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Stability Analysis of Degenerate Einstein Model of Brownian Motion

Isanka Hevage Orcid Logo, Akif Ibraguimov Orcid Logo, Zeev Sobol Orcid Logo

American Journal of Applied Mathematics, Volume: 12, Issue: 5, Pages: 118 - 132

Swansea University Author: Zeev Sobol Orcid Logo

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DOI (Published version): 10.11648/j.ajam.20241205.12

Abstract

Recent advancements in stochastic processes have uncovered a paradox associated with the Einstein model of Brownian motion of random particles, which diffuse in the media with no boundary . The classical model developed by Einstein provide diffusion coefficient which does not depend on numbers of pa...

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Published in: American Journal of Applied Mathematics
ISSN: 2330-0043 2330-006X
Published: Science Publishing Group 2024
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Based on this model one can predict the propagation speed of particles movement, conflicting with the second law of thermodynamics. We justify that within Einstein paradigm this issue can be resolved. For that we revisited approach proposed by Einstein, and significantly modified his ideas by introducing inverse Kolmogorov equation, with coefficient degenerating as concentration of the particle of interest vanishes. The modified model successfully resolves paradox affiliated to classical Brownian motion model by introducing a concentration-dependent diffusion matrix, establishing a finite propagation speed. Proposed model utilize but of inverse Kolmogorov stochastic parabolic equation and propose sufficient condition (Hypotheses 1.1) for degeneracy of diffusion coefficient, which guarantee finite speed of propagation inside domain of diffusion. This paper outlines the necessary conditions for this property through a counterexample, which provide infinite speed of propagation for the solution of the equation, with diffusion coefficient, which degenerate as concentration vanishes but with lower speed than in (Hypotheses 1.1). The second part focuses on the stability analysis of the solution of the degenerate Einstein model in case when boundary condition are crucial. We considered degenerate Einstein model in the boundary domain with Dirichlet boundary conditions. Our model bridge degenerate Brownian equation in the bulk of media with boundary of the domain. We with detail investigate stability of the problem with perturbed boundary Data, which vanishes with time. A functional dependence is introduced on the solution that satisfies a specific ordinary differential inequality. 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spelling v2 67945 2024-10-09 Stability Analysis of Degenerate Einstein Model of Brownian Motion f318e4c186ab19e3d3d3591a2e075d03 0000-0003-4862-427X Zeev Sobol Zeev Sobol true false 2024-10-09 MACS Recent advancements in stochastic processes have uncovered a paradox associated with the Einstein model of Brownian motion of random particles, which diffuse in the media with no boundary . The classical model developed by Einstein provide diffusion coefficient which does not depend on numbers of particles(concentration) and does not degenerate. Based on this model one can predict the propagation speed of particles movement, conflicting with the second law of thermodynamics. We justify that within Einstein paradigm this issue can be resolved. For that we revisited approach proposed by Einstein, and significantly modified his ideas by introducing inverse Kolmogorov equation, with coefficient degenerating as concentration of the particle of interest vanishes. The modified model successfully resolves paradox affiliated to classical Brownian motion model by introducing a concentration-dependent diffusion matrix, establishing a finite propagation speed. Proposed model utilize but of inverse Kolmogorov stochastic parabolic equation and propose sufficient condition (Hypotheses 1.1) for degeneracy of diffusion coefficient, which guarantee finite speed of propagation inside domain of diffusion. This paper outlines the necessary conditions for this property through a counterexample, which provide infinite speed of propagation for the solution of the equation, with diffusion coefficient, which degenerate as concentration vanishes but with lower speed than in (Hypotheses 1.1). The second part focuses on the stability analysis of the solution of the degenerate Einstein model in case when boundary condition are crucial. We considered degenerate Einstein model in the boundary domain with Dirichlet boundary conditions. Our model bridge degenerate Brownian equation in the bulk of media with boundary of the domain. We with detail investigate stability of the problem with perturbed boundary Data, which vanishes with time. A functional dependence is introduced on the solution that satisfies a specific ordinary differential inequality. The investigation explores the solution&amp;apos;s dependence on the boundary and initial data of the original problem, demonstrating asymptotic stability under various conditions. These results have practical applications in understanding stochastic processes and its dependence on the boundary Data within bounded domains. Journal Article American Journal of Applied Mathematics 12 5 118 132 Science Publishing Group 2330-0043 2330-006X Stability analysis, degenerate PDEs, particle localization, finite speed of propagation 19 9 2024 2024-09-19 10.11648/j.ajam.20241205.12 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University Another institution paid the OA fee The authors express their sincere appreciation to Dr. Luan Hoang for his immense contribution and rigorous guidance Research of Akif Ibragimov, professor emeritus of TTU and cso of institute of the Oil and Gas, was partially supported through project 122022800272-4. 2024-10-09T11:39:35.4134248 2024-10-09T11:28:40.1973429 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Isanka Hevage 0000-0002-5520-4030 1 Akif Ibraguimov 0000-0001-6827-8007 2 Zeev Sobol 0000-0003-4862-427X 3 67945__32567__772639ea4f294c91b2017086bef81c9b.pdf 67945.VOR.pdf 2024-10-09T11:36:50.1765117 Output 320586 application/pdf Version of Record true © The Author(s), 2024. This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (CC BY 4.0). true eng http://creativecommons.org/licenses/by/4.0/
title Stability Analysis of Degenerate Einstein Model of Brownian Motion
spellingShingle Stability Analysis of Degenerate Einstein Model of Brownian Motion
Zeev Sobol
title_short Stability Analysis of Degenerate Einstein Model of Brownian Motion
title_full Stability Analysis of Degenerate Einstein Model of Brownian Motion
title_fullStr Stability Analysis of Degenerate Einstein Model of Brownian Motion
title_full_unstemmed Stability Analysis of Degenerate Einstein Model of Brownian Motion
title_sort Stability Analysis of Degenerate Einstein Model of Brownian Motion
author_id_str_mv f318e4c186ab19e3d3d3591a2e075d03
author_id_fullname_str_mv f318e4c186ab19e3d3d3591a2e075d03_***_Zeev Sobol
author Zeev Sobol
author2 Isanka Hevage
Akif Ibraguimov
Zeev Sobol
format Journal article
container_title American Journal of Applied Mathematics
container_volume 12
container_issue 5
container_start_page 118
publishDate 2024
institution Swansea University
issn 2330-0043
2330-006X
doi_str_mv 10.11648/j.ajam.20241205.12
publisher Science Publishing Group
college_str Faculty of Science and Engineering
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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
document_store_str 1
active_str 0
description Recent advancements in stochastic processes have uncovered a paradox associated with the Einstein model of Brownian motion of random particles, which diffuse in the media with no boundary . The classical model developed by Einstein provide diffusion coefficient which does not depend on numbers of particles(concentration) and does not degenerate. Based on this model one can predict the propagation speed of particles movement, conflicting with the second law of thermodynamics. We justify that within Einstein paradigm this issue can be resolved. For that we revisited approach proposed by Einstein, and significantly modified his ideas by introducing inverse Kolmogorov equation, with coefficient degenerating as concentration of the particle of interest vanishes. The modified model successfully resolves paradox affiliated to classical Brownian motion model by introducing a concentration-dependent diffusion matrix, establishing a finite propagation speed. Proposed model utilize but of inverse Kolmogorov stochastic parabolic equation and propose sufficient condition (Hypotheses 1.1) for degeneracy of diffusion coefficient, which guarantee finite speed of propagation inside domain of diffusion. This paper outlines the necessary conditions for this property through a counterexample, which provide infinite speed of propagation for the solution of the equation, with diffusion coefficient, which degenerate as concentration vanishes but with lower speed than in (Hypotheses 1.1). The second part focuses on the stability analysis of the solution of the degenerate Einstein model in case when boundary condition are crucial. We considered degenerate Einstein model in the boundary domain with Dirichlet boundary conditions. Our model bridge degenerate Brownian equation in the bulk of media with boundary of the domain. We with detail investigate stability of the problem with perturbed boundary Data, which vanishes with time. A functional dependence is introduced on the solution that satisfies a specific ordinary differential inequality. The investigation explores the solution&amp;apos;s dependence on the boundary and initial data of the original problem, demonstrating asymptotic stability under various conditions. These results have practical applications in understanding stochastic processes and its dependence on the boundary Data within bounded domains.
published_date 2024-09-19T11:39:33Z
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