XXI.—The Statistical Theory of Stiff Chains

Abstract

SynopsisThe paper is concerned with the distributional properties of Markoff chains in two and three dimensions where the transition probability for the length of a step and its orientation relative to that of the previous step is specified.The discrete two-dimensional chain of n steps is first discussed, and by the use of moving axes an equation relating characteristic functions of the end-point distribution for successive values of n is obtained. The corresponding differential equation for the limiting chain with continuous first derivatives is given and asymptotic solutions for long chains are found.The three-dimensional chain is similarly treated in terms of moving axes, and the limiting continuous chain is again discussed. Finally the same methods are applied to the discrete chain of equal steps to obtain the asymptotic form of the end-point distribution for long chains.