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 $N$. As a function of temperature $T$, and for large $N$, we identify: (i) a quantum regime, for $T sim hbaromega$, where quantum fluctuations dominate and (ii) a thermal regime, for $T sim N hbar omega$, 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 $T$ increases. J. Grela, S.N. Majumdar, G. Schehr, Phys. Rev. Lett., 119, 130601 (2017) A. Grabsch, S.N. Majumdar, G. Schehr, C. Texier, SciPost Phys. 4, 014 (2018)

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