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Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed

AKIF IBRAGIMOV, Zeev Sobol Orcid Logo, ISANKA HEVAGE

Turkish Journal of Mathematics, Volume: 47, Issue: 3, Pages: 934 - 948

Swansea University Author: Zeev Sobol Orcid Logo

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DOI (Published version): 10.55730/1300-0098.3404

Abstract

The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion, leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix is de...

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Published in: Turkish Journal of Mathematics
ISSN: 1300-0098
Published: The Scientific and Technological Research Council of Turkey (TUBITAK-ULAKBIM) - DIGITAL COMMONS JOURNALS 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa63034
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Abstract: The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion, leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix is dependent on the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentrations. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality, and the Ladyzhenskaya-Ural’tseva iteration lemma.
Keywords: Nonlinear partial differential equations, degenerate parabolic equations, Einstein paradigm, finite propagation speed
College: Faculty of Science and Engineering
Issue: 3
Start Page: 934
End Page: 948

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