First-passage times to anisotropic partially reactive targets

Narrow escape with imperfect reactions

The imperfect narrow escape problem considers the mean first passage time (MFPT) of a Brownian particle through one of several small, partially reactive traps on an otherwise reflecting boundary within a confining domain. Mathematically, this problem is equivalent to Poisson's equation with mixed Neumann-Robin boundary conditions. Here, we obtain this MFPT in general three-dimensional domains by using strong localized perturbation theory in the small trap limit. These leading-order results involve factors, which are analogous to electrostatic capacitances, and we use Brownian local time theory and kinetic Monte Carlo (KMC) algorithms to rapidly compute these factors. Furthermore, we use a heuristic approximation of such a capacitance to obtain a simple, approximate MFPT, which is valid for any trap reactivity. In addition, we develop KMC algorithms to efficiently simulate the full problem and find excellent agreement with our asymptotic approximations.

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.

Imperfect narrow escape problem

We consider the kinetics of the imperfect narrow escape problem, i.e., the time it takes for a particle diffusing in a confined medium of generic shape to reach and to be adsorbed by a small, imperfectly reactive patch embedded in the boundary of the domain, in two or three dimensions. Imperfect reactivity is modeled by an intrinsic surface reactivity κ of the patch, giving rise to Robin boundary conditions. We present a formalism to calculate the exact asymptotics of the mean reaction time in the limit of large volume of the confining domain. We obtain exact explicit results in the two limits of large and small reactivities of the reactive patch, and a semianalytical expression in the general case. Our approach reveals an anomalous scaling of the mean reaction time as the inverse square root of the reactivity in the large-reactivity limit, valid for an initial position near the extremity of the reactive patch. We compare our exact results with those obtained within the "constant flux approximation"; we show that this approximation turns out to give exactly the next-to-leading-order term of the small-reactivity limit, and provides a good approximation of the reaction time far from the reactive patch for all reactivities, but not in the vicinity of the boundary of the reactive patch due to the above-mentioned anomalous scaling. These results thus provide a general framework to quantify the mean reaction times for the imperfect narrow escape problem.

Diffusion toward a nanoforest of absorbing pillars

Spiky coatings (also known as nanoforests or Fakir-like surfaces) have found many applications in chemical physics, material sciences, and biotechnology, such as superhydrophobic materials, filtration and sensing systems, and selective protein separation, to name but a few. In this paper, we provide a systematic study of steady-state diffusion toward a periodic array of absorbing cylindrical pillars protruding from a flat base. We approximate a periodic cell of this system by a circular tube containing a single pillar, derive an exact solution of the underlying Laplace equation, and deduce a simple yet exact representation for the total flux of particles onto the pillar. The dependence of this flux on the geometric parameters of the model is thoroughly analyzed. In particular, we investigate several asymptotic regimes, such as a thin pillar limit, a disk-like pillar, and an infinitely long pillar. Our study sheds light onto the trapping efficiency of spiky coatings and reveals the roles of pillar anisotropy and diffusional screening.

Encounter-based approach to diffusion with resetting

An encounter-based approach consists in using the boundary local time as a proxy for the number of encounters between a diffusing particle and a target to implement various surface reaction mechanisms on that target. In this paper, we investigate the effects of stochastic resetting onto diffusion-controlled reactions in bounded confining domains. We first discuss the effect of position resetting onto the propagator and related quantities; in this way, we retrieve a number of earlier results but also provide complementary insights into them. Second, we introduce boundary local time resetting and investigate its impact. Curiously, we find that this type of resetting does not alter the conventional propagator governing the diffusive dynamics in the presence of a partially reactive target with a constant reactivity. In turn, the generalized propagator for other surface reaction mechanisms can be significantly affected. Our general results are illustrated for diffusion on an interval with reactive end points. Further perspectives and some open problems are discussed.

Enhancing search efficiency through diffusive echo

Despite having been studied for decades, first passage processes remain an active area of research. In this contribution we examine a particle diffusing in an annulus with an inner absorbing boundary and an outer reflective boundary. We obtain analytic expressions for the joint distribution of the hitting time and the hitting angle in two and three dimensions. For certain configurations we observe a ``diffusive echo", i.e. two well-defined maxima in the first passage time distribution to a targeted position on the absorbing boundary. This effect, which results from the interplay between the starting location and the environmental constraints, may help to significantly increase the efficiency of the random search by generating a high, sustained flux to the targeted position over a short period. Finally, we examine the corresponding one-dimensional system for which there is no well-defined echo. In a confined system, the flux integrated over all target positions always displays a shoulder. This does not, however, guarantee the presence of an echo in the joint distribution.