Work fluctuations in a time-dependent harmonic potential: rigorous results and beyond the overdamped limit

TL;DR

This paper rigorously analyzes the stochastic dynamics of a Brownian particle in a time-dependent harmonic potential, revealing complex work distribution features and extending results beyond the overdamped approximation.

Contribution

It provides exact relations for the work distribution in a time-dependent harmonic trap, including inertial effects, and uncovers novel oscillatory phenomena in nonequilibrium work.

Findings

Work distribution has an exponential tail with a power-law prefactor.

Inertial effects cause oscillatory features and pseudo locking-unlocking transitions.

Exact solutions are discussed for overdamped cases.

Abstract

We investigate the stochastic motion of a Brownian particle in the harmonic potential with a time-dependent force constant. It may describe the motion of a colloidal particle in an optical trap where the potential well is formed by a time-dependent field. We use the path integral formalism to solve the Langevin equation and the associated Fokker-Planck (Kramers) equation. Rigorous relations are derived to generate the probability density function for the time-dependent nonequilibrium work production beyond the overdamped limit. We find that the work distribution exhibits an exponential tail with a power-law prefactor, accompanied by an interesting oscillatory feature (multiple pseudo locking-unlocking transitions) due to the inertial effect. Some exactly solvable cases are also discussed in the overdamped limit.