Macroscopic dynamical fluctuations in Kac ring model

TL;DR

This paper investigates the fluctuations in macroscopic paths of the Kac ring model, deriving the generating function and demonstrating the fluctuation relation resulting from microscopic reversibility.

Contribution

It derives the generating function for macroscopic paths and establishes the fluctuation relation in the Kac ring model, linking microscopic reversibility to macroscopic fluctuations.

Findings

Small deviations follow a discrete-time Ornstein-Uhlenbeck process

Fluctuation relation of the rate function is proven

Microscopic reversibility influences macroscopic fluctuation behavior

Abstract

We study dynamical fluctuations in the macroscopic paths around the most probable path of the Kac ring model, which is a simple deterministic and reversible dynamical system exhibiting the macroscopic irreversible relaxation. We derive the form of the generating function for macroscopic paths and show that the small deviations are described by a discrete-time Ornstein-Uhlenbeck process. We also argue that the microscopic reversibility leads to the fluctuation relation of the rate function, and prove it based on the form of the generating function.