Bose-Einstein Condensation of a Gaussian Random Field in the Thermodynamic Limit

TL;DR

This paper establishes criteria for Bose-Einstein condensation of Gaussian fields in any dimension, revealing two types of condensates and analyzing their properties, including at the critical transition point.

Contribution

It derives the BEC criterion for Gaussian fields based on covariance functions and characterizes the nature of condensates, including normal and anomalous types, across dimensions.

Findings

Existence of BEC for certain covariance functions in all dimensions.

Identification of Gaussian and non-Gaussian condensates.

Analysis of fluctuations at the transition point.

Abstract

We derive the criterion for the Bose-Einstein condensation (BEC) of a Gaussian field $\phi$ (real or complex) in the thermodynamic limit. The field is characterized by its covariance function and the control parameter is the intensity $u=\|\phi\|_2^2/V$, where $V$ is the volume of the box containing the field. We show that for any dimension $d$ (including $d=1$), there is a class of covariance functions for which $\phi$ exhibits a BEC as $u$ is increased through a critical value $u_c$. In this case, we investigate the probability distribution of the part of $u$ contained in the condensate. We show that depending on the parameters characterizing the covariance function and the dimension $d$, there can be two distinct types of condensate: a Gaussian distributed "normal" condensate with fluctuations scaling as $1/\sqrt{V}$, and a non Gaussian distributed "anomalous" condensate. A detailed analysis of the anomalous condensate is performed for a one-dimensional system ($d=1$). Extending this one-dimensional analysis to exactly the point of transition between normal and anomalous condensations, we find that the condensate at the transition point is still Gaussian distributed but with anomalously large fluctuations scaling as $\sqrt{\ln(L)/L}$, where $L$ is the system length. The conditional spectral density of $\phi$, knowing $u$, is given for all the regimes (with and without BEC).