Random-search efficiency in a bounded interval with spatially heterogeneous diffusion coefficient
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arXiv
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| Format: | Preprint |
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2023
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| author | Menon Jr, L. Santos, M. A. F. dos Anteneodo, C. |
| author_facet | Menon Jr, L. Santos, M. A. F. dos Anteneodo, C. |
| contents | We consider random walkers searching for a target in a bounded one-dimensional heterogeneous environment, in the interval $[0,L]$, where diffusion is described by a space-dependent diffusion coefficient $D(x)$. Boundary conditions are absorbing at the position of the target (set at $x=0$) and reflecting at the border $x=L$. We calculate and compare the estimates of efficiency $\varepsilon_1=\langle 1/ t\rangle$ and $\varepsilon_2=1/\langle t \rangle$. For the Stratonovich framework of the multiplicative random process, both measures are analytically calculated for arbitrary $D(x)$. For other interpretations of the stochastic integrals (e.g., Itô and anti-Itô), we get general results for $\varepsilon_2$, while $\varepsilon_1$ is obtained for particular forms of $D(x)$. The impact of the diffusivity profile on these measures of efficiency is discussed. Symmetries and peculiar properties arise when the search starts at the border ($x_0=L$), in particular, heterogeneity spoils the efficiency of the search within the Stratonovich framework, while for other interpretations the searcher can perform better in certain heterogeneous diffusivity profiles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_00818 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Random-search efficiency in a bounded interval with spatially heterogeneous diffusion coefficient Menon Jr, L. Santos, M. A. F. dos Anteneodo, C. Statistical Mechanics We consider random walkers searching for a target in a bounded one-dimensional heterogeneous environment, in the interval $[0,L]$, where diffusion is described by a space-dependent diffusion coefficient $D(x)$. Boundary conditions are absorbing at the position of the target (set at $x=0$) and reflecting at the border $x=L$. We calculate and compare the estimates of efficiency $\varepsilon_1=\langle 1/ t\rangle$ and $\varepsilon_2=1/\langle t \rangle$. For the Stratonovich framework of the multiplicative random process, both measures are analytically calculated for arbitrary $D(x)$. For other interpretations of the stochastic integrals (e.g., Itô and anti-Itô), we get general results for $\varepsilon_2$, while $\varepsilon_1$ is obtained for particular forms of $D(x)$. The impact of the diffusivity profile on these measures of efficiency is discussed. Symmetries and peculiar properties arise when the search starts at the border ($x_0=L$), in particular, heterogeneity spoils the efficiency of the search within the Stratonovich framework, while for other interpretations the searcher can perform better in certain heterogeneous diffusivity profiles. |
| title | Random-search efficiency in a bounded interval with spatially heterogeneous diffusion coefficient |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2311.00818 |