On the direct calculation from the Langevin equation of the Kerr effect and higher-order nonlinear responses of an assembly of dipolar molecules

Abstract

It is shown how the Kerr effect and higher-order nonlinear responses of an assembly of non-interacting dipolar molecules in the presence of a driving field may be directly calculated from the Langevin equation thus bypassing the usual method of solution based on the Smoluchowski equation entirely. It is also shown that the hierarchy of differential-difference equations for the ensemble averages which arises out of the averaging procedure for the Langevin equation is identical to that obtained by separating the variables in the Smoluchowski equation. The calculation of the transition matrix for this hierarchy when written as an infinite set of ordinary simultaneous differential equations is identical to finding the eigenvalues of the Smoluchowski equation where the solution of that equation may be treated as a Sturm-Liouville problem. The averaging procedure for the Langevin equation effectively replaces that nonlinear equation by an infinite set of linear differential equations for the average values. If inertial effects are included it is shown that the averaged Langevin equation is identical to Brinkman's hierarchy of differential-difference equations obtained by separating the variables in the Fokker-Planck-Kramers equation.