The mathematics of Brownian motion and Johnson noise
作者:
Daniel T. Gillespie,
期刊:
American Journal of Physics
(AIP Available online 1996)
卷期:
Volume 64,
issue 3
页码: 225-240
ISSN:0002-9505
年代: 1996
DOI:10.1119/1.18210
出版商: American Association of Physics Teachers
关键词: BROWNIAN MOVEMENT;DIFFERENTIAL CALCULUS;FLUCTUATIONS;LANGEVIN EQUATION;MARKOV PROCESS;NOISE;STOCHASTIC PROCESSES;05.40;02.
数据来源: AIP
摘要:
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 fluctuation–dissipation formulas, auto‐covariance transport formulas, spectral density formulas, Nyquist’s formula, the notions of white and 1/f2noise, 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.
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