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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

REFERENCES TO LITERATURE

Chandrasekhar, S., 1943. “Stochastic Problems in Physics and Astronomy”, Rev. Mod. Phys., xv, 1–89.CrossRefGoogle Scholar
Einstein, A., 1905. “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen”, Ann. der Phys., XVII, 549–560.CrossRefGoogle Scholar
Fürth, R., 1920. “Die Brownsche Bewegung bei Berücksichtigung einer Persistenz der Bewegungsrichtung. Mit Anwendungen auf die Bewegung lebender Infusorien”, Zeits. f. Phys., II, 244–256.CrossRefGoogle Scholar
Kac, M., 1945. “Random Walk in the Presence of Absorbing Barriers”, Ann. Math. Statistics, xvi, 62–67.CrossRefGoogle Scholar
Kac, M., 1947. “Random Walk and the Theory of Brownian Motion”, Amer. Math. Mon., LIV, 369–391.Google Scholar
Ornstein, L. S., 1918. “On the Brownian Motion”, Proc. Kon. Ned. Akad. Wet. Amsterdam, xxi, 96–108.Google Scholar
Prigogine, I., 1947. Étude Thermodynamique des Phénomènes irréversibles, Dunod-Paris.Google Scholar
Rayleigh, Lord, 1919. “On the Problem of Random Vibrations, and of Random Flight in One, Two, or Three Dimensions”, Phil. Mag., XXXVII, 321–347.CrossRefGoogle Scholar
Rayleigh, Lord, Scientific Papers.Google Scholar
Smoluchowski, M. V., 1906. “Zur kinetischen Theorie der Brownschen Mole kularbewegung und der Suspensionen”, Ann. der Phys., xxi, 756–780.Google Scholar
Smoluchowski, M. V., 1923. Ostwalds Klassiker, No. 207, Leipzig.Google Scholar