Statistical Mechanics for the Effective Hamiltonian

To note the nature of the problem, we begin with a reminder concerning the entropy in the case in which no correlation between the occupancies of adjacent sites is assumed. In this limit, the entropy reduces to that of the ideal entropy already revealed in eqn (3.89). In preparation for the notation that will emerge in our discussion of the cluster variation method, we revisit the analysis culminating in eqn (3.89). Recall from chap. 3 that the entropy of a system characterized by a series of discrete states with probabilities pi is given by 

In the present case, there are 2 such states (i.e. the various occupancies of the N sites of the lattice by A and B atoms), each of which has a probability P cri,02, , on). This notation refers to the probability that site 1 has occupation number site 2 has occupation number 02 and so on. We introduce the alternative notation P oi]), which again signifies the probability of a given state characterized by the set of occupation numbers ai. In light of this more explicit notation, we can rewrite eqn (6.31) as 

Point Approximation. The simplest approximation is to advance the assumption that the probability may be written as P( cTi ) = Piicri)Piicri) TV(o v). Conceptually, what this means is that we have abandoned the treatment of correlations between the occupancies of different sites. As a result, certain configurations are given much higher statistical weight than they really deserve. On the other hand, once this form for the probabilities has been accepted and using the fact that Pj(l) + Pi(—1) = 1, the sum in eqn (6.32) can be simplified to the form 

This is no great surprise. We have recovered the result of eqn (3.89), but now seen as the first in a series of possible approximations. We will denote the present approximation as the point approximation. 

Pair Approximation. At the next level of approximation, we can insist that our probabilities P oi]) reproduce not only site occupancies with correct statistical weight, but also the distribution of AA, AB and BB bonds. We begin from a kinematic perspective by describing the relations between the number of A and B atoms Na and Nb), the number of AA, AB and BB bonds Naa, Nab and Nbb) and the total number of sites, N. Note that we have not distinguished AB and BA bonds. These ideas are depicted in fig. 6.21. What we note is that for our one-dimensional model, every A atom is associated with two bonds, the identity of which must be either AA or AB. Similar remarks can be made for the B atoms. 

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