Quantum effects in the system of Boltzmann hard spheres

The quantum contribution to the energy of the Boltzmann gas of solid spheres turns out to be practically constant to the highest temperatures, when the De Broglie thermal wavelength is only a small fraction of the diameter of the solid sphere. Accordingly, the heat capacity of the system is not much different from the classical value of 3/2 k everywhere except for the region of the lowest temperatures, where the dependence of the heat capacity of the system on temperature has a Debye appearance, but with a very low "Debye" temperature of the order of several degrees.
The quantum crystal-liquid equilibrium line of the “Boltzmann” system of hard spheres coincides with the classical one except for the region of very low temperatures. In the system under consideration, high-temperature quantum effects appear in a kind of “naked” form, while in the case of more realistic systems or models they can be masked by the complex behavior of the other components of the total energy.