Self-consistent equation for an interacting Bose gas

TL;DR

This paper derives a self-consistent equation for the density of an interacting Bose gas using polymer and Mayer graph techniques, analyzing mean-field limits and corrections near Bose-Einstein condensation.

Contribution

It introduces a novel self-consistent relation between density and chemical potential for interacting Bose gases, valid within the Mayer series convergence range and explores corrections beyond mean-field theory.

Findings

Established the self-consistent density relation using Mayer graphs.

Proved that in the mean-field limit only tree diagrams contribute.

Derived bounds and resummations for corrections near Bose-Einstein condensation.

Abstract

We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential $V(r)$ such that $0<\int d\br V(r) = a<\infty$. Expressing the partition function by the Feynman-Kac functional integral yields a classical-like polymer representation of the quantum gas. With Mayer graph summation techniques, we demonstrate the existence of a self-consistent relation $\rho (\mu)=F(\mu-a\rho(\mu))$ between the density $\rho $ and the chemical potential $\mu$, valid in the range of convergence of Mayer series. The function $F$ is equal to the sum of all rooted multiply connected graphs. Using Kac's scaling $V_{\gamma}(\br)=\gamma^{3}V(\gamma r)$ we prove that in the mean-field limit $\gamma\to 0$ only tree diagrams contribute and function $F$ reduces to the free gas density. We also investigate how to extend the validity of the self-consistent relation beyond the convergence radius of Mayer series (vicinity of Bose-Einstein condensation) and study dominant corrections to mean field. At lowest order, the form of function $F$ is shown to depend on single polymer partition function for which we derive lower and upper bounds and on the resummation of ring diagrams which can be analytically performed.