Large deviations for ideal quantum systems

TL;DR

This paper analyzes the large deviation behavior of particle density fluctuations in ideal quantum systems, establishing the exponential decay of untypical densities and confirming the central limit theorem for small fluctuations.

Contribution

It provides a rigorous proof connecting density fluctuations in quantum gases to thermodynamic potentials via large deviation principles.

Findings

Density fluctuations follow a large deviation principle related to pressure.

Unusual densities occur with exponentially small probability.

Small fluctuations obey the central limit theorem.

Abstract

We consider a general d-dimensional quantum system of non-interacting particles, with suitable statistics, in a very large (formally infinite) container. We prove that, in equilibrium, the fluctuations in the density of particles in a subdomain of the container are described by a large deviation function related to the pressure of the system. That is, untypical densities occur with a probability exponentially small in the volume of the subdomain, with the coefficient in the exponent given by the appropriate thermodynamic potential. Furthermore, small fluctuations satisfy the central limit theorem.