Computing the entropy can be done in a similar way. As the density operator has the same form as the one describing the thermal equilibrium of an ideal gas, we can use for a system described by the formulas obtained for the entropy of a system without interactions.
β. One particle, reduced density operator
Let us compute the average value of with the density operator :
where the distribution function fβ is noted for fermions, and for bosons:
(39)
When the system is described by the density operator the average populations of the individual states are therefore determined by the usual Fermi-Dirac or Bose-Einstein distributions. From now on, and to simplify the notation, we shall write simply |θk〉 for the kets .
We can introduce a “one-particle reduced density operator” (1) by2:
where the 1 enclosed in parentheses and the subscript 1 on the left-hand side emphasize we are dealing with an operator acting in the one-particle state space (as opposed to that acts in the Fock space); needless to say, this subscript has nothing to do with the initial numbering of the particles, but simply refers to any single particle among all the system particles. The diagonal elements of (1) are the individual state populations. With this operator, we can compute the average value over of any one-particle operator :
(41)
as we now show. Using the expression (B-12) of Chapter XV for any one-particle operator3, as well as (38), we can write:
As we shall see, the density operator (1) is quite useful since it allows obtaining in a simple way all the average values that come into play in the Hartree-Fock computations. Our variational calculations will simply amount to varying (1). This operator presents, in a certain sense, all the properties of the variational density operator chosen in (28) in the Fock space. It plays the same role4 as the projector PN (which also represents the essence of the variational N-particle ket) played in Complement EXV. In a general way, one can say that the basic principle of the Hartree-Fock method is to reduce the binary correlation functions of the system to products of single-particle correlation functions (more details on this point will be given in § 2-b of Complement CXVI).
The average value of the operator for the total particle number is written:
Both functions and increase as a function of μ and, for any given temperature, the total particle number is controlled by the chemical potential. For a large physical system whose energy levels are very close, the orbital part of the discrete sum in (44) can be replaced by an integral. Figure 1 of Complement BXV shows the variations of the Fermi-Dirac and Bose-Einstein distributions. We also mentioned that for a boson system, the chemical potential could not exceed the lowest value e0 of the energies el; when it approaches that value, the population of the corresponding level diverges, which is the Bose-Einstein condensation phenomenon we will come back to in the next complement. For fermions, on the other hand, the chemical potential has no upper boundary, as, whatever its value, the population of states having an energy lower than μ cannot exceed 1.
ϒ. Two particles, distribution functions
We now consider an arbitrary two-particle operator Ĝ and compute its average value with the density operator . The general expression of a symmetric two-particle operator
