Effective coupling between two Brownian particles

TL;DR

This paper investigates how a specially modeled environment can induce a tunable, nonlinear effective interaction between two independent Brownian particles, revealing new insights into their coupled dynamics.

Contribution

It extends the traditional bath of oscillators model to generate an environment-dependent, nonlinear effective coupling between two Brownian particles, ensuring translational invariance.

Findings

Derived effective equations of motion for the coupled particles

Identified an explicit form of the interaction potential

Demonstrated the model's ability to produce environment-induced coupling

Abstract

We use the system-plus-reservoir approach to study the dynamics of a system composed of two independent Brownian particles. We present an extension of the well-known model of a bath of oscillators which is capable of inducing an effective coupling between the two particles depending on the choice made for the spectral function of the bath oscillators. The coupling is non-linear in the variables of interest and an exponential dependence on these variables is imposed in order to guarantee the translational invariance of the model if the two particles are not subject to any external potential. The effective equations of motion for the particles are obtained by the Laplace transform method and besides recovering all the local dynamical properties for each particle we end up with an effective interaction potential between them. We explicitly analyze one of its possible forms.