One-dimensional Ising model: Kinetic studies

Abstract

We investigate methods to deal with the mathematical diffculties inherent in the general model. Two approaches are taken, namely closure and extrapolation from short to large chains. Exact solutions on short chains indicate that the closure approximation employed earlier for infinite chains is satisfactory under a variety of conditions and should provide a valuable computational technique. Moreover, analytic results are presented, which elucidate the reasons behind the success of the numerical approximations. At present it appears that the extrapolation calculations, employing cyclic chains, were not carried far enough to predict the infinite chain results obtained by the earlier approximate calculations. The rigorous short chain studies are of intrisic interest since they yield rarely available data. They also suggest and provide a check on certain special cases of the general kinetics. In particular, a model where only one nucleation event is allowed, very closely approximates the general results in highly cooperative cyclic chains, at least for small chain lengths. A possible tractable generalization of the single nucleation model is suggested. Copyright © 1974 American Institute of Physics.