Kinetic energy of a trapped Fermi gas at finite temperature

Abstract

We study the statistics of the kinetic (or equivalently potential) energy for $N$ non-interacting fermions in a $1d$ harmonic trap of frequency $\omega$, at finite temperature $T$. Remarkably, we find an exact solution for the full distribution of the kinetic energy, at any temperature $T$ and for any $N$, using a non-trivial mapping to an integrable Calogero-Moser-Sutherland model. As a function of temperature $T$, and for large $N$, we identify: (i) a quantum regime, for $T\sim \hslash \omega$, where quantum fluctuations dominate and (ii) a thermal regime, for $T\sim N\hslash \omega$, governed by thermal fluctuations. We show how the mean, the variance as well as the large deviation function associated with the distribution of the kinetic energy cross over from the quantum to the thermal regime as temperature increases.

5 pages + 9 pages of supplementary material, 3 figures, typos corrected

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Cited by in corpus (14)

• Noninteracting fermions in a trap and random matrix theory
• The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases
• Survival Probability of Random Walks and Lévy Flights on a Semi-Infinite Line
• Theory of Non-Interacting Fermions and Bosons in the Canonical Ensemble
• Quantum partition of energy for a free Brownian particle: Impact of dissipation
• Non-interacting fermions in hard-edge potentials
• Fluctuations of observables for free fermions in a harmonic trap at finite temperature
• Free fermions and $\alpha$-determinantal processes
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• General truncated linear statistics for the top eigenvalues of random matrices
• Narain transform for spectral deformations of random matrix models
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"