On the distribution of the total energy of a system of non-interacting fermions: random matrix and semiclassical estimates

We consider a single particle spectrum as given by the eigenvalues of the W igner-Dyson ensembles of random matrices, and fill consecutive single parti cle levels with n fermions. Assuming that the fermions are non-interacting, we show that the distribution of the total energy is Gaussian and its vari ance grows as n(2)log n in the large-n limit. Next to leading order correct ions are also computed. Some related quantities are discussed, in particula r the nearest neighbor spacing autocorrelation function. Canonical and gran d canonical approaches are considered and compared in detail. A semiclassic al formula describing, as a function of n, a non-universal behavior of the variance of the total energy starting at a critical number of particles is also obtained. it is illustrated with the particular case of single particl e energies given by the imaginary part of the zeros of the Riemann zeta fun ction on the critical line. (C) 1999 Elsevier Science B.V. All rights reser ved.