Numerical solutions for brownian motion of particles in a periodic potential

Abstract

The Kramers equation for one dimensional motion of brownian particles in a multiwell cosine potential is solved by a series expansion method followed by numerical solution of the resulting set of linear differential equations to obtain angular velocity and orientational autocorrelation functions (a.c.f.s) in the Laplace domain. The former, inverted by analytic function fitting, take the correct forms at zero and large (harmonic limit) potentials and intermediate behaviours compare well with those found using other numerical techniques. Solutions evaluated using delta function initial conditions are of interest in the study of pendula or of chemical dissociation, while solutions evaluated under equilibrium initial conditions and with imaginary Laplace variable iw are important in dielectric spectroscopy. Application of the theory both to superionic conductivity and dielectric-far-I.R. absorption of rotator phases and liquids is demonstrated.