Universal kinetics of imperfect reactions in confinement

Optimizing Work Extraction in the Presence of an Entropic Potential: An Entropic Stochastic Resonance

We study the nontrivial thermodynamic responses of an overdamped Brownian system driven by an unbiased driving force when the particle is confined inside a bilobal irregular structure. The spatial irregularity of the confinement results in an effective entropic bistable potential along the direction of transport. We calculate the thermodynamic response functions in terms of the averaged work done and the absorbed heat over a cycle of driving. We find that the thermodynamic responses are influenced by the nonlinearity of the effective entropic potential, the frequency of the external periodic driving force, and the random thermal fluctuations in a nontrivial way. In the presence of an optimal amount of thermal noise and a favoring driving frequency, the process exhibits a resonance-like precedent in terms of both output work and absorbed heat. We explore the conditions to get best synchronized work extraction (or absorbed heat), which can be utilized as a potential quantifier of an entropic stochastic resonance phenomenon. Finally, we identify a hallmark of entropy dominance over an analogous energy-driven scenario in terms of output work.

Searching for a partially absorbing target by a run-and-tumble particle in a confined space

A random search of a partially absorbing target by a run-and-tumble particle in a confined one-dimensional space is investigated. We analytically obtain the mean searching time, which shows a nonmonotonic behavior as a function of the self-propulsion speed of the active particle, indicating the existence of an optimal speed, when the absorption strength of the target is finite. In the limit of large and small absorption strengths, respectively, asymptotes of the mean searching time and the optimal speed are found. We also demonstrate that the first-passage problem of a diffusive run-and-tumble particle in high dimensions can be mapped into a one-dimensional problem with a partially absorbing target. Finally, as a practical application exploiting the existence of the optimal speed, we propose a filtering device to extract active particles with a desired speed and evaluate how the resolution of the filtering device depends on the absorption strength.

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.

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.

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.

Spectral theory of diffusion in partially absorbing media

A probabilistic framework for studying single-particle diffusion in partially absorbing media has recently been developed in terms of an encounter-based approach. The latter computes the joint probability density (generalized propagator) for particle position X t and a Brownian functional U t that specifies the amount of time the particle is in contact with a reactive component M . Absorption occurs as soon as U t crosses a randomly distributed threshold (stopping time). Laplace transforming the propagator with respect to U t leads to a classical boundary value problem (BVP) in which the reactive component has a constant rate of absorption z , where z is the corresponding Laplace variable. Hence, a crucial step in the encounter-based approach is finding the inverse Laplace transform. In the case of a reactive boundary ∂ M , this can be achieved by solving a classical Robin BVP in terms of the spectral decomposition of a Dirichlet-to-Neumann (D-to-N) operator on ∂ M . In this paper, we develop the analogous construction in the case of a reactive substrate M . In particular, we show that the Laplace transformed propagator can be computed in terms of the spectral decomposition of a pair of D-to-N operators on ∂ M . However, inverting the Laplace transform with respect to z is considerably more involved. We illustrate the theory by considering the D-to-N operators for some simple geometries.

Depletion of resources by a population of diffusing species

Depletion of natural and artificial resources is a fundamental problem and a potential cause of economic crises, ecological catastrophes, and death of living organisms. Understanding the depletion process is crucial for its further control and optimized replenishment of resources. In this paper, we investigate a stock depletion by a population of species that undergo an ordinary diffusion and consume resources upon each encounter with the stock. We derive the exact form of the probability density of the random depletion time, at which the stock is exhausted. The dependence of this distribution on the number of species, the initial amount of resources, and the geometric setting is analyzed. Future perspectives and related open problems are discussed.