Large deviations in rarefied quantum gases

Abstract

The probability of observing a large deviation (LD) in the number of particles in a region $\Lambda$ in a dilute quantum gas contained in a much larger region $V$ is shown to decay as $\exp[-|\Lambda|\Delta F]$, where $|\L|$ is the volume of $\Lambda$ and $\Delta F$ is the change in the appropriate free energy density, the same as in classical systems. However, in contrast with the classical case, where this formula holds at all temperatures and chemical potentials our proof is restricted to rarefied gases, both for the typical and observed density, at least for Bose or Fermi systems. The case of Boltzmann statistics with a bounded repulsive potential can be treated at all temperatures and densities. Fermions on a lattice in any dimension, or in the continuum in one dimension, can be treated at all densities and temperatures if the interaction is small enough (depending on density and temperature), provided one assumes periodic boundary conditions.