Energy fluctuations of a finite free-electron Fermi gas

We discuss the energy distribution of free-electron Fermi-gas, a problem with a textbook solution of Gaussian energy fluctuations in the limit of a large system. We find that for a small system, characterized solely by its heat capacity $C$, the distribution can be solved analytically, and it is both skewed and it vanishes at low energies, exhibiting a sharp drop to zero at the energy corresponding to the filled Fermi sea. The results are relevant from the experimental point of view, since the predicted non-Gaussian effects become pronounced when $C/k_{\text{B}}\lesssim {10}^3$ ($k_{\text{B}}$ is the Boltzmann constant), a regime that can be easily achieved for instance in mesoscopic metallic conductors at sub-kelvin temperatures.

Figure 1

Distribution $P\left(u\right)$, where $u=\beta \delta E/\sqrt{c}$, where $\delta E$ is the deviation of the energy of the Fermi-gas from its mean, for a few different values of $c=C/k_{\text{B}}$ (solid lines). Dashed line is the Gaussian distribution.