Higher Order Corrections to Density and Temperature of Fermions from Quantum Fluctuations

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 {\epsilon}f . Quadrupole and particle multiplicity fluctuations relations are derived in terms of T . The relevant Fermi integrals are numerically solved for any values of T and compared to the analytical approximations. The classical limit is obtained, as expected, in the limit of large temperatures and small densities. We propose simple analytical formulas which reproduce the numerical results, valid for all values of T . The entropy can also be easily derived from quantum fluctuations and give important insight for the behavior of the system near a phase transition. A comparison of the quantum entropy to the entropy derived from the ratio of the number of deuterons to neutrons gives a very good agreement especially when the density of the system is very low.