VII. IDEAL FERMI–DIRAC SYSTEMS

VII.1. Density of the states and equation of state. As we have seen at the end of the previous Chapter, the granpotential for an ideal Fermi-Dirac gas of non-interacting molecules with single molecule hamiltonian H ̂ is pro-vided by βΩ(β, µ;a) = Tr ln 1+ exp{βµ − βĤ(a)}; (1.1) here the system is supposed to be enclosed in a vessel of volume V, in thermal equilibrium at temperature T and chemical potential µ ∈ R, in the presence of external fields, collectively labeled by the external parameters a = (a1, a2,..., ar), which actually also include those parameters corresponding to the presence of the repulsive barriers keeping the molecules inside the vessel. Let us also suppose that the spectrum of the one-molecule hamiltonian is bounded from below. According to general thermodynamics, we obtain the average density of molecules-i.e. the grancanonical average of the number of molecules per unit volume- in the form 〈N〉