Mean field dynamics of a quantum tracer particle interacting with a boson gas

TL;DR

This paper derives generalized Hartree equations describing the mean-field dynamics of a heavy quantum tracer particle interacting with a boson gas, proving well-posedness for weak interactions in the large particle limit.

Contribution

It introduces a rigorous derivation of mean-field equations for a quantum tracer particle coupled to a boson gas, extending understanding of such quantum many-body systems.

Findings

Derivation of generalized Hartree equations in the large particle limit

Proof of global well-posedness for the associated Cauchy problem

Validation of the mean-field approximation for weak interactions

Abstract

We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in ${\mathbb R}^3$. The pair interactions of the bosons are of mean-field type, with coupling strength proportional to $\frac1N$ where $N$ is the expected particle number. Assuming that the mass of the tracer particle is proportional to $N$, we derive generalized Hartree equations in the limit $N\rightarrow\infty$. Moreover, we prove the global well-posedness of the associated Cauchy problem for sufficiently weak interaction potentials.