second entropy, and care is taken to distinguish general relations from relations that hold only in the steady state, 6 which distinction is overlooked in most prior works. Accordingly, the nonequilibrium theory is cast in a form deliberately analogous to the ordinary equilibrium theory.…”

“…27,28 Almost all these thermodynamic Lagrangians are based on the original functional for the rate of first entropy production given by Onsager and Machlup. 16,[18][19][20] The Onsager-Machlup functional was quantitatively tested with computer simulations in the preceding paper in this series, 6 and it was found to be deficient at finite times. 16,[18][19][20] The Onsager-Machlup functional was quantitatively tested with computer simulations in the preceding paper in this series, 6 and it was found to be deficient at finite times.…”