Retainability of canonical distributions for a Brownian particle controlled by a time-dependent harmonic potential

TL;DR

This paper explores conditions under which canonical distributions can be maintained for a Brownian particle in a time-dependent harmonic potential, revealing differences between overdamped and underdamped regimes and effects of temperature control.

Contribution

It provides a detailed analysis of retainability conditions for canonical distributions in both overdamped and underdamped cases with time-dependent potentials and temperatures.

Findings

Canonical distributions are retainable in both regimes when temperature is constant.

Retainability is limited to overdamped case when temperature varies with potential.

Retainability depends on the specific form of the potential beyond harmonic cases.

Abstract

The retainability of canonical distributions for a Brownian particle controlled by a time-dependent harmonic potential is investigated in the overdamped and underdamped situations, respectively. Because of different time scales, the overdamped and underdamped Langevin equations (as well as the corresponding Fokker-Planck equations) lead to distinctive restrictions on protocols maintaining canonical distributions. Two special cases are analyzed in details: First, a Brownian particle is controlled by a time-dependent harmonic potential and embedded in medium with constant temperature; Second, a Brownian particle is controlled by a time-dependent harmonic potential and embedded in a medium whose temperature is tuned together with the potential stiffness to keep a constant effective temperature of the Brownian particle. We find that the canonical distributions are usually retainable for both the overdamped and underdamped situations in the former case. However, the canonical distributions are retainable merely for the overdamped situation in the latter case. We also investigate general time-dependent potentials beyond the harmonic form and find that the retainability of canonical distributions depends sensitively on the specific form of potentials.