Random-search efficiency in a bounded interval with spatially heterogeneous diffusion coefficient

Abstract

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 ${\epsilon }_1=⟨1/t⟩$ and ${\epsilon }_2=1/⟨t⟩$. 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 ${\epsilon }_2$, while ${\epsilon }_1$ 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.

References in corpus (19)

• Intermittent search strategies
• Lévy walks
• Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes
• Infinite Ergodic Theory for Heterogeneous Diffusion Processes
• Optimizing the search for resources by sharing information: Mongolian gazelles as a case study
• Optimizing Brownian escape rates by potential shaping
• Neutral theory and scale-free neural dynamics
• Space-dependent diffusion with stochastic resetting: A first-passage study
• Threshold $q$-voter model
• Optimization and universality of Brownian search in quenched heterogeneous media
• Optimization of random search processes in the presence of an external bias
• Comparison of pure and combined search strategies for single and multiple targets
• Universal singularities of anomalous diffusion in the Richardson class
• How a finite potential barrier decreases the mean first passage time
• Critical patch size reduction by heterogeneous diffusion
• Efficiency of random search with space-dependent diffusivity
• A sluggish random walk with subdiffusive spread
• How Smart Should a Forager Be?
• Target Search of a Protein on DNA in the Presence of Position-dependent Bias

Cited by in corpus (1)

• Random search with resetting in heterogeneous environments