The thermalized Fokker–Planck equation

The Fokker–Planck equation for a thermalized ensemble of Brownian particles is exactly solved by a method of (nontruncated) moments, using the expansion of the velocity distribution in Hermite polynomial eigenfunctions. We explicitly studied only the cases of linear and quadratic external potentials bounded by a reflecting wall. The double Laplace transforms (with respect to position and time) of the moments of the velocity distribution were obtained from the generating function, which was found to be a functional only of the density of particles. The closed equation for the density can be derived for the thermalized system. The solution of this equation (obtained by means of a Laplace transform in time) is analyzed in the long and short time limits. The (truncated) moment solution for a nonthermalized problem is also discussed. In the long time limit, the solutions for both thermalized and nonthermalized systems approach the Smoluchowski diffusive approximation.