The mathematics of Brownian motion and Johnson noise

@article{citeulike:2196616, 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 fluctuation\&\#150;dissipation formulas, auto-covariance transport formulas, spectral density formulas, Nyquist's formula, the notions of white and 1/f2 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. \©1996 American Association of Physics Teachers.}, author = {Gillespie, Daniel T. }, citeulike-article-id = {2196616}, doi = {10.1119/1.18210}, journal = {American Journal of Physics}, keywords = {o-u}, number = {3}, pages = {225--240}, posted-at = {2008-01-05 08:53:21}, priority = {2}, publisher = {AAPT}, title = {The mathematics of Brownian motion and Johnson noise}, url = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal\&id=AJPIAS000064000003000225000001\&idtype=cvips\&gifs=yes}, volume = {64}, year = {1996} }

TY - JOUR ID - citeulike:2196616 L3 - citeulike-article-id:2196616 TI - The mathematics of Brownian motion and Johnson noise JF - American Journal of Physics VL - 64 IS - 3 SP - 225 EP - 240 PB - AAPT N2 - 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/f2 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. ©1996 American Association of Physics Teachers. KW - o-u AU - Gillespie, Daniel PY - 1996/// UR - http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000064000003000225000001&idtype=cvips&gifs=yes ER -