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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.10682 |
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| _version_ | 1866929431027646464 |
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| author | Hevage, Isanka Garli Ibraguimov, Akif Sobol, Zeev |
| author_facet | Hevage, Isanka Garli Ibraguimov, Akif Sobol, Zeev |
| contents | Our Recent advancements in stochastic processes have illuminated a paradox associated with the Einstein model of Brownian motion. The model predicts an infinite propagation speed, conflicting with the second law of thermodynamics. The modified model successfully resolves the issue, establishing a finite propagation speed by introducing a concentration-dependent diffusion matrix. In this paper, we outline the necessary conditions for this property through a counter-example. The second part of the paper focuses on the stability analysis of the solution of the degenerate Einstein model. We introduce a functional dependence on the solution that satisfies a specific ordinary differential inequality. Our investigation explores the solution'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 within bounded domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10682 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability Analysis of Degenerate Einstein Model of Brownian Motion Hevage, Isanka Garli Ibraguimov, Akif Sobol, Zeev Analysis of PDEs Probability Classical Physics Fluid Dynamics 35K65, 76R50, 35K92, 35C06 Our Recent advancements in stochastic processes have illuminated a paradox associated with the Einstein model of Brownian motion. The model predicts an infinite propagation speed, conflicting with the second law of thermodynamics. The modified model successfully resolves the issue, establishing a finite propagation speed by introducing a concentration-dependent diffusion matrix. In this paper, we outline the necessary conditions for this property through a counter-example. The second part of the paper focuses on the stability analysis of the solution of the degenerate Einstein model. We introduce a functional dependence on the solution that satisfies a specific ordinary differential inequality. Our investigation explores the solution'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 within bounded domains. |
| title | Stability Analysis of Degenerate Einstein Model of Brownian Motion |
| topic | Analysis of PDEs Probability Classical Physics Fluid Dynamics 35K65, 76R50, 35K92, 35C06 |
| url | https://arxiv.org/abs/2312.10682 |