On Bose-Einstein condensation in one-dimensional noninteracting Bose gases in the presence of soft Poisson obstacles

TL;DR

This paper proves that Bose-Einstein condensation occurs in one-dimensional noninteracting Bose gases with Poisson random potentials, showing that macroscopic occupation of the ground state or multiple states is highly probable under certain conditions.

Contribution

It establishes the occurrence of Bose-Einstein condensation in one-dimensional systems with soft Poisson obstacles, including conditions for type-I generalized BEC at positive temperatures.

Findings

BEC occurs with high probability for large potential strength

Type-I g-BEC is probable with many macroscopically occupied states

BEC persists in the limit of infinitely strong or slowly increasing potentials

Abstract

We study Bose-Einstein condensation (BEC) in one-dimensional noninteracting Bose gases in Poisson random potentials on $\mathbb R$ with single-site potentials that are nonnegative, compactly supported, and bounded measurable functions in the grand-canonical ensemble at positive temperatures in the thermodynamic limit. For particle densities that are larger than a critical one, we prove the following: With arbitrarily high probability when choosing the fixed strength of the random potential sufficiently large, BEC where only the ground state is macroscopically occupied occurs. If the strength of the Poisson random potential converges to infinity in a certain sense but arbitrarily slowly, then this kind of BEC occurs in probability and in the $r$th mean, $r \ge 1$. Furthermore, in Poisson random potentials of any fixed strength an arbitrarily high probability for type-I g-BEC is also obtained by allowing sufficiently many one-particle states to be macroscopically occupied.