Overdamped and underdamped Langevin equations in the interpretation of experiments and simulations

Abstract

The Brownian motion (BM) is not only a natural phenomenon but also a fundamental concept in several scientific fields. The mathematical description of the BM for students of various disciplines is most often based on Langevin's equation with the Stokes friction force and the random force modeling Brownian particle (BP) collisions with surrounding molecules. For many phenomena, such a description is insufficient, as it assumes an infinitesimal correlation time of random force. This shortcoming is overcome by the generalized Langevin equation (GLE), which is now one of the most widely used equations in physics. In the present work, we offer a simple way of solving this equation, consisting of its transformation into an integro-differential equation for the mean square displacement of the BP, which is then effectively solved using the Laplace transform (LT). We demonstrate the use of this method to solve both the standard Langevin equation and the GLE for the BP in an external harmonic field. We analyze the cases of overdamped (when frictional forces prevail over inertial forces and the BP mass is considered zero in the equation) and underdamped (inertial effects are not neglected) equations. We show under what conditions an overdamped solution can be used instead of complicated solutions of the underdamped equation. We also demonstrate the effectiveness of the use of the LT on a microscopic approach to the derivation of the GLE. Graduate students are offered several problems in which the internal shortcomings of the overdamped Langevin equations manifest themselves.