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 Wigner-Dyson ensembles of random matrices, and fill consecutive single particle levels with n fermions. Assuming that the fermions are non-interacting, we show that the distribution of the total energy is Gaussian and its variance grows as n(2)log n in the large-n limit. Next to leading order corrections are also computed. Some related quantities are discussed, in particular the nearest neighbor spacing autocorrelation function. Canonical and grand canonical approaches are considered and compared in detail. A semiclassical 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 particle energies given by the imaginary part of the zeros of the Riemann zeta function on the critical line. (C) 1999 Elsevier Science B.V. All rights reserved.