General theory of Onsager symmetries for perturbations of equilibrium and nonequilibrium steady states

Abstract

The theory of Onsager symmetry is reconsidered from the point of view of its application to nonequilibrium, possibly turbulent steady states. A dynamical formalism based on correlation and response functions is used; understanding of its relationship to more conventional approaches based on entropy production enables one to resolve various confusions about the proper use of the theory, even near thermal equilibrium. Previous claims that ‘‘kinematic’’ flows must be excluded from considerations of Onsager symmetry are refuted by showing that suitably defined reversible and irreversible parts of the Onsager matrix separately obey the appropriate symmetry; fluctuating hydrodynamics serves as an example. It is shown that Onsager symmetries are preserved under arbitrary covariant changes of variables; the Weinhold metric is used as a fundamental tensor. Covariance is used to render moot the controversy over the proper choice of fluxes and forces in neoclassical plasma transport theory. The fundamental distinction between the fully contravariant Onsager matrix Lij and its mixed representation Lij is emphasized and used to explain why some previous workers have failed to find Onsager symmetry around turbulent steady states. The generalized Onsager theorem of Dufty and Rubí [Phys. Rev. A 36, 222 (1987)] is reviewed. An explicitly soluble Langevin problem is shown to violate Onsager’s original symmetry but to obey the generalized theorem. The physical content of the generalized Onsager symmetry is discussed from the point of view of Nosé–Hoover dynamics. A set of extended Graham–Haken potential conditions are derived for Fokker–Planck models and shown to be consistent with the generalized Onsager relations. Finally, for quite general, possibly turbulent steady states it is argued that realizable Markovian statistical closures with underlying Langevin representations must also obey the generalized theorem. In the special case in which all state variables have even parity and there are no external parameters that change sign under time reversal, the steady-state energy balance fully determines the Onsager matrix, which is guaranteed to be symmetric.