Universal Fermi distribution of semiclassical nonequilibrium Fermi states

When a classical device suddenly perturbs a degenerate Fermi gas, a semiclassical nonequilibrium Fermi state arises. Similar states appear when a charge is transferred into a Fermi gas through a localized contact. Semiclassical Fermi states are characterized by a Fermi energy or Fermi momentum that slowly depends on space and/or time. We show that the Fermi distribution of a semiclassical Fermi state has a universal nature. It is described by Airy functions regardless of the details of the perturbation.

Figure 1

Modulated Fermi edge. The Wigner functions are almost 1 below the edge (shaded area) and vanish above the edge, featuring a universal character in the “surf.”

Figure 2

Universal behavior of the Wigner function, Eq. (3).

Figure 3

Universal behavior of the Fermi number, Eq. (4).

Figure 4

Universal asymptotes of the Fermi number. The graph (red/gray curve) shows the Fermi number in units of $\hslash /\ell$ for the example $\langle \rho \left(x\right)\rangle ={\rho }_0+(n/\ell )\cos (x/\ell )$ computed from Eq. (17). Black curves are the asymptotic forms obtained from the universal Fermi number formula (4) depicted and magnified on Fig. 3.