Variational principles of irreversible processes

Abstract

This article reviews developments of variational principles in the study of irreversible processes during the past three decades or so. The variational principles we consider here are related to entropy production. The purpose of this article is to explicate that we can formulate a variational principle which relates the transport coefficients to microscopic dynamics of fluctuations. The quantum variational principle restricts the nonequilibrium density matrix to a class conforming to the requirement demanded by the second law of thermodynamics. These are various kinds of variational principles according to different stages of a macroscopic system. The three stages are known, which are dynamical, kinetic, and thermodynamical stages. The relationships among these variational principles are discussed from the point of view of the contraction of information about irrelevant components. Nakano's variational principle has close similarity to the Lippmann-Schwinger theory of scattering, in which some incoming and outgoing disturbances have to be considered in a pair. It is also shown that the variational principle of Onsager's type can be reformulated in the form of Hamilton's principle if a generalization of Hamilton's principle proposed by Djukic and Vujanovic is used. A variational principle in the diagrammatic method is also reviewed, which utilizes the generalized Ward-Takahashi relations.