Tracer particles coupled to an interacting boson gas

Abstract

In this work, we investigate the mean-field limit of a model consisting in $m\ge 1$ tracer particles, coupled to an interacting boson field. We assume the mass of the tracer particles and the expected number of bosons to be of the same order of magnitude $N\ge 1$ and we investigate the $N\to \infty$ limit. In particular, we show that the limiting system can be effectively described by a pair of variables $({\text{X}}_t,{\phi }_t)\in {\mathbb{R}}^{3m}\times H^1({\mathbb{R}}^3)$ that solve a mean-field equation. Our methods are based on proving estimates for the number of bosonic particles in a suitable \textit{fluctuation state} ${\Omega }_{N,t}$. The main diffculty of the problem comes from the fact that the interaction with the tracer particles can create or destroy bosons for states close to the vacuum.

36 pages, 0 figures