Published In
American Journal of Physics
Document Type
Article
Publication Date
8-1-2009
Subjects
Langevin equations, Random variables, Brownian motion processes
Abstract
The energy of a mechanical system subjected to a random force with zero mean increases irreversibly and diverges with time in the absence of friction or dissipation. This random heating effect is usually encountered in phenomenological theories formulated in terms of stochastic differential equations, the epitome of which is the Langevin equation of Brownian motion. We discuss a simple discrete impulsive model that captures the essence of random heating and Brownian motion. The model may be regarded as a discrete analog of the Langevin equation, although it is developed ab initio. Its analysis requires only simple algebraic manipulations and elementary averaging concepts, but no stochastic differential equations (or even calculus). The irreversibility in the model is shown to be a consequence of a natural causal stochastic condition that is closely analogous to Boltzmann's molecular chaos hypothesis in the kinetic theory of gases. The model provides a simple introduction to several ostensibly more advanced topics, including random heating, molecular chaos, irreversibility, Brownian motion, the Langevin equation, and fluctuation-dissipation theorems.
DOI
10.1119/1.3213526
Persistent Identifier
http://archives.pdx.edu/ds/psu/7665
Citation Details
Ramshaw, J. D. (2010). A discrete impulsive model for random heating and Brownian motion. American Journal Of Physics, 78(1), 9-13.
Description
Copyright 2010 American Association of Physics Teachers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Association of Physics Teachers. The following article appeared in American Journal of Physics, 78(1), 9-13.