Level density of a Bose gas: beyond the saddle point approximation

TL;DR

This paper develops a uniform approximation method for calculating the many-body density of states in a Bose gas, especially effective when the ground state is highly populated, surpassing the limitations of the saddle point approximation.

Contribution

It introduces a new uniform formula for the many-body density of states that remains valid in regimes where traditional saddle point methods fail.

Findings

The uniform formula improves accuracy over the standard second order method.

Application to the 1D harmonic oscillator demonstrates the method's effectiveness.

Results leverage number theory to enhance the approximation.

Abstract

The present article is concerned with the use of approximations in the calculation of the many-body density of states (MBDS) of a system with total energy E, composed by N bosons. In the mean-field framework, an integral expression for MBDS, which is proper to be performed by asymptotic expansions, can be derived. However, the standard second order steepest descent method cannot be applied to this integral when the ground-state is sufficiently populated. Alternatively, we derive a uniform formula for MBDS, which is potentially able to deal with this regime. In the case of the one-dimensional harmonic oscillator, using results found in the number theory literature, we show that the uniform formula improves the standard expression achieved by means of the second order method.