Review of a Simplified Approach to study the Bose gas at all densities

TL;DR

This paper reviews a simplified theoretical framework for studying the Bose gas across all densities, capturing key physical properties and phase behaviors, with analytical and numerical insights into effective equations.

Contribution

It introduces and discusses a simplified approach that models the Bose gas at all densities using effective one-particle equations, connecting to known estimates and exploring new phases.

Findings

Reproduces Bogolyubov's estimates at low density

Captures mean-field behavior at high density

Identifies a liquid-like phase at intermediate densities

Abstract

In this paper, we will review the results obtained thus far by Eric A. Carlen, Elliott H. Lieb and me on a Simplified Approach to the Bose gas. The Simplified Approach yields a family of effective one-particle equations, which capture some non-trivial physical properties of the Bose gas at both low and high densities, and even some of the behavior at intermediate densities. In particular, the Simplified Approach reproduces Bogolyubov's estimates for the ground state energy and condensate fraction at low density, as well as the mean-field estimate for the energy at high densities. We will also discuss a phase that appears at intermediate densities with liquid-like properties. The simplest of the effective equations in the Simplified Approach can be studied analytically, and we will review several results about it; the others are so far only amenable to numerical analysis, and we will discuss several numerical results. We will start by reviewing some results and conjectures on the Bose gas, and then introduce the Simplified Approach and its derivation from the Bose gas. We will then discuss the predictions of the Simplified Approach and compare these to results and conjectures about the Bose gas. Finally, we will discuss a few open problems about the Simplified Approach.