Effective Dynamics of a Tracer Particle Interacting with an Ideal Bose Gas

TL;DR

This paper analyzes the quantum dynamics of a heavy tracer particle interacting with an ideal Bose gas, deriving an effective classical description and quantifying its accuracy in the mean-field limit.

Contribution

It introduces a rigorous comparison between quantum and effective classical dynamics for a tracer particle in an ideal Bose gas, bridging thermodynamic and mean-field limits.

Findings

Quantum dynamics closely approximated by classical equations as gas density increases

Effective Newtonian and wave equations accurately describe the system in the mean-field limit

Results enable interchangeability of thermodynamic and mean-field limits in this context

Abstract

We study a system consisting of a heavy quantum particle, called tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density $\to\infty$). Our estimates allow us to interchange the thermodynamic with the mean-field limit.