# Kinetic energy of a trapped Fermi gas at finite temperature

## TL;DR

This paper derives an exact distribution of the kinetic energy for a finite number of non-interacting fermions in a 1D harmonic trap at any temperature, revealing quantum and thermal fluctuation regimes and their crossover.

## Contribution

It provides an exact solution for the kinetic energy distribution at finite temperature using a mapping to an integrable model, highlighting quantum and thermal regimes.

## Key findings

- Exact distribution of kinetic energy at any temperature and particle number
- Identification of quantum and thermal fluctuation regimes
- Analysis of crossover behavior between regimes

## 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 $T \sim \hbar \omega$, 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 temperature increases.