Random walks with sign reversals and the intrinsic angular momentum behaviour on lattices

Abstract

Classical uniaxial rotators on lattices are considered. The assumption is made that the rotators exchange angular momentum only in a certain sector of relative orientations (collision sector) because of the finite range of the anisotropic interaction potential; the latter may in particular be a hard core potential (instantaneous collisions). Since the axes of the rotators are supposed to be parallel but not collinear, during each collision the exchange of angular momentum occurs with sign reversal, i.e. is of the cog-wheel type. In a first approximation all correlations between successive collisions are neglected. This amounts to considering a random walk on a lattice in which the walker carries with him a variable whose sign is changed at each step. Then, one is led to distinguish two types of lattices, depending upon the possibility for returning to the initial site after an odd number of steps. For two-dimensional lattices, the asymptotic solution of the autocorrelation function of the angular momentum is analytically given, and its behaviour at shorter times is obtained by a numerical method. For the odd lattices, the asymptotic behaviour is generally exponential, whereas there exist algebraic long tails for the even lattices. Finally, coming back to the problem where the successive collisions are not independent, some simulation results for two-dimensional lattices of soft core diatomics are described.