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Energy fluctuation and discontinuity of specific heat

Specific heat per particle ($c_v$) of an ideal gas, in many occasions, is interpreted as energy fluctuation per particle ($\triangleε^2$) of the ideal gas through the relation: $\triangleε^2=kT^2c_v$, where $k$ is the Boltzmann constant and $T$ is the temperature. This relationship is true only in the classical limit, and deviates significantly in the quantum degenerate regime. We have analytically explored quantum to classical crossover of this relationship, in particular, for 3-D free Bose and Fermi gases. We also have explored the same for harmonically trapped cases. We have obtained a hump of $\triangleε^2/kT^2c_v^{(\text{cl})}$ around the condensation point for 3-D harmonically trapped Bose gas. We have discussed the possibility of occurring phase transition with discontinuity of heat capacity from existence of such a hump for other Bose and Fermi systems.