Diffusive escape through a narrow opening: new insights into a classic problem

Abstract

We study the mean first exit time $T_{\ve}$ of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of an angular size $\ve$. Focusing on the effects of an energy/entropy barrier at the EW, and of the long-range interactions (LRI) with the boundary on the diffusive search for the EW, we develop a self-consistent approximation to derive for $T_{\ve}$ a general expression, akin to the celebrated Collins-Kimball relation in chemical kinetics and accounting for both rate-controlling factors in an explicit way. Our analysis reveals that the barrier-induced contribution to $T_{\ve}$ is the dominant one in the limit $\ve \to 0$, implying that the narrow escape problem is not "diffusion-limited" but rather "barrier-limited". We present the small-$\ve$ expansion for $T_{\ve}$, in which the coefficients in front of the leading terms are expressed via some integrals and derivatives of the LRI potential. On example of a triangular-well potential, we show that $T_{\ve}$ is non-monotonic with respect to the extent of the attractive LRI, being minimal for the ones having an intermediate extent, neither too concentrated on the boundary nor penetrating too deeply into the bulk. Our analytical predictions are in a good agreement with the numerical simulations.

17 pages, 6 figures, Supplemental Materials

References in corpus (21)

• Intermittent search strategies
• Optimal search strategies for hidden targets
• Entropic transport: Kinetics, scaling and control mechanisms
• Optimal reaction time for surface-mediated diffusion
• Distance matters: the impact of gene proximity in bacterial gene regulation
• Entropic particle transport: higher order corrections to the Fick-Jacobs diffusion equation
• Kinetics of stochastically-gated diffusion-limited reactions and geometry of random walk trajectories
• In vivo facilitated diffusion model
• Universal formula for the mean first passage time in planar domains
• Mean first-passage time of surface-mediated diffusion in spherical domains
• Hydrodynamically enforced entropic trapping of Brownian particles
• Efficient search by optimized intermittent random walks
• Exit time distribution in spherically symmetric two-dimensional domains
• Narrow-escape times for diffusion in microdomains with a particle-surface affinity: Mean-field results
• Biased Brownian motion in extreme corrugated tubes
• Kinetics of active surface-mediated diffusion in spherically symmetric domains
• Exact mean exit time for surface-mediated diffusion
• Optimization and universality of Brownian search in quenched heterogeneous media
• Entropically-induced asymmetric passage times of charged tracers across corrugated channels
• Optimization of random search processes in the presence of an external bias
• Narrow-escape-time problem: the imperfect trapping case

Cited by in corpus (44)

• Diffusion-limited reactions in dynamic heterogeneous media
• Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control
• Paradigm shift in diffusion-mediated surface phenomena
• 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 escape problem for mortal walkers
• Active colloids in the context of chemical kinetics
• 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
• Interlacing Relaxation and First-Passage Phenomena in Reversible Discrete and Continuous Space Markovian Dynamics
• Duality Between Relaxation and First Passage in Reversible Markov Dynamics: Rugged Energy Landscapes Disentangled
• A probabilistic approach to extreme statistics of Brownian escape times in dimensions 1, 2, and 3
• Competition between slow and fast regimes for extreme first passage times of diffusion
• Narrow escape of interacting diffusing particles
• Surface Hopping Propagator: An Alternative Approach to Diffusion-Influenced Reactions
• Extreme first passage times for random walks on networks
• Preface: New trends in first-passage methods and applications in the life sciences and engineering
• Statistics of boundary encounters by a particle diffusing outside a compact planar domain
• 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
• Depletion of Resources by a Population of Diffusing Species
• First-encounter time of two diffusing particles in confinement
• Search efficiency in the Adam-Delbrück reduction-of-dimensionality scenario versus direct diffusive search
• Statistics of diffusive encounters with a small target: Three complementary approaches
• First-passage times to anisotropic partially reactive targets
• First-passage times of multiple diffusing particles with reversible target-binding kinetics
• Distribution of first-reaction times with target regions on boundaries of shell-like domains
• Extreme statistics of anomalous subdiffusion following a fractional Fokker-Planck equation: Subdiffusion is faster than normal diffusion
• Reversible Target-Binding Kinetics of Multiple Impatient Particles
• Steady-state reaction rate of diffusion-controlled reactions in sheets
• An encounter-based approach to the escape problem
• Reversible reactions controlled by surface diffusion on a sphere
• Active microrheology in corrugated channels
• Optimal potentials for diffusive search strategies
• Escape dynamics of a self-propelled nanorod from circular confinements with narrow openings
• First-encounter time of two diffusing particles in two- and three-dimensional confinement
• The narrow escape problem in a circular domain with radial piecewise constant diffusivity
• Narrow escape problem in two-shell spherical domains
• A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes
• Splitting probabilities for dynamics in corrugated channels: passive VS active Brownian motion
• The distribution of the maximum of independent resetting Brownian motions
• An encounter-based approach for restricted diffusion with a gradient drift
• Encounter-based approach to target search problems: a review
• Mean first-passage time to a small absorbing target in three-dimensional elongated domains