Symmetry classes of classical stochastic processes
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910870876979200 |
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| author | Sá, Lucas Ribeiro, Pedro Prosen, Tomaž Bernard, Denis |
| author_facet | Sá, Lucas Ribeiro, Pedro Prosen, Tomaž Bernard, Denis |
| contents | We perform a systematic symmetry classification of the Markov generators of classical stochastic processes. Our classification scheme is based on the action of involutive symmetry transformations of a real Markov generator, extending the Bernard-LeClair scheme to the arena of classical stochastic processes and leading to a set of up to fifteen allowed symmetry classes. We construct families of solutions of arbitrary matrix dimensions for five of these classes with a simple physical interpretation of particles hopping on multipartite graphs. In the remaining classes, such a simple construction is prevented by the positivity of entries of the generator particular to classical stochastic processes, which imposes a further requirement beyond the usual symmetry classification constraints. We partially overcome this difficulty by resorting to a stochastic optimization algorithm, finding specific examples of generators of small matrix dimensions in six further classes, leaving the existence of the final four allowed classes an open problem. Our symmetry-based results unveil new possibilities in the dynamics of classical stochastic processes: Kramers degeneracy of eigenvalue pairs, dihedral symmetry of the spectra of Markov generators, and time reversal properties of stochastic trajectories and correlation functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_17955 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetry classes of classical stochastic processes Sá, Lucas Ribeiro, Pedro Prosen, Tomaž Bernard, Denis Statistical Mechanics Disordered Systems and Neural Networks Mathematical Physics We perform a systematic symmetry classification of the Markov generators of classical stochastic processes. Our classification scheme is based on the action of involutive symmetry transformations of a real Markov generator, extending the Bernard-LeClair scheme to the arena of classical stochastic processes and leading to a set of up to fifteen allowed symmetry classes. We construct families of solutions of arbitrary matrix dimensions for five of these classes with a simple physical interpretation of particles hopping on multipartite graphs. In the remaining classes, such a simple construction is prevented by the positivity of entries of the generator particular to classical stochastic processes, which imposes a further requirement beyond the usual symmetry classification constraints. We partially overcome this difficulty by resorting to a stochastic optimization algorithm, finding specific examples of generators of small matrix dimensions in six further classes, leaving the existence of the final four allowed classes an open problem. Our symmetry-based results unveil new possibilities in the dynamics of classical stochastic processes: Kramers degeneracy of eigenvalue pairs, dihedral symmetry of the spectra of Markov generators, and time reversal properties of stochastic trajectories and correlation functions. |
| title | Symmetry classes of classical stochastic processes |
| topic | Statistical Mechanics Disordered Systems and Neural Networks Mathematical Physics |
| url | https://arxiv.org/abs/2406.17955 |