Higher order corrections to density and temperature of fermions from quantum fluctuations

Abstract

A novel method to determine the density and temperature of a system based on quantum Fermionic fluctuations is generalized to the limit where the reached temperature $T$ is large compared to the Fermi energy ${\ensuremath{\varepsilon}}_{f}$. Quadrupole and particle multiplicity fluctuation relations are derived in terms of $\frac{T}{{\ensuremath{\varepsilon}}_{f}}$. The relevant Fermi integrals are numerically solved for any values of $\frac{T}{{\ensuremath{\varepsilon}}_{f}}$ and compared to the analytical approximations. The classical limit is obtained, as expected, for large temperatures and low densities. The entropy can also be easily derived from quantum fluctuations and gives important insight into the behavior of the system near a phase transition.