## Bose-Einstein Condensation of a Gaussian Random Field in the Thermodynamic Limit    [PDF]

Philippe Mounaix, Satya N. Majumdar, Abhimanyu Banerjee
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).
View original: http://arxiv.org/abs/1111.3229