Role of Topology in Relaxation of One-Dimensional Stochastic Processes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909089578090496 |
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| author | Sawada, Taro Sone, Kazuki Hamazaki, Ryusuke Ashida, Yuto Sagawa, Takahiro |
| author_facet | Sawada, Taro Sone, Kazuki Hamazaki, Ryusuke Ashida, Yuto Sagawa, Takahiro |
| contents | Stochastic processes are commonly used models to describe dynamics of a wide variety of nonequilibrium phenomena ranging from electrical transport to biological motion. The transition matrix describing a stochastic process can be regarded as a non-Hermitian Hamiltonian. Unlike general non-Hermitian systems, the conservation of probability imposes additional constraints on the transition matrix, which can induce unique topological phenomena. Here, we reveal the role of topology in relaxation phenomena of classical stochastic processes. Specifically, we define a winding number that is related to topology of stochastic processes and show that it predicts the existence of a spectral gap that characterizes the relaxation time. Then, we numerically confirm that the winding number corresponds to the system-size dependence of the relaxation time and the characteristic transient behavior. One can experimentally realize such topological phenomena in magnetotactic bacteria and cell adhesions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09832 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Role of Topology in Relaxation of One-Dimensional Stochastic Processes Sawada, Taro Sone, Kazuki Hamazaki, Ryusuke Ashida, Yuto Sagawa, Takahiro Statistical Mechanics Mesoscale and Nanoscale Physics Stochastic processes are commonly used models to describe dynamics of a wide variety of nonequilibrium phenomena ranging from electrical transport to biological motion. The transition matrix describing a stochastic process can be regarded as a non-Hermitian Hamiltonian. Unlike general non-Hermitian systems, the conservation of probability imposes additional constraints on the transition matrix, which can induce unique topological phenomena. Here, we reveal the role of topology in relaxation phenomena of classical stochastic processes. Specifically, we define a winding number that is related to topology of stochastic processes and show that it predicts the existence of a spectral gap that characterizes the relaxation time. Then, we numerically confirm that the winding number corresponds to the system-size dependence of the relaxation time and the characteristic transient behavior. One can experimentally realize such topological phenomena in magnetotactic bacteria and cell adhesions. |
| title | Role of Topology in Relaxation of One-Dimensional Stochastic Processes |
| topic | Statistical Mechanics Mesoscale and Nanoscale Physics |
| url | https://arxiv.org/abs/2301.09832 |