Matrix theory of correlations in a lattice. Part I

Abstract

The statistical mechanics of some crystalline systems may be reduced to statistical correlations between objects which are the unit cells of a fictitious lattice. The correlations are deduced from postulates according to which some configurations of the cells are incompatible with some configurations of the neighbouring cells; if, on the other hand, configurations of neighbours are compatible with each other, their probabilities are to combine by multiplication. By these postulates matrices are implicitly defined such that the probability distribution for a chain of cells is found by forming the powers of a matrix. A similar approach to the statistics of a lattice involves infinite matrices. It does not seem practicable to give explicit expressions for these matrices. If appropriate conditions are complied with, the correlations in a chain are accounted for by adjusting the mean probability coefficients of the cells and for the rest regarding the cells as statistically independent. In this case the infinite matrices may be replaced by the outer power of finite matrices. As result an equation is given by means of which the thermodynamical energy may be calculated as function of temperature.