Two simulation methods of Brownian motion

Abstract

Einstein-Smoluchowski's theory and Langevin's theory are two main theories to describe Brownian motion. The Einstein-Smoluchowski theory regards Brownian motion as random walking of Brownian particles, while the Langevin theory establishes a random differential equation describing the motion of Brownian particles. Both theories involve random numbers, i.e., only the statistical results can be discussed. Based on the above theories, this paper presents the corresponding simulations, which verifies the statistical conclusions as well as compares their similarities, differences, and applicable conditions. These results offer a guideline for further studies on Brownian motion.