Interpretation of generalized Langevin equations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911077672943616 |
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| author | Sabin-Miller, David Abrams, Daniel M. |
| author_facet | Sabin-Miller, David Abrams, Daniel M. |
| contents | Many real-world systems exhibit ``noisy'' evolution in time; interpreting their finitely-sampled behavior as arising from continuous-time processes (in the Itô or Stratonovich sense) has led to significant success in modeling and analysis in a wide variety of fields. Yet such interpretation hinges on a fundamental linear separation of randomness from determinism in the underlying dynamics. Here we propose some theoretical systems which resist easy and self-consistent interpretation into this well-defined class of equations, requiring an expansion of the interpretive framework. We argue that a wider class of stochastic differential equations, where evolution depends nonlinearly on a random or effectively-random quantity, may be consistently interpreted and in fact exhibit finite-time stochastic behavior in line with an equivalent Itô process, at which point many existing numerical and analytical techniques may be used. We put forward a method for this conversion, and demonstrate its use on both a toy system and on a system of direct physical relevance: the velocity of a meso-scale particle suspended in a turbulent fluid. This work enables the theoretical and numerical examination of a wide class of mathematical models which might otherwise be oversimplified due to a lack of appropriate tools. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03781 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Interpretation of generalized Langevin equations Sabin-Miller, David Abrams, Daniel M. Mathematical Physics Dynamical Systems Probability Adaptation and Self-Organizing Systems Many real-world systems exhibit ``noisy'' evolution in time; interpreting their finitely-sampled behavior as arising from continuous-time processes (in the Itô or Stratonovich sense) has led to significant success in modeling and analysis in a wide variety of fields. Yet such interpretation hinges on a fundamental linear separation of randomness from determinism in the underlying dynamics. Here we propose some theoretical systems which resist easy and self-consistent interpretation into this well-defined class of equations, requiring an expansion of the interpretive framework. We argue that a wider class of stochastic differential equations, where evolution depends nonlinearly on a random or effectively-random quantity, may be consistently interpreted and in fact exhibit finite-time stochastic behavior in line with an equivalent Itô process, at which point many existing numerical and analytical techniques may be used. We put forward a method for this conversion, and demonstrate its use on both a toy system and on a system of direct physical relevance: the velocity of a meso-scale particle suspended in a turbulent fluid. This work enables the theoretical and numerical examination of a wide class of mathematical models which might otherwise be oversimplified due to a lack of appropriate tools. |
| title | Interpretation of generalized Langevin equations |
| topic | Mathematical Physics Dynamical Systems Probability Adaptation and Self-Organizing Systems |
| url | https://arxiv.org/abs/2210.03781 |