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XIX.—A Generalization of the Classical Random-walk problem, and a Simple Model of Brownian Motion Based Thereon

Published online by Cambridge University Press:  14 February 2012

G. Klein
G. Klein
Affiliation:
Birkbeck College, London, now at Université Libre de Bruxelles*.
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Synopsis

Suggested by the analogy between the classical one-dimensional random-walk and the approximate (diffusion) theory of Brownian motion, a generalization of the random-walk is proposed to serve as a model for the more accurate description of the phenomenon. Using the methods of the calculus of finite differences, some general results are obtained concerning averages based on a time-varying bivariate discrete probability distribution in which the variates stand in the particular relation of “position” and “velocity.” These are applied to the special cases of Brownian motion from initial thermal equilibrium, and from arbitrary initial kinetic energy. In the latter case the model describes accurately quantized Brownian motion of two energy states, one of zero energy.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 63 , Issue 3 , 1952 , pp. 268 - 279
Copyright
Copyright © Royal Society of Edinburgh 1952

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References

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