Analysis of the Linear Cooperative Problem as a Markoff Process

Abstract

The equilibrium statistical structure of a linear chain of interacting elements, each capable of $\alpha$ states, is determined by utilizing the theory of the entropy of a Markoff process to express the free energy of the chain as a function of its transition probabilities. Minimization of the free energy then leads to equations for the transition probabilities. The condition that these equations have a nontrivial solution permits the determination of the free energy from the largest eigenvalue of a certain matrix A, already familiar in the matrix method of cooperative phenomena. The corresponding eigenvector (together with the elements $A_{\mathrm{ij}}$) is shown to determine the statistical structure of the chain. Previous discussion of the cooperative chain, in terms of a Markoff process, have been based on the assumption that the transition probabilities are given by normalized Boltzmann factors. This assumption is shown to be erroneous for most of the cases to which it has been applied.