Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature

TL;DR

This paper analyzes the distribution of the furthest fermions in a trapped Fermi gas, revealing universal Gumbel laws in higher dimensions and Tracy-Widom in one dimension, with extensions to finite temperature.

Contribution

It provides an analytical characterization of the maximal distance and momentum distributions in a trapped Fermi gas, highlighting universality and dimensional dependence.

Findings

In 1D, the maximal distance distribution converges to Tracy-Widom.

In higher dimensions, the distribution converges to Gumbel law.

Results are extended to low finite temperatures.

Abstract

We consider $N$ non-interacting fermions in an isotropic $d$-dimensional harmonic trap. We compute analytically the cumulative distribution of the maximal radial distance of the fermions from the trap center at zero temperature. While in $d=1$ the limiting distribution (in the large $N$ limit), properly centered and scaled, converges to the squared Tracy-Widom distribution of the Gaussian Unitary Ensemble in Random Matrix Theory, we show that for all $d>1$, the limiting distribution converges to the Gumbel law. These limiting forms turn out to be universal, i.e., independent of the details of the trapping potential for a large class of isotropic trapping potentials. We also study the position of the right-most fermion in a given direction in $d$ dimensions and, in the case of a harmonic trap, the maximum momentum, and show that they obey similar Gumbel statistics. Finally, we generalize these results to low but finite temperature.