A drifting Markov process on the circle, with physical applications

Abstract

A discrete-time Markov process for directions in the plane is introduced. The random direction at any time is influenced both by the direction at the preceding time and by a target direction that depends on the current time. The deviations of the directions from their targets have a limiting distribution as time tends to infinity, and asymptotic approximations to the limiting probability density function are given both for the case of weak dependence and for the case of strong dependence. Tests of independence and of independence of increments are presented. Various types of behaviour of the process are illustrated by simulations. Some physical applications, notably concerning the commensurate versus incommensurate nature of solid materials, are discussed. The application of the process is illustrated also by the analysis of some data on semi-diurnal wind directions.