Brownian motion and generalized Lifson-Jackson formula in quasi-periodic systems
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912343089217536 |
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| author | Yang, Sang Sun, Juyuan Guo, Guangcan Gong, Ming |
| author_facet | Yang, Sang Sun, Juyuan Guo, Guangcan Gong, Ming |
| contents | Brownian motion in periodic potentials has been widely investigated in statistical physics and related interdisciplinary fields. In the overdamped regime, it has been well-known that the diffusion constant $D^*$ is given by the Lifson-Jackson (LJ) formula. With a tilted potential, this model can exhibit giant diffusion. In this work, we start from the basic argument that since any quasi-periodic potential can be approximated accurately using a periodic potential, this formula and the associated physics should also apply to the quasi-periodic potential after some proper redefinition. We derive $D^*$ from the Smoluchowski equation using the fact that its asymptotic solution is a product of a Boltzmann weight and a Gaussian envelope function. Then we analytically calculate $D^*$ in terms of Bessel functions. Finally, we study the giant diffusion with quasi-periodic potentials, generalize the corresponding formula to the condition with tilted potential under the same argument, and calculate $D^*$ analytically. This work generalizes the Brownian motion from periodic potentials to the much broader quasi-periodic potentials, which should have applications in interdisciplinary fields in physics, chemistry, engineering, and life sciences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16527 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Brownian motion and generalized Lifson-Jackson formula in quasi-periodic systems Yang, Sang Sun, Juyuan Guo, Guangcan Gong, Ming Statistical Mechanics Brownian motion in periodic potentials has been widely investigated in statistical physics and related interdisciplinary fields. In the overdamped regime, it has been well-known that the diffusion constant $D^*$ is given by the Lifson-Jackson (LJ) formula. With a tilted potential, this model can exhibit giant diffusion. In this work, we start from the basic argument that since any quasi-periodic potential can be approximated accurately using a periodic potential, this formula and the associated physics should also apply to the quasi-periodic potential after some proper redefinition. We derive $D^*$ from the Smoluchowski equation using the fact that its asymptotic solution is a product of a Boltzmann weight and a Gaussian envelope function. Then we analytically calculate $D^*$ in terms of Bessel functions. Finally, we study the giant diffusion with quasi-periodic potentials, generalize the corresponding formula to the condition with tilted potential under the same argument, and calculate $D^*$ analytically. This work generalizes the Brownian motion from periodic potentials to the much broader quasi-periodic potentials, which should have applications in interdisciplinary fields in physics, chemistry, engineering, and life sciences. |
| title | Brownian motion and generalized Lifson-Jackson formula in quasi-periodic systems |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2504.16527 |