**Author**: G. Gallavotti,
J.L. Lebowitz, V. Mastropietro

**Title**: *
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.

**Key words: **Large deviations, Quantum
Statistics, Fermi statistics, Bose statistics

Giovanni Gallavotti

Fisica, Universita' di Roma 1

P.le Moro 2

00185 Roma, Italia

tel +39-06-4991-4370

fax +39-06-4957697

em giovanni.gallavotti@roma1.infn.it

http://ipparco.roma1.infn.it

Joel L. Lebowitz

Rutgers University

110 Frelinghuysen Rd.

Piscataway, NJ 08854, USA

Vieri Mastropietro

Universita' di Roma 2, Dip. Matematica

Via Fontanile di Carcaricola

00133 Roma, Italia