BXV Ideal gas in thermal equilibrium; quantum distribution functions
1 1 Grand canonical description of a system without interactions 1-a Density operator 1-b Grand canonical partition function, grand potential
2 2 Average values of symmetric one-particle operators 2-a Fermion distribution function 2-b Boson distribution function 2-c Common expression 2-d Characteristics of Fermi-Dirac and Bose-Einstein distributions
3 3 Two-particle operators 3-a Fermions 3-b Bosons 3-c Common expression
4 4 Total number of particles 4-a Fermions 4-b Bosons
5 5 Equation of state, pressure 5-a Fermions 5-b Bosons
This complement studies the average values of one- or two-particle operators for an ideal gas, in thermal equilibrium. It includes a discussion of several useful properties of the Fermi-Dirac and Bose-Einstein distribution functions, already introduced in Chapter XIV.
To describe thermal equilibrium, statistical mechanics often uses the grand canonical ensemble, where the particle number may fluctuate, with an average value fixed by the chemical potential μ (cf. Appendix VI, where you will find a number of useful concepts for reading this complement). This potential plays, with respect to the particle number, a role similar to the role the inverse of the temperature term β = 1/kBT plays with respect to the energy (kB is the Boltzmann constant). In quantum statistical mechanics, Fock space is a good choice for the grand canonical ensemble as it easily allows changing the total number of particles. As a direct application of the results of §§ B and C of Chapter XV, we shall compute the average values of symmetric one- or two-particle operators for a system of identical particles in thermal equilibrium.
We begin in § 1 with the density operator for non-interacting particles, and then show in §§ 2 and 3 that the average values of the symmetric operators may be expressed in terms of the Fermi-Dirac and Bose-Einstein distribution functions, increasing their application range and hence their importance. In § 5, we shall study the equation of state for an ideal gas of fermions or bosons at temperature T and contained in a volume
1. Grand canonical description of a system without interactions
We first recall how a system of non-interacting particles is described, in quantum statistical mechanics, by the grand canonical ensemble; more details on this subject can be found in Appendix VI, § 1-c.
1-a. Density operator
Using relations (42) and (43) of Appendix VI, we can write the grand canonical density operator ρeq (whose trace has been normalized to 1) as:
where Z is the grand canonical partition function:
In these relations, β = 1/(kBT) is the inverse of the absolute temperature T multiplied by the Boltzmann constant kB, and μ, the chemical potential (which may be fixed by a large reservoir of particles). Operators Ĥ and
Assuming the particles do not interact, equation (B-1) of Chapter XV allows writing the system Hamiltonian Ĥ as a sum of one-particle operators, in each subspace having a total number of particles equal to N:
(3)
Let us call {|uk〉} the basis of the individual states that are the eigenstates of the operator
(4)
where the ek, are the eigenvalues of
We shall now compute the average values of all the one- or two-particle operators for a system described by the density operator (1).
