Non-equilibrium fluctuations for linear diffusion dynamics

TL;DR

This paper develops a comprehensive theoretical framework for analyzing non-equilibrium fluctuations in high-dimensional linear diffusion systems driven by non-conservative forces and correlated noise, providing exact solutions and revealing a dynamic phase transition.

Contribution

It introduces a generalized thermodynamic theory for NEQ fluctuations in multi-dimensional diffusion, deriving exact distributions and identifying a novel dynamic phase transition in work fluctuations.

Findings

Exact time-dependent probability distribution for NEQ systems

Explicit low-order cumulants of work production

Discovery of a dynamic phase transition in work fluctuation tail shape

Abstract

We present the theoretical study on non-equilibrium (NEQ) fluctuations for diffusion dynamics in high dimensions driven by a linear drift force. We consider a general situation in which NEQ is caused by two conditions: (i) drift force not derivable from a potential function and (ii) diffusion matrix not proportional to the unit matrix, implying non-identical and correlated multi-dimensional noise. The former is a well-known NEQ source and the latter can be realized in the presence of multiple heat reservoirs or multiple noise sources. We develop a statistical mechanical theory based on generalized thermodynamic quantities such as energy, work, and heat. The NEQ fluctuation theorems are reproduced successfully. We also find the time-dependent probability distribution function exactly as well as the NEQ work production distribution $P({\mathcal W})$ in terms of solutions of nonlinear differential equations. In addition, we compute low-order cumulants of the NEQ work production explicitly. In two dimensions, we carry out numerical simulations to check out our analytic results and also to get $P({\mathcal W})$. We find an interesting dynamic phase transition in the exponential tail shape of $P({\mathcal W})$, associated with a singularity found in solutions of the nonlinear differential equation. Finally, we discuss possible realizations in experiments.