Central limit theorem for Bose gases interacting through singular potentials

TL;DR

This paper proves a central limit theorem for fluctuations in large bosonic systems with singular interactions, showing that the quantum fluctuations around the mean-field evolve normally in the limit.

Contribution

It establishes a central limit theorem for quantum fluctuations in Bose gases with singular potentials, extending understanding of their statistical behavior.

Findings

Fluctuations follow a multivariate normal distribution in the large N limit.

The dynamics are well approximated by a quadratic fluctuation around a nonlinear Schrödinger equation.

The results apply to initial data with Bose-Einstein condensation.

Abstract

We consider a system of $N$ bosons in the limit $N \rightarrow \infty$, interacting through singular potentials. For initial data exhibiting Bose-Einstein condensation, the many-body time evolution is well approximated through a quadratic fluctuation dynamics around a cubic non-linear Schr\"odinger equation of the condensate wave function. We show that these fluctuations satisfy a (multi-variate) central limit theorem.