[Paper Review] An encounter-based approach for restricted diffusion with a gradient drift

Denis S. Grebenkov|arXiv (Cornell University)|Oct 23, 2021

Diffusion and Search Dynamics74 references47 citations

TL;DR

This paper introduces an encounter-based approach for restricted diffusion with a gradient drift by extending the Dirichlet-to-Neumann operator to derive a spectral decomposition of the full propagator—the joint probability density of particle position and boundary local time. The method enables exact computation of reaction kinetics, survival probability, and first-passage time statistics under drift, revealing how drift enhances boundary encounters and alters local time statistics in confined geometries.

ABSTRACT

We develop an encounter-based approach for describing restricted diffusion with a gradient drift towards a partially reactive boundary. For this purpose, we introduce an extension of the Dirichlet-to-Neumann operator and use its eigenbasis to derive a spectral decomposition for the full propagator, i.e., the joint probability density function for the particle position and its boundary local time. This is the central quantity that determines various characteristics of diffusion-influenced reactions such as conventional propagators, survival probability, first-passage time distribution, boundary local time distribution, and reaction rate. As an illustration, we investigate the impact of a constant drift onto the boundary local time for restricted diffusion on an interval. More generally, this approach accesses how external forces may influence the statistics of encounters of a diffusing particle with the reactive boundary.

Motivation & Objective

• To develop a systematic framework for modeling restricted diffusion with a gradient drift toward a partially reactive boundary.
• To incorporate boundary local time as a central stochastic variable to describe particle encounters with reactive surfaces.
• To extend the Dirichlet-to-Neumann operator for spectral decomposition of the full propagator in the presence of drift.
• To enable exact computation of key reaction-diffusion quantities such as survival probability, first-passage time, and reaction rates.
• To analyze how external forces (drift) quantitatively influence the statistics of boundary encounters in confined geometries.

Proposed method

• Introduces an extended Dirichlet-to-Neumann operator that incorporates both position and boundary local time dynamics.
• Derives a spectral decomposition of the full propagator using eigenfunctions of the extended operator.
• Uses the joint probability density of position and boundary local time as the central quantity for all reaction-diffusion characteristics.
• Applies the spectral method to solve the Fokker-Planck equation with Robin-type boundary conditions under a constant drift.
• Employs stochastic simulations and analytical solutions on a one-dimensional interval to validate the approach.
• Reconciles the Skorokhod stochastic differential equation with the Fokker-Planck formalism via boundary local time as a dynamical variable.

Experimental results

Research questions

• RQ1 How does a constant drift affect the distribution of boundary local time in restricted diffusion on an interval?
• RQ2 Can the full propagator for diffusion with drift be spectrally decomposed using an extended Dirichlet-to-Neumann operator?
• RQ3 How do external forces modify the statistics of particle encounters with a partially reactive boundary?
• RQ4 What is the impact of drift on the survival probability and first-passage time distribution in confined geometries?
• RQ5 Can the encounter-based approach generalize beyond the Robin boundary condition to model encounter-dependent reactivity?

Key findings

• The spectral decomposition of the full propagator is derived using eigenfunctions of an extended Dirichlet-to-Neumann operator, enabling exact solutions for the joint position-local time distribution.
• A constant drift significantly alters the boundary local time distribution: positive drift (toward boundary) increases local time, while negative drift (away) reduces it, with explicit formulas derived for the interval.
• The survival probability and first-passage time distribution are analytically computed and shown to depend non-trivially on the drift strength and boundary reactivity.
• The method reveals that drift enhances the frequency and duration of boundary encounters, quantified through the mean and variance of boundary local time.
• The approach generalizes beyond the Robin boundary condition, allowing modeling of encounter-dependent reactivity and non-Markovian surface reactions.
• In the limit of strong drift, the system transitions from diffusive to drift-dominated dynamics, with boundary local time scaling as ℓt ∼ t for positive drift and approaching zero for strong negative drift.