Geometric theory of (extended) time-reversal symmetries in stochastic processes -- Part I: finite dimension
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| author | O'Byrne, Jérémy Cates, Michael E. |
| author_facet | O'Byrne, Jérémy Cates, Michael E. |
| contents | In this article, we analyze three classes of time-reversal of a Markov process with Gaussian noise on a manifold. We first unveil a commutativity constraint for the most general of these time-reversals to be well defined. Then we give a triad of necessary and sufficient conditions for the stochastic process to be time-reversible. While most reversibility conditions in the literature require knowledge of the stationary probability, our conditions do not, and therefore can be analytically checked in a systematic way. We then show that the mathematical objects whose cancellation is required by our reversibility conditions play the role of independent sources of entropy production. Furthermore, we give a geometric interpretation of the so-called irreversible cycle-affinity as the vorticity of a certain vector field for a Riemannian geometry given by the diffusion tensor. We also discuss the relation between the time-reversability of the stochastic process and that of an associated deterministic dynamics: its Stratonovitch average. Finally, we show that a suitable choice of a reference measure - that can be considered as a prior or a gauge, depending on the context - allows to study a stochastic process in a way that is both coordinate-free and independent of the prescription used to define stochastic integrals. When this reference measure plays the role of a gauge choice, we interpret our previous results through the lens of gauge theory and prove them to be gauge-invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04217 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric theory of (extended) time-reversal symmetries in stochastic processes -- Part I: finite dimension O'Byrne, Jérémy Cates, Michael E. Statistical Mechanics Soft Condensed Matter In this article, we analyze three classes of time-reversal of a Markov process with Gaussian noise on a manifold. We first unveil a commutativity constraint for the most general of these time-reversals to be well defined. Then we give a triad of necessary and sufficient conditions for the stochastic process to be time-reversible. While most reversibility conditions in the literature require knowledge of the stationary probability, our conditions do not, and therefore can be analytically checked in a systematic way. We then show that the mathematical objects whose cancellation is required by our reversibility conditions play the role of independent sources of entropy production. Furthermore, we give a geometric interpretation of the so-called irreversible cycle-affinity as the vorticity of a certain vector field for a Riemannian geometry given by the diffusion tensor. We also discuss the relation between the time-reversability of the stochastic process and that of an associated deterministic dynamics: its Stratonovitch average. Finally, we show that a suitable choice of a reference measure - that can be considered as a prior or a gauge, depending on the context - allows to study a stochastic process in a way that is both coordinate-free and independent of the prescription used to define stochastic integrals. When this reference measure plays the role of a gauge choice, we interpret our previous results through the lens of gauge theory and prove them to be gauge-invariant. |
| title | Geometric theory of (extended) time-reversal symmetries in stochastic processes -- Part I: finite dimension |
| topic | Statistical Mechanics Soft Condensed Matter |
| url | https://arxiv.org/abs/2402.04217 |