Fluctuations in the Level Density of a Fermi Gas

We present a theory that accurately describes the counting of excited states of a noninteracting fermionic gas. At high excitation energies the results reproduce Bethe’s theory. At low energies oscillatory corrections to the many-body density of states, related to shell effects, are obtained. The fluctuations depend nontrivially on energy and particle number. Universality and connections with Poisson statistics and random matrix theory are established for regular and chaotic single-particle motion.

Figure 1

The logarithm of the many-body density of states of $A$ noninteracting fermions in a rectangular billiard of sides $a=\sqrt{(1+\sqrt{5})/2}$ and $b=a^{-1}$. Dots, numerical computation at three different excitation energies; solid curves, theoretical prediction (15); dashed curves, smooth part Eq. (3).