Work distribution in time-dependent logarithmic-harmonic potential: exact results and asymptotic analysis

TL;DR

This paper derives exact and asymptotic results for the work distribution of a Brownian particle in a time-dependent logarithmic-harmonic potential, revealing tail behaviors and special cases like the breathing parabola.

Contribution

It provides an exact solution for the work distribution in a complex potential and analyzes its asymptotic behavior, extending understanding of stochastic work in time-dependent systems.

Findings

Exact solution for work distribution in specific potential

Asymptotic tail behavior for small and large work values

Work distribution for the breathing parabola model

Abstract

We investigate the distribution of work performed on a Brownian particle in a time-dependent asymmetric potential well. The potential has a harmonic component with time-dependent force constant and a time-independent logarithmic barrier at the origin. For arbitrary driving protocol, the problem of solving the Fokker-Planck equation for the joint probability density of work and particle position is reduced to the solution of the Riccati differential equation. For a particular choice of the driving protocol, an exact solution of the Riccati equation is presented. Asymptotic analysis of the resulting expression yields the tail behavior of the work distribution for small and large work values. In the limit of vanishing logarithmic barrier, the work distribution for the breathing parabola model is obtained.