A random walk approach to spiral motion

Abstract

The paper deals with random spiral motion described in polar coordinates: the trajectory of a particle is described by the distance r from a given point as a function of the angle theta relative to a given direction. A stochastic description of the spiral shapes generated by the random variations of r and theta is suggested, based on the following assumptions: (a) the ray r is made up of additive contributions corresponding to certain jump processes; (b) the angle phi , between two jumps is a random variable selected from a given probability law g( phi ) d phi , with finite or infinite moments; (c) the contributions rho 1, rho 2, ... of the different jumps to the ray r are independent random variables selected from a probability law p( rho ) d rho with finite moments. An expression for the generating functional of a random spiral r( theta ) is given in the form of an infinite series which can be used to evaluate the stochastic properties of the ray r. The asymptotic properties of the random spiral shapes depend on the function g( phi ) d phi : if all moments of the angle between two jumps exist and are finite, the average shape is a linear Archimedean spiral (r( theta )) approximately theta as theta to infinity , and the dispersion of the ray increases linearly with the angle ( Delta r2( theta )) approximately theta as theta to infinity . If g( phi ) has a long tail g( phi ) approximately phi -(1+H) as phi to infinity with 1>H>0, the average shape is a nonlinear (fractal) Archimedean spiral (r( theta )) approximately theta H as phi to infinity and the fluctuations of the ray have an intermittent behaviour ( Delta r2( theta )) approximately theta 2H as phi to infinity . A complete analysis is possible in a Markovian-like case for which the angle between two jumps is exponentially distributed. In this case a closed expression tor the generating functional is available and all cumulants of the ray can be computed exactly: the mth cumulant which expresses the correlations among the rays at different angles theta 1, ..., theta m is proportional to the minimum angle min ( theta 1, ..., theta m) and the intermittent behaviour is missing. An alternative stochastic description is suggested based on the assumption that the number of jumps in a given angle interval is distributed according to an inhomogeneous Poisson law. This model is also analytically tractable; it is less restrictive in the sense that the average shape corresponds to a broader class of spirals including the linear and the Fractal Archimedean and the logarithmic ones; however, it cannot be used to describe the intermittent behaviour.