Extension of the path-probability method beyond the pair approximation. Triangle approximation

The path-probability method of irreversible statistical mechanics is applied to a system undergoing a first-order order-disorder transformation. For this purpose, the path-probability method has been worked out for the first time in a degree of approximation higher than the pair approximation. The simplest case, i.e., the triangle approximation for an $A_{2}B$-type two-dimensional system, is chosen to show the new features and complexities which arise. The treatment is compared with the previously published derivation for the pair approximation. It is shown that a higher degree of approximation than the pair approximation can be worked out successfully, that the new approximation essentially retains the previously emphasized advantages of the path-probability method, and that calculations with the triangle approximation can be carried out routinely. The extension of the degree of approximation to the tetrahedron approximation, which is appropriate for fcc alloys, is discussed.