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The nonzero minimum of the diffusion parameter and the uncertainty principle for a Brownian particle

Journal article
Authors Eugen Mamontov
Magnus Willander
Published in Modern Physics Letters B (MPLB)
Volume 16
Issue 13
Pages 467 - 471
ISSN 0217-9849
Publication year 2002
Published at Department of Physics (GU)
Pages 467 - 471
Language en
Links dx.doi.org/10.1142/S021798490200402...
Subject categories Applied mathematics, Mesoscopic physics, Mathematical physics, Statistical physics

Abstract

The limits of applicability of many classical (non-quantum-mechanical) theories are not sharp. These theories are sometimes applied to the problems which are, in their nature, not very well suited for that. Two of the most widely used classical approaches are the theory of diffusion stochastic process and Itô's stochastic differential equations. It includes the Brownian-motion treatment as the basic particular case. The present work shows that, for quantum-mechanical reasons, the diffusion parameter of a Brownian particle cannot be arbitrarily small since it has a nonzero minimum value. This fact leads to the version of Heisenberg's uncertainty principle for a Brownian particle which is obtained in the precise mathematical form of a limit inequality. These quantitative results can help to properly apply the theories associated with Brownian-particle modelling. The consideration also discusses a series of works of other authors.

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