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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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first_indexed 2023-03-27T17:05:12Z
last_indexed 2023-04-14T03:23:45Z
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spelling v2 63034 2023-03-27 Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed f318e4c186ab19e3d3d3591a2e075d03 0000-0003-4862-427X Zeev Sobol Zeev Sobol true false 2023-03-27 SMA 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. Journal Article Turkish Journal of Mathematics 47 3 934 948 The Scientific and Technological Research Council of Turkey (TUBITAK-ULAKBIM) - DIGITAL COMMONS JOURNALS 1300-0098 Nonlinear partial differential equations, degenerate parabolic equations, Einstein paradigm, finite propagation speed 17 3 2023 2023-03-17 10.55730/1300-0098.3404 http://dx.doi.org/10.55730/1300-0098.3404 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University Not Required 2023-04-25T12:12:18.1622290 2023-03-27T17:48:00.1377513 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics AKIF IBRAGIMOV 1 Zeev Sobol 0000-0003-4862-427X 2 ISANKA HEVAGE 3 63034__27028__5a8df7544a7a470db16eaed2f0b70fff.pdf 63034.pdf 2023-04-13T11:19:57.1264763 Output 319843 application/pdf Version of Record true This work is licensed under a Creative Commons Attribution 4.0 International License true eng https://creativecommons.org/licenses/by/4.0/
title Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed
spellingShingle Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed
Zeev Sobol
title_short Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed
title_full Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed
title_fullStr Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed
title_full_unstemmed Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed
title_sort Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed
author_id_str_mv f318e4c186ab19e3d3d3591a2e075d03
author_id_fullname_str_mv f318e4c186ab19e3d3d3591a2e075d03_***_Zeev Sobol
author Zeev Sobol
author2 AKIF IBRAGIMOV
Zeev Sobol
ISANKA HEVAGE
format Journal article
container_title Turkish Journal of Mathematics
container_volume 47
container_issue 3
container_start_page 934
publishDate 2023
institution Swansea University
issn 1300-0098
doi_str_mv 10.55730/1300-0098.3404
publisher The Scientific and Technological Research Council of Turkey (TUBITAK-ULAKBIM) - DIGITAL COMMONS JOURNALS
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
url http://dx.doi.org/10.55730/1300-0098.3404
document_store_str 1
active_str 0
description 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.
published_date 2023-03-17T12:12:17Z
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score 11.035634

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