Perturbative expansion of irreversible works in symmetric and asymmetric processes

TL;DR

This paper develops a perturbative method to calculate mean work in Fokker-Planck systems, revealing how symmetry influences thermodynamic properties and identifying a critical deformation time affecting work differences.

Contribution

It introduces a new perturbative formula for mean work applicable to degenerate Fokker-Planck operators, analyzing symmetry effects in thermodynamic processes.

Findings

Symmetric and asymmetric potential deformations show maximal work difference at a critical time.

The critical deformation time for work is shorter than that for energy change due to hysteresis.

The method extends to systems with degenerate eigenvalues of the Fokker-Planck operator.

Abstract

The systematic expansion method of the solution of the Fokker-Planck equation is developed by generalizing the formulation proposed in [J. Phys. A50, 325001 (2017)]. Using this method, we obtain a new formula to calculate the mean work perturbatively which is applicable to systems with degeneracy in the eigenvalues of the Fokker-Planck operator. This method enables us to study how the geometrical symmetry affects thermodynamic description of a Brownian particle. To illustrate the application of the derived theory, we consider the Fokker-Planck equation with a two-dimensional harmonic potential. To investigate the effect of symmetry of the potential, we study thermodynamic properties in symmetric and asymmetric deformation processes of the potential: the rotational symmetry of the harmonic potential is held in the former, but it is broken in the latter. Optimized deformations in these processes are defined by minimizing mean works. Comparing these optimized processes, we find that the difference between the symmetric and asymmetric processes is maximized when the deformation time of the potential is given by a critical time which is characterized by the relaxation time of the Fokker-Planck equation. This critical time in the mean work is smaller than that of the change of the mean energy because of the hysteresis effect in the irreversible processes.