A framework of nonequilibrium statistical mechanics. I. Role and type of fluctuations

Abstract

Understanding the fluctuations by which phenomenological evolution equations with thermodynamic structure can be enhanced is the key to a general framework of nonequilibrium statistical mechanics. These fluctuations provide an idealized representation of microscopic details. We consider fluctuation-enhanced equations associated with Markov processes and elaborate the general recipes for evaluating dynamic material properties, which characterize force-flux constitutive laws, by statistical mechanics. Markov processes with continuous trajectories are conveniently characterized by stochastic differential equations and lead to Green-Kubo-type formulas for dynamic material properties. Markov processes with discontinuous jumps include transitions over energy barriers with the rates calculated by Kramers. We describe a unified approach to Markovian fluctuations and demonstrate how the appropriate type of fluctuations (continuous versus discontinuous) is reflected in the mathematical structure of the phenomenological equations.

arXiv commentThis paper and its continuation (II) replace the previous paper arXiv:1809.07253. Now the content and the presentation style are more targeted to physicists, with particular emphasis on the applications to specific physical systems. The theoretical and the practical aspects have been separated from each other in the two papers

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