Eigenvalues and eigenfunctions of the Kramers equation. Application to the Brownian motion of a pendulum

Abstract

The Brownian motion of s particle in a potential is described by a Fokker-Planck equation with two variables (position and velocity), i.e. by the Kramers equation. Eigenvalues and eigenfunctions can be obtained by a matrix continued fraction method. This method is applied to the Brownian motion of a pendulum, where the force is proportional to sin x. The real and complex eigenvalues with small real parts are plotted as a function of the damping constant. In the low friction limit the real parts of the complex eigenvalues are proportional to the square of the friction constant, whereas the real eigenvalues are proportional to the friction constant itself. In the high friction limit the eigenvalues are real and the lowest ones are proportional to the inverse friction constant. In the intermediate friction regime a complicated mixture of real and complex eigenvalues is found. The same method is also applied to the Boltzmann equation with the BGK collision operator.