Additivity of multiple heat reservoirs in Langevin equation

TL;DR

This study uses molecular dynamics simulations to verify whether the effects of multiple heat reservoirs on a Brownian system can be accurately modeled as additive, confirming general additivity with minor deviations.

Contribution

The paper provides the first numerical validation of the additive approach for multiple heat reservoirs in Langevin equations, confirming its general validity.

Findings

Additivity of heat reservoir effects is confirmed in simulations.

Minor deviations in dissipation and noise are statistically insignificant.

Steady-state properties adhere to the additive model perfectly.

Abstract

The Langevin equation greatly simplifies the mathematical expression of the effects of thermal noise by using only two terms, a dissipation term, and a random-noise term. The Langevin description was originally applied to a system in contact with a single heat reservoir; however, many recent studies have also adopted a Langevin description for systems connected to multiple heat reservoirs. This is accomplished through the introduction of a simple summation for the dissipation and random-noise terms associated with each reservoir. However, the validity of this simple addition has been the focus of only limited discussion and has raised several criticisms. Moreover, this additive description has never been either experimentally or numerically verified, rendering its validity is still an open question. Here, we perform molecular dynamics simulations for a Brownian system in simultaneous contact with multiple heat reservoirs to check the validity of this additive approach. Our simulation results confirm that the effect of multiple heat reservoirs is additive in general. A very small deviation in the total amount of dissipation and associated noise is found, but seems not significant within statistical errors. We find that the steady-state properties satisfy the additivity perfectly and are not affected by this deviation.