An old efficient approach to anomalous Brownian motion - Condensed Matter > Statistical MechanicsReport as inadecuate




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Abstract: A number of random processes in various fields of science is described byphenomenological equations containing a stochastic force, the best knownexample being the Langevin equation LE for the Brownian motion BM ofparticles. Long ago Vladimirsky 1942 proposed a simple method for solvingsuch equations. The method, based on the classical Gibbs statistics, consistsin converting the stochastic LE into a deterministic one, and is applicable tolinear equations with any kind of memory. When the memory effects are takeninto account in the description of the BM, the mean square displacement of theparticle at long times can exhibit an -anomalous- different from that in theEinstein theory time dependence. In the present paper we show how some generalproperties of such anomalous BM can be easily derived using the Vladimirskyapproach. The method can be effectively used in solving many of the problemscurrently considered in the literature. We apply it to the description of theBM when the memory kernel in the Volterra-type integro-differential LEexponentially decreases with the time. The problem of the hydrodynamic BM of acharged particle in an external magnetic field is also solved.



Author: V. Lisy, J. Tothova

Source: https://arxiv.org/







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