Asymptotics of the Entropy Production Rate for d-Dimensional Ornstein–Uhlenbeck Processes

Abstract

In the context of non-equilibrium statistical physics, the entropy production rate is an important concept to describe how far a specific state of a system is from its equilibrium state. In this paper, we establish a central limit theorem and a moderate deviation principle for the entropy production rate of d-dimensional Ornstein-Uhlenbeck processes, by the techniques of functional inequalities such as Poincaré inequality and log-Sobolev inequality. As an application, we obtain a law of iterated logarithm for the entropy production rate.

The Law of the Iterated Logarithm for a Class of SPDEs

“…We note that this version is also given by ( 18) and ( 19) with S n − nµ replaced by X n with µ = 0 and σ 2 = 1. For examples of this form see for instance [26,35].…”

“…For instance, P. Baldi [2], G. Divanji, K. Vidyalaxmi [14], B. Jing, Q. Shao, Q. Wang [26], and A. Mogul'skii [28] used their large deviations results to prove LIL, whereas, Y. Chen, L. Liu [6] and R. Wang, L. Xu [35] applied their theorem on moderate deviations. LIL has useful applications in other fields including finance (see for example [24,41]).…”

The large deviation principle and steady-state fluctuation theorem for the entropy production rate of a stochastic process in magnetic fields

“…Proof. (45) and ( 46) are immediate from the analysis above and (39). Routine calculus based upon (33), in combination with (38) and (40), gives (47)…”

Large deviations for quadratic functionals of stable Gauss-Markov chains and entropy production

“…The use of the entropy production rate to quantify the irreversibility of a stochastic process was proposed by Kurchan [1] and in more generality by Lebowitz, Spohn, and Maes [2,3], who extended the seminal work by Gallavotti and Cohen [4] in the context of deterministic dynamical systems. Since then, the entropy production rate has become a basic topic in non-equilibrium statistical physics [5][6][7][8][9][10][11][12][13]. The entropy production rate came out with a supposed symmetry associated with its large fluctuations, which in fact was discovered by Gallavotti and Cohen [4] prompted by results of computer simulations [14].…”