Computational Cost of Brownian Motion
نویسنده
چکیده
Brownian motion, or random motion in some number of dimensions, occurs frequently in the study of particle theory and fractals. It was first observed as pollen moving in a fluid in 1827 by Robert Brown and formalized mathematically in 1905 by Albert Einstein. Brownian motion has applications in biology, biophysics, cellular biology, and stellar dynamics. Numerous algorithms have been created that claim to generate Brownian motion on a computer, but the inherent computational complexities and empirical accuracy of these models have not been discussed. An accurate and cheap model could serve to solve open problems such as the narrow escape problem in biology, or used to simulate large scale Brownian motion that occurs when large stellar bodies respond to gravitational forces from nearby bodies. In this research, two algorithms used to generate Brownian motion, brown noise developed by C.W. Gardiner and midpoint displacement developed by Fournier, Fussell and Carpenter at SIGGRAPH 1982, are implemented and assessed. Programs to generate the motion in one and two dimensions have been implemented and assessed via convergence in mean and variance utilizing several different GNUplot functions. This research also addresses the suitable number of samples to create a normal standard distribution from a uniform distribution using the Central Limit Theorem, and draws a relationship between Einstein’s equation and the mathematical definition of Brownian motion. The experimental results from this work confirm the relationships defined by Einstein, and shows that the midpoint displacement algorithm has a smaller mean square displacement versus brown noise.
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