Additivity of multiple heat reservoirs in the Langevin equation

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.