One-Dimensional Cooperative Kinetic Model. Equilibrium Solution for Finite Chains

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Abstract
In connection with a cooperative kinetics on linear lattices, the well‐known equilibrium solution for the infinite Ising model is readily derived by the use of detailed balancing. The natural variables in this approach are conditional probabilities. For nearest‐neighbor interactions and a two‐state model, the independent variables are two conditional probabilities for doublets. The simplicity of the procedure suggests various generalizations. We consider a finite chain where the above probabilities become functions of location within the chain. They are obtained as the solution of a set of elementary difference equations. A second such set then yields the singlet frequencies as a function of position and thus determines the frequency of any sequence as a function of location, length, and composition. These quantities are also required in connection with solutions of the coupled kinetic equations. Our results lead to expressions derived by others by explicit use of a partition function, evaluated by matrix methods.