Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 73

The grand canonical potential of a fermion ideal gas is given by (9). Equation (14) indicates that, for a system at thermal equilibrium, this grand potential is equal to the opposite of the product of the volume and the pressure P. We thus have:

(68)

(where the second equality is valid in the limit of large volumes). Simplifying by , we get the pressure of a fermion system contained in a box of macroscopic dimension:

(69)

with:

(70)

where x has been defined in (53).

To obtain the equation of state, we must find a relation between the pressure P, the volume , and the temperature T of the physical system, assuming the particle number to be fixed. We have, however, used the grand canonical ensemble (cf. Appendix VI), where the temperature is determined by the parameter β and the volume is fixed, but where the particle number can vary: its average value is a function of a parameter, the chemical potential μ (for fixed values of μ and ). Mathematically, the pressure P appears as a function of , T and μ and not as the function of , T and the particle number we were looking for. We can nevertheless vary μ, and obtain values of the pressure and particle number of the system and consequently explore, point by point, the equation of state in this parametric form. To obtain an explicit form of the equation of state would require the elimination of the chemical potential using both (47) and (69); there is generally no algebraic solution, and people just use the parametric form of the equation of state, which allows computing all the possible state variables. There also exists a “virial expansion” in powers of the fugacity eβμ, which allows the explicit elimination of μ at all the successive orders; its description is beyond the scope of this book.