On the Ising problem and Mayer's cluster sums

Abstract

When Mayer’s imperfect-gas formalism is applied to the Ising problem emphasis is focused on certain irreducible cluster sums, β k . It is shown that there is no contribution to β k unless the k +1 systems concerned occupy a set of lattice sites which are themselves either ( a ) a single point, ( b ) a pair of neighbouring points or ( c ) multiply-connected on the lattice. Each β k is the sum of such contributions. The familiar quasi-chemical approximation results from neglecting all contributions to the β k from multipty-connected sets of occupied lattice sites. A second approximation, involving the smallest of such multiply-connected sets of occupied sites, can always be carried through explicitly.