Study of the correlated walks with reflecting walls

摘要

Exact expressions for the arrival probabilities with direction are obtained for correlated walks on an infinite line. The probability distribution exhibits a diffusive maximum, similar to that characteristic of random walks,andarunawaycomponentwhich is associated with free passage (no scattering). For symmetric step probabilities, the arrival probabilities for a finite line bounded by reflecting walls are expressible in terms of freehyphen;space probabilities. The evolution of the system of probabilities is studied in terms of the BoltzmannHfunction. The system approaches equilibrium monotonically. In general, there exists an optimum degree of correlation between successive steps at which the randomization in space and direction proceeds most rapidly. At lower correlation the system moves like a wave packet with dissipation. The randomization in space is aided by the reflecting walls (and by the periodic boundary).