arXiv subjects

Denis S. Grebenkov

Publications and source records attributed to Denis S. Grebenkov.

At least 37 records · Page 2Linked to original sources

arXiv2025

Mixed Steklov-Neumann problem: asymptotic analysis and applications to diffusion-controlled reactions

We consider the mixed Steklov-Neumann spectral problem for the modified Helmholtz equation in a bounded domain when the Steklov condition is imposed on a connected subset of the smooth boundary. In order to deduce the asymptotic behavior in the limit when the size of the subset goes to zero, we reformulate the original problem in terms of an integral operator whose kernel is the restriction of a suitable Green's function (or pseudo-Green's function) to the subset. Its singular behavior on the boundary yields the asymptotic formulas for the eigenvalues and eigenfunctions of the Steklov-Neumann problem. While this analysis remains at a formal level, it is supported by extensive numerical results for two basic examples: an arc on the boundary of a disk and a spherical cap on the boundary of a ball. Solving the original Steklov-Neumann problem numerically in these domains, we validate the asymptotic formulas and reveal their high accuracy, even when the subset is not small. A straightforward application of these spectral results to first-passage processes and diffusion-controlled reactions is presented. We revisit the small-target limit of the mean first-reaction time on perfectly or partially reactive targets. The effect of multiple failed reaction attempts is quantified by a universal function for the whole range of reactivities. Moreover, we extend these results to more sophisticated surface reactions that go beyond the conventional narrow escape problem.

physics.comp-ph↗

arXiv2025

Improved boundary homogenization for a sphere with an absorbing cap of arbitrary size

Finding accurate approximations for the effective reactivity of a structured spherical target with a circular absorbing patch of arbitrary size is a long-standing problem in chemical physics. In this Communication, we reveal limitations of the empirical approximation proposed in [J. Chem. Phys. 145, 214101 (2016)]. We show that the original approximation fails at large patch surface fractions $σ$ and propose a simple amendment. The improved approximation is validated against a semi-analytical solution and is shown to be accurate over the entire range of $σ$ from $0$ to $1$. This approximation also determines the probability of reaction on the patch and the capacitance of such a structured target.

physics.chem-ph↗

arXiv2025

Local persistence exponent and its log-periodic oscillations

We investigate the local persistence exponent of the survival probability of a particle diffusing near an absorbing self-similar boundary. We show by extensive Monte Carlo simulations that the local persistence exponent exhibits log-periodic oscillations over a broad range of timescales. We determine the period and mean value of these oscillations in a family of Koch snowflakes of different fractal dimensions. The effect of the starting point and its local environment on this behavior is analyzed in depth by a simple yet intuitive model. This analysis uncovers how spatial self-similarity of the boundary affects the diffusive dynamics and its temporal characteristics in complex systems.

cond-mat.stat-mech↗

arXiv2025

Boundary local time on wedges and prefractal curves

We investigate the boundary local time on polygonal boundaries such as finite generations of the Koch snowflake. To reveal the role of angles, we first focus on wedges and obtain the mean boundary local time, its variance, and the asymptotic behavior of its distribution. Moreover, we establish the coupled partial differential equations for higher-order moments. Next, we propose an efficient multi-scale Monte Carlo approach to simulate the boundary local time, as well as the escape duration and position of the associated reaction event on a polygonal boundary. This numerical approach combines the walk-on-spheres algorithm in the bulk with an approximate solution of the escape problem from a sector. We apply it to investigate how the statistics of the boundary local time depends on the geometric complexity of the Koch snowflake. Eventual applications to diffusion-controlled reactions on partially reactive boundaries, including the asymptotic behavior of the survival probability, are discussed.

cond-mat.stat-mech↗

arXiv2025

Fastest first-passage time statistics for time-dependent particle injection

A common scenario in a variety of biological systems is that multiple particles are searching in parallel for an immobile target located in a bounded domain, and the fastest among them that arrives to the target first triggers a given desirable or detrimental process. The statistics of such extreme events -- the \textit{fastest\/} first-passage to the target -- is well-understood by now through a series of theoretical analyses, but exclusively under the assumption that all $N$ particles start \textit{simultaneously\/}, i.e., all are introduced into the domain instantly, by $δ$-function-like pulses. However, in many practically important situations this is not the case: in order to start their search, the particles often have to enter first into a bounded domain, e.g., a cell or its nucleus, penetrating through gated channels or nuclear pores. This entrance process has a random duration so that the particles appear in the domain sequentially and with a time delay. Here we focus on the effect of such an extended-in-time injection of multiple particles on the fastest first-passage time (fFPT) and its statistics. We derive the full probability density function $H_N(t)$ of the fFPT with an arbitrary time-dependent injection intensity of $N$ particles. Under rather general assumptions on the survival probability of a single particle and on the injection intensity, we derive the large-$N$ asymptotic formula for the mean fFPT, which is quite different from that obtained for the instantaneous $δ$-pulse injection. The extended injection is also shown to considerably slow down the convergence of $H_N(t)$ to the large-$N$ limit -- the Gumbel distribution -- so that the latter may be inapplicable in the most relevant settings with few tens to few thousands of particles.

cond-mat.stat-mech↗

arXiv2025

Imperfect diffusion-controlled reactions on a torus and on a pair of balls

We employ a general spectral approach based on the Steklov eigenbasis to describe imperfect diffusion-controlled reactions on bounded reactive targets in three dimensions. The steady-state concentration and the total diffusive flux onto the target are expressed in terms of the eigenvalues and eigenfunctions of the exterior Steklov problem. In particular, the eigenvalues are shown to provide the geometric lengthscales of the target that are relevant for diffusion-controlled reactions. Using toroidal and bispherical coordinates, we propose an efficient procedure for analyzing and solving numerically this spectral problem for an arbitrary torus and a pair of balls, respectively. A simple two-term approximation for the diffusive flux is established and validated. Implications of these results in the context of chemical physics and beyond are discussed.

physics.chem-ph↗

arXiv2025

The Steklov problem for exterior domains: asymptotic behavior and applications

We investigate the spectral properties of the Steklov problem for the modified Helmholtz equation $(p-Δ) u = 0$ in the exterior of a compact set, for which the positive parameter $p$ ensures exponential decay of the Steklov eigenfunctions at infinity. We obtain the small-$p$ asymptotic behavior of the eigenvalues and eigenfunctions and discuss their features for different space dimensions. These results find immediate applications to the theory of stochastic processes and unveil the long-time asymptotic behavior of probability densities of various first-passage times in exterior domains. Theoretical results are validated by solving the exterior Steklov problem by a finite-element method with a transparent boundary condition.

math-ph↗

arXiv2025

Escape-from-a-layer approach for simulating the boundary local time in Euclidean domains

We propose an efficient numerical approach to simulate the boundary local time of reflected Brownian motion, as well as the time and position of the associated reaction event on a smooth boundary of a Euclidean domain. This approach combines the standard walk-on-spheres algorithm in the bulk with the approximate solution of the escape problem in a boundary layer. In this way, the most time-consuming simulation of multiple reflections on the boundary is replaced by an equivalent escape event. We validate the proposed escape-from-a-layer approach by comparing simulated statistics of the boundary local time with exact results known for simple domains (a disk, a circular annulus, a sphere, a spherical shell) and with the numerical results obtained by a finite-element method in more sophisticated domains. This approach offers a powerful tool for simulating reflected Brownian motion in multi-scale confinements such as porous media or biological environments, and for solving the related partial differential equations. Its applications in the context of diffusion-controlled reactions in chemical physics are discussed.

physics.comp-ph↗

arXiv2024

First-passage times to a fractal boundary: local persistence exponent and its log-periodic oscillations

We investigate the statistics of the first-passage time (FPT) to a fractal self-similar boundary of the Koch snowflake. When the starting position is fixed near the absorbing boundary, the FPT distribution exhibits an apparent power-law decay over a broad range of timescales, culminated by an exponential cut-off. By extensive Monte Carlo simulations, we compute the local persistence exponent of the survival probability and reveal its log-periodic oscillations in time due to self-similarity of the boundary. The effect of the starting point onto this behavior is analyzed in depth. Theoretical bounds on the survival probability are derived from the analysis of diffusion in a circular sector. Physical rationales for the refined structure of the survival probability are presented.

cond-mat.stat-mech↗

arXiv2024

Adsorption and permeation events in molecular diffusion

How many times a diffusing molecule can permeate across a membrane or be adsorbed on a substrate? We employ the encounter-based approach to find the statistics of adsorption or permeation events for molecular diffusion in a general confining medium. Various features of these statistics are illustrated for two practically relevant cases of a flat boundary and a spherical confinement. Some applications of these fundamental results are discussed.

physics.chem-ph↗

arXiv2024

Spectral properties of the Dirichlet-to-Neumann operator for spheroids

We study the spectral properties of the Dirichlet-to-Neumann operator and the related Steklov problem in spheroidal domains ranging from a needle to a disk. An explicit matrix representation of this operator for both interior and exterior problems is derived. We show how the anisotropy of spheroids affects the eigenvalues and eigenfunctions of the operator. As examples of physical applications, we discuss diffusion-controlled reactions on spheroidal partially reactive targets and the statistics of encounters between the diffusing particle and the spheroidal boundary.

math-ph↗

arXiv2024

Escape from textured adsorbing surfaces

The escape dynamics of sticky particles from textured surfaces is poorly understood despite importance to various scientific and technological domains. In this work, we address this challenge by investigating the escape time of adsorbates from prevalent surface topographies, including holes/pits, pillars, and grooves. Analytical expressions for the probability density function and the mean of the escape time are derived. A particularly interesting scenario is that of very deep and narrow confining spaces within the surface. In this case, the joint effect of the entrapment and stickiness prolongs the escape time, resulting in an effective desorption rate that is dramatically lower than that of the untextured surface. This rate is shown to abide a universal scaling law, which couples the equilibrium constants of adsorption with the relevant confining length scales. While our results are analytical and exact, we also present an approximation for deep and narrow cavities based on an effective description of one dimensional diffusion that is punctuated by motionless adsorption events. This simple and physically motivated approximation provides high-accuracy predictions within its range of validity and works relatively well even for cavities of intermediate depth. All theoretical results are corroborated with extensive Monte-Carlo simulations.

cond-mat.stat-mech↗

arXiv2024

Spectral properties of the Bloch-Torrey operator in three dimensions

We consider the Bloch-Torrey operator, $-Δ+ igx$, that governs the time evolution of the transverse magnetization in diffusion magnetic resonance imaging (dMRI). Using the matrix formalism, we compute numerically the eigenvalues and eigenfunctions of this non-Hermitian operator for two bounded three-dimensional domains: a sphere and a capped cylinder. We study the dependence of its eigenvalues and eigenfunctions on the parameter $g$ and on the shape of the domain (its eventual symmetries and anisotropy). In particular, we show how an eigenfunction drastically changes its shape when the associated eigenvalue crosses a branch (or exceptional) point in the spectrum. Potential implications of this behavior for dMRI are discussed.

math-ph↗

arXiv2023

Diffusion-controlled reactions: an overview

We review the milestones in the century-long development of the theory of diffusion-controlled reactions. Starting from the seminal work by von Smoluchowski who recognized the importance of diffusion in chemical reactions, we discuss perfect and imperfect surface reactions, their microscopic origins, and the underlying mathematical framework. Single-molecule reaction schemes, anomalous bulk diffusions, reversible binding/unbinding kinetics and many other extensions are presented. An alternative encounter-based approach to diffusion-controlled reactions is introduced, with emphasis on its advantages and potential applications. Some open problems and future perspectives are outlined.

physics.chem-ph↗

arXiv2023

A numerical study of the Dirichlet-to-Neumann operator in planar domains

We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on the domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for various simply-connected planar domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate several conjectures with relevance in spectral geometry.

math.NA↗

arXiv2023

Encounter-based approach to target search problems: a review

In this review, we present the encounter-based approach to target search problems, in which the diffusive dynamics is described by the joint probability of the position of the particle and the number of its encounters with a given target set. The knowledge of the statistics of encounters allows one to implement various mechanisms of reactions on the target set, beyond conventional reaction schemes. We formulate this approach for three relevant settings: discrete random walks, Brownian motion with bulk reactions, and reflected Brownian motion with surface reactions. In all cases, we discuss the advantages of this approach, its recent applications and possible extensions.

cond-mat.stat-mech↗

arXiv2023

Escape of a Sticky Particle

Adsorption to a surface, reversible-binding, and trapping are all prevalent scenarios where particles exhibit "stickiness". Escape and first-passage times are known to be drastically affected, but detailed understanding of this phenomenon remains illusive. To tackle this problem, we develop an analytical approach to the escape of a diffusing particle from a domain of arbitrary shape, size, and surface reactivity. This is used to elucidate the effect of stickiness on the escape time from a slab domain: revealing how adsorption and desorption rates affect the mean and variance, and providing a novel approach to infer these rates from measurements. Moreover, as any smooth boundary is locally flat, slab results are leveraged to devise a numerically efficient scheme for simulating sticky boundaries in arbitrary domains. Generalizing our analysis to higher dimensions reveals that the mean escape time abides a general structure that is independent of the dimensionality of the problem. This letter thus offers a new starting point for analytical and numerical studies of stickiness and its role in escape, first-passage, and diffusion-controlled reactions.

cond-mat.stat-mech↗

arXiv2023

Boundary homogenization for target search problems

In this review, we describe several approximations in the theory of Laplacian transport near complex or heterogeneously reactive boundaries. This phenomenon, governed by the Laplace operator, is ubiquitous in fields as diverse as chemical physics, hydrodynamics, electrochemistry, heat transfer, wave propagation, self-organization, biophysics, and target search. We overview the mathematical basis and various applications of the effective medium approximation and the related boundary homogenization when a complex heterogeneous boundary is replaced by an effective much simpler boundary. We also discuss the constant-flux approximation, the Fick-Jacobs equation, and other mathematical tools for studying the statistics of first-passage times to a target. Numerous examples and illustrations are provided to highlight the advantages and limitations of these approaches.

physics.chem-ph↗