Microcanonical temperature for a classical field: application to Bose-Einstein condensation

TL;DR

This paper develops a rigorous method to determine the temperature and chemical potential of a classical Bose field using the projected Gross-Pitaevskii equation, revealing how interactions affect the critical temperature for Bose-Einstein condensation.

Contribution

It introduces an exact mapping of the PGPE to Hamilton's equations and adapts statistical mechanics techniques to measure temperature and chemical potential in classical Bose fields.

Findings

The method agrees with previous temperature measurement techniques.

The critical temperature increases with interaction strength in a lattice Bose gas.

The temperature shift in the continuum limit matches Monte Carlo results.

Abstract

We show that the projected Gross-Pitaevskii equation (PGPE) can be mapped exactly onto Hamilton's equations of motion for classical position and momentum variables. Making use of this mapping, we adapt techniques developed in statistical mechanics to calculate the temperature and chemical potential of a classical Bose field in the microcanonical ensemble. We apply the method to simulations of the PGPE, which can be used to represent the highly occupied modes of Bose condensed gases at finite temperature. The method is rigorous, valid beyond the realms of perturbation theory, and agrees with an earlier method of temperature measurement for the same system. Using this method we show that the critical temperature for condensation in a homogeneous Bose gas on a lattice with a UV cutoff increases with the interaction strength. We discuss how to determine the temperature shift for the Bose gas in the continuum limit using this type of calculation, and obtain a result in agreement with more sophisticated Monte Carlo simulations. We also consider the behaviour of the specific heat.