Statistics of diffusive encounters with a small target: Three complementary approaches

Abstract

Diffusive search for a static target is a common problem in statistical physics with numerous applications in chemistry and biology. We look at this problem from a different perspective and investigate the statistics of encounters between the diffusing particle and the target. While an exact solution of this problem was recently derived in the form of a spectral expansion over the eigenbasis of the Dirichlet-to-Neumann operator, the latter is generally difficult to access for an arbitrary target. In this paper, we present three complementary approaches to approximate the probability density of the rescaled number of encounters with a small target in a bounded confining domain. In particular, we derive a simple fully explicit approximation, which depends only on a few geometric characteristics such as the surface area and the harmonic capacity of the target, and the volume of the confining domain. We discuss the advantages and limitations of three approaches and check their accuracy. We also deduce an explicit approximation for the distribution of the first-crossing time, at which the number of encounters exceeds a prescribed threshold. Its relations to common first-passage time problems are discussed.

References in corpus (41)

• Intermittent search strategies
• First-passage times in complex scale-invariant media
• Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities
• Geometry-controlled kinetics
• Geometrical structure of Laplacian eigenfunctions
• Diffusion-limited reactions in dynamic heterogeneous media
• Optimal reaction time for surface-mediated diffusion
• First passage time distribution in heterogeneity controlled kinetics: going beyond the mean first passage time
• Paradigm shift in diffusion-mediated surface phenomena
• Survival of an evasive prey
• Capture of the Lamb: Diffusing Predators Seeking a Diffusing Prey
• Mortality, Redundancy, and Diversity in Stochastic Search
• Kinetics of stochastically-gated diffusion-limited reactions and geometry of random walk trajectories
• Universal formula for the mean first passage time in planar domains
• Diffusive escape through a narrow opening: new insights into a classic problem
• Exploration and Trapping of Mortal Random Walkers
• Full distribution of first exit times in the narrow escape problem
• Towards a full quantitative description of single-molecule reaction kinetics in biological cells
• The Dirichlet-to-Neumann operator via hidden compactness
• The escape problem for mortal walkers
• From single-particle stochastic kinetics to macroscopic reaction rates: fastest first-passage time of $N$ random walkers
• Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces
• Explicit Construction of the Brownian Self-Transport Operator
• Computation of the Mean First-Encounter Time Between the Ends of a Polymer Chain
• Partially Reflected Brownian Motion: A Stochastic Approach to Transport Phenomena
• Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains
• Surface Hopping Propagator: An Alternative Approach to Diffusion-Influenced Reactions
• The two-particle problem in comb-like structures
• Survival probability of stochastic processes beyond persistence exponents
• Statistics of boundary encounters by a particle diffusing outside a compact planar domain
• Slow Encounters of Particle Pairs in Branched Structures
• Joint distribution of multiple boundary local times and related first-passage time problems with multiple targets
• The narrow capture problem: an encounter-based approach to partially reactive targets
• First-encounter time of two diffusing particles in confinement
• Depletion of Resources by a Population of Diffusing Species
• First-passage times to anisotropic partially reactive targets
• Distribution of first-reaction times with target regions on boundaries of shell-like domains
• Exact encounter times for many random walkers on regular and complex networks
• Narrow Escape, Part III: Riemann surfaces and non-smooth domains
• Capture of a diffusive prey by multiple predators in confined space
• First-encounter time of two diffusing particles in two- and three-dimensional confinement

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