We investigate the probability distribution of the quantum fluctuations of
thermodynamic functions of finite, ballistic, phase-coherent Fermi gases.
Depending on the chaotic or integrable nature of the underlying classical
dynamics, on the thermodynamic function considered, and on temperature, we find
that the probability distributions are dominated either (i) by the local
fluctuations of the single-particle spectrum on the scale of the mean level
spacing, or (ii) by the long-range modulations of that spectrum produced by the
short periodic orbits. In case (i) the probability distributions are computed
using the appropriate local universality class, uncorrelated levels for
integrable systems and random matrix theory for chaotic ones. In case (ii) all
the moments of the distributions can be explicitly computed in terms of
periodic orbit theory, and are system-dependent, non-universal, functions. The
dependence on temperature and number of particles of the fluctuations is
explicitly computed in all cases, and the different relevant energy scales are
displayed.Comment: 24 pages, 7 figures, 5 table