Abstract
Random walk processes are an important class of stochastic processes. They have many applications in physics, computer science, ecology, economics, and other fields. A random walk is a sequence of successive random steps. In this chapter we study Markovian discrete time models. The time evolution of a system is described in terms of a N-dimensional vector, which can be, for instance, the position of a molecule in a liquid or the price of a fluctuating stock. In a first experiment we compare one-dimensional walks with constant or varying step sizes. The mean square distance is compared to the central limit theorem. A three-dimensional walk is used to simulate the freely jointed chain model in polymer physics. In a computer experiment we calculate the gyration tensor which is relevant to scattering experiments. We discuss the simplified Hookean spring model to simulate the force–extension relation. In a further experiment we study Brownian motion in a harmonic potential.
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References
K. Pearson, The problem of the Random Walk. Nature 72, 294 (1905)
A.A. Markov, in Theory of Algorithms [Translated by Jacques J. Schorr-Kon and PST staff] Imprint (Academy of Sciences of the USSR, Moscow, 1954) [Jerusalem, Israel Program for Scientific Translations, 1961; available from Office of Technical Services, United States Department of Commerce] Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, vol. 42. Original title: Teoriya algorifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60–51085.]
A.A. Markov, in Extension of the limit theorems of probability theory to a sum of variables connected in a chain reprinted in Appendix B ed. by R. Howard. Dynamic Probabilistic Systems, vol. 1 (Wiley, Markov Chains, 1971)
P. Fluekiger, H.P. Luethi, S. Portmann, J.Weber, MOLEKEL 4.0 (Swiss National Supercomputing Centre CSCS, Manno, Switzerland, 2000)
W.L. Mattice, U.W. Suter, Conformational Theory of Large Molecules (Wiley Interscience, New York, NY, 1994). ISBN 0-471-84338-5
R. Brown, Phil. Mag. 4, 161 (1828)
A. Einstein, Investigations on the Theory of Brownian Movement (Dover, New York, NY, 1956)
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© 2010 Springer-Verlag Berlin Heidelberg
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Scherer, P.O. (2010). Random Walk and Brownian Motion. In: Computational Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13990-1_14
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DOI: https://doi.org/10.1007/978-3-642-13990-1_14
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