Stationary and Transient Work-Fluctuation Theorems for a Dragged Brownian Particle

TL;DR

This paper theoretically derives and analyzes stationary and transient work-fluctuation theorems for a dragged Brownian particle, extending previous experimental results and proposing new experimental conditions for observing these theorems.

Contribution

It provides explicit theoretical expressions for work-fluctuation theorems under arbitrary harmonic force motion, including conditions for their validity and a novel long-time relation between the theorems.

Findings

The transient fluctuation theorem holds at all times.

The stationary fluctuation theorem holds asymptotically.

A long-time relation between the theorems was identified.

Abstract

Recently Wang et al. carried out a laboratory experiment, where a Brownian particle was dragged through a fluid by a harmonic force with constant velocity of its center. This experiment confirmed a theoretically predicted work related integrated (I) Transient Fluctuation Theorem (ITFT), which gives an expression for the ratio for the probability to find positive or negative values for the fluctuations of the total work done on the system in a given time in a transient state. The corresponding integrated stationary state fluctuation theorem (ISSFT) was not observed. Using an overdamped Langevin equation and an arbitrary motion for the center of the harmonic force, all quantities of interest for these theorems and the corresponding non-integrated ones (TFT and SSFT, resp.) are theoretically explicitly obtained in this paper. While the (I)TFT is satisfied for all times, the (I)SSFT only holds asymptotically in time. Suggestions for further experiments with arbitrary velocity of the harmonic force and in which also the ISSFT could be observed, are given. In addition, a non-trivial long-time relation between the ITFT and the ISSFT was discovered, which could be observed experimentally, especially in the case of a resonant circular motion of the center of the harmonic force.