Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature. (English) Zbl 1457.82392
Summary: 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.
MSC:
| 82D05 | Statistical mechanics of gases |
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