Large Deviation Function for a Driven Underdamped Particle in a Periodic Potential

TL;DR

This paper investigates the fluctuations of current in a driven underdamped particle within a periodic potential using large deviation theory, deriving explicit bounds and computational methods for the large deviation function.

Contribution

It introduces a new explicit expression for the large deviation functional in underdamped dynamics and compares bounds with overdamped cases, providing computational techniques.

Findings

Derived an explicit large deviation functional for underdamped dynamics.

Provided bounds on the large deviation function and compared with overdamped cases.

Numerical case study demonstrating the tightness of bounds for a cosine potential.

Abstract

Employing large deviation theory, we explore current fluctuations of underdamped Brownian motion for the paradigmatic example of a single particle in a one dimensional periodic potential. Two different approaches to the large deviation function of the particle current are presented. First, we derive an explicit expression for the large deviation functional of the empirical phase space density, which replaces the level 2.5 functional used for overdamped dynamics. Using this approach, we obtain several bounds on the large deviation function of the particle current. We compare these to bounds for overdamped dynamics that have recently been derived motivated by the thermodynamic uncertainty relation. Second, we provide a method to calculate the large deviation function via the cumulant generating function. We use this method to assess the tightness of the bounds in a numerical case study for a cosine potential.