Non-equilibrium Onsager-Machlup theory

Abstract

This paper proposes a simple mathematical model of non-stationary and non-linear stochastic dynamics, which approximates a (globally) non-stationary and non-linear stochastic process by its locally (or \emph{"piecewise"}) stationary version. Profiting from the elegance and simplicity of both, the exact mathematical model referred to as the Ornstein-Uhlenbeck stochastic process (which is \emph{globally} stationary, Markov and Gaussian) and of the Lyapunov criterion associated with the stability of stationarity, we show that the proposed non-linear non-stationary model provides a natural extension of the Onsager-Machlup theory of equilibrium thermal fluctuations, to the realm of non-stationary, non-linear, and non-equilibrium processes. As an illustrative application, we then apply the extended non-equilibrium Onsager-Machlup theory, to the description of thermal fluctuations and irreversible relaxation processes in liquids, leading to the main exact equations employed to construct the \emph{non-equilibrium self-consistent generalized Langevin equation} (NE-SCGLE) theory of irreversible processes in liquids. This generic theory has demonstrated that the most intriguing and long-unsolved questions of the glass and the gel transitions are understood as a natural consequence of the second law of thermodynamics, enunciated in terms of the proposed piecewise stationary stochastic mathematical model.