A numerical approach to study the Kramers equation for finite geometries: boundary conditions and potential fields

Abstract

The Kramers equation for the phase-space function, which models the dynamics of an underdamped Brownian particle, is the subject of our study. Numerical solutions of this equation for natural boundaries (unconfined geometries) have been well reported in the literature. But not much has been done on the Kramers equation for finite (confining) geometries which require a set of additional constraints imposed on the phase-space function at physical boundaries. In this paper we present numerical solutions for the Kramers equation with a variety of potential fields—namely constant, linear, harmonic and periodic—in the presence of fully absorbing and fully reflecting boundary conditions (BCs). The choice of the numerical method and its implementation take into consideration the type of BCs, in order to avoid the use of ghost points or artificial conditions. We study and assess the conditions under which the numerical method converges. Various aspects of the solutions for the phase-space function are presented with figures and discussed in detail.