Properties of the dissipation functions for passive and active systems

TL;DR

This paper derives the dissipation functions for passive and active systems using Langevin equations, verifies the fluctuation theorem numerically, and explores their relation to entropy production.

Contribution

It provides explicit expressions for dissipation functions in passive and active systems and analyzes their properties and relation to entropy production.

Findings

Dissipation function for passive systems depends only on initial and final states.

Numerical verification of the fluctuation theorem for a 1D passive system.

Active systems' work relates to entropy production under time-reversal symmetry.

Abstract

The dissipation function for a system is defined as the natural logarithm of the ratio between probabilities of a trajectory and its time-reversed trajectory, and its probability distribution follows a well-known relation called the fluctuation theorem. Using the generic Langevin equations, we derive the expressions of the dissipation function for passive and active systems. For passive systems, the dissipation function depends only on the initial and the final values of the dynamical variables of the system, not on the trajectory of the system. Furthermore, it does not depend explicitly on the reactive or dissipative coupling coefficients of the generic Langevin equations. In addition, we study a 1D case numerically to verify the fluctuation theorem with the form of the dissipation function we obtained. For active systems, we define the work done by active forces along a trajectory. If the probability distribution of the dynamical variables is symmetric under time reversal, in both cases, the average rate of change of the dissipation function with trajectory duration is nothing but the average entropy production rate of the system and reservoir.