Abstract
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.
Highlights
Phenomenological evolution equations with a thermodynamic structure can be enhanced by adding fluctuations
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
As our central contribution, we present the consequences of this correspondence for nonequilibrium statistical mechanics, by answering the following two questions. (i) Given a microscopic model and a meaningful coarse-graining map, how can one find the thermodynamic structure of the macroscopic, phenomenological equation? (ii) In the opposite direction, given a phenomenological equation and its thermodynamic structure, how can we add fluctuations? For these procedures we use the terms “coarse-graining” and “noise enhancement.”
Summary
Phenomenological evolution equations with a thermodynamic structure can be enhanced by adding fluctuations. We start from a class of phenomenological evolution equations for isolated systems with a particular thermodynamic structure (Section 2) To enhance these phenomenological equations, we do not restrict ourselves to continuous Gaussian noise but rather allow for general Markov processes, including jump processes. We try to recognize the proper types of noise to be used for enhancing phenomenological equations in the thermodynamic structure of those equations This attempt leads us to the very general fluctuation-dissipation theorem that is at the heart of this paper (Section 3). We analyze the noise resulting from more microscopic descriptions to extract the detailed form of more macroscopic, phenomenological evolution equations, which is a key task of nonequilibrium statistical mechanics (Section 4). We strive for a multiscale approach with reassuring mutual consistency between different levels of description
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