The effective diffusion constant of stochastic processes with spatially periodic noise
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914328007933952 |
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| author | Giordano, Stefano Blossey, Ralf |
| author_facet | Giordano, Stefano Blossey, Ralf |
| contents | We discuss the effective diffusion constant $D_{\it eff}$ for stochastic processes with spatially-dependent noise. Starting from a stochastic process given by a Langevin equation, different drift-diffusion equations can be derived depending on the choice of the discretization rule $ 0 \leq α\leq 1$. We initially study the case of periodic heterogeneous diffusion without drift and we determine a general result for the effective diffusion coefficient $D_{\it eff}$, which is valid for any value of $α$. We study the case of periodic sinusoidal diffusion in detail and we find a relationship with Legendre functions. Then, we derive $D_{\it eff}$ for general $α$ in the case of diffusion with periodic spatial noise and in the presence of a drift term, generalizing the Lifson-Jackson theorem. Our results are illustrated by analytical and numerical calculations on generic periodic choices for drift and diffusion terms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_10813 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The effective diffusion constant of stochastic processes with spatially periodic noise Giordano, Stefano Blossey, Ralf Statistical Mechanics We discuss the effective diffusion constant $D_{\it eff}$ for stochastic processes with spatially-dependent noise. Starting from a stochastic process given by a Langevin equation, different drift-diffusion equations can be derived depending on the choice of the discretization rule $ 0 \leq α\leq 1$. We initially study the case of periodic heterogeneous diffusion without drift and we determine a general result for the effective diffusion coefficient $D_{\it eff}$, which is valid for any value of $α$. We study the case of periodic sinusoidal diffusion in detail and we find a relationship with Legendre functions. Then, we derive $D_{\it eff}$ for general $α$ in the case of diffusion with periodic spatial noise and in the presence of a drift term, generalizing the Lifson-Jackson theorem. Our results are illustrated by analytical and numerical calculations on generic periodic choices for drift and diffusion terms. |
| title | The effective diffusion constant of stochastic processes with spatially periodic noise |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2407.10813 |