When and why is the random force in Brownian motion a Gaussian process
Related Experiment Video
Updated: Jul 19, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
1Instituut-Lorentz, University of Leiden, P.O. Box 9506, 2300 RA Leiden, the Netherlands.
Abstract:
It is shown that the assumptions of causality and time-reversal invariance severely restrict the possibility to describe the fluctuations of a variable in a non-linear Markovian system using a Langevin equation. In fact a theorem is proven which implies that with the aforementioned assumptions a Langevin force which is independent of the state of the system is necessarily Gaussian and white. The theorem furthermore implies that such a description is only possible if the socalled "systematic force" is proportional to the derivative of the logarithm of the equilibrium distribution of the variable. Our analysis is given for a system with one variable which may be either even or odd under time reversal.
Related Concept Videos
Gauss's Law
Central-Force Motion
Random Variables
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
For example, let X = the...
Entropy Change in Reversible Processes
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
Distribution of Molecular Speeds
Gauss's Law: Problem-Solving
