The mathematics of Brownian motion and Johnson noise
Abstract
One reason why Brownian motion and Johnson noise are difficult subjects to teach is that their mathematical requirements transcend the capabilities of ordinary differential calculus. Presented here is an exposition of the needed generalization of calculus, namely continuous Markov process theory, in a form that should be accessible to advanced physics undergraduates. It is shown how this mathematical framework enables one to give clear, concise derivations of all the principal results of Brownian motion and Johnson noise, including fluctuationdissipation formulas, autocovariance transport formulas, spectral density formulas, Nyquist's formula, the notions of white and 1/f^{2} noise, and an accurate numerical simulation algorithm. An added benefit of this exposition is a clearer view of the mathematical connection between the two very different approaches to Brownian motion taken by Einstein and Langevin in their pioneering papers of 1905 and 1908.
 Publication:

American Journal of Physics
 Pub Date:
 March 1996
 DOI:
 10.1119/1.18210
 Bibcode:
 1996AmJPh..64..225G
 Keywords:

 05.40.+j