Interfacial territory covered by surface-mediated diffusion

Constructing efficient strategies for the process optimization by restart

Optimization of the mean completion time of random processes by restart is a subject of active theoretical research in statistical physics and has long found practical application in computer science. Meanwhile, one of the key issues remains largely unsolved: how to construct a restart strategy for a process whose detailed statistics are unknown to ensure that the expected completion time will reduce? Addressing this query here we propose several constructive criteria for the effectiveness of various protocols of noninstantaneous restart in the mean completion time problem and in the success probability problem. Being expressed in terms of a small number of easily estimated statistical characteristics of the original process (MAD, median completion time, low-order statistical moments of completion time), these criteria allow informed restart decision based on partial information.

Narrow escape problem in two-shell spherical domains

Intracellular transport in living cells is often spatially inhomogeneous with an accelerated effective diffusion close to the cell membrane and a ballistic motion away from the centrosome due to active transport along actin filaments and microtubules, respectively. Recently it was reported that the mean first passage time (MFPT) for transport to a specific area on the cell membrane is minimal for an optimal actin cortex width. In this paper, we ask whether this optimization in a two-compartment domain can also be achieved by passive Brownian particles. We consider a Brownian motion with different diffusion constants in the two shells and a potential barrier between the two, and we investigate the narrow escape problem by calculating the MFPT for Brownian particles to reach a small window on the external boundary. In two and three dimensions, we derive asymptotic expressions for the MFPT in the thin cortex and small escape region limits confirmed by numerical calculations of the MFPT using the finite-element method and stochastic simulations. From this analytical and numeric analysis, we finally extract the dependence of the MFPT on the ratio of diffusion constants, the potential barrier height, and the width of the outer shell. The first two are monotonous, whereas the last one may have a minimum for a sufficiently attractive cortex, for which we propose an analytical expression of the potential barrier height matching very well the numerical predictions.

Overlap of two Brownian trajectories: exact results for scaling functions

We consider two random walkers starting at the same time t=0 from different points in space separated by a given distance R. We compute the average volume of the space visited by both walkers up to time t as a function of R and t and dimensionality of space d. For d<4, this volume, after proper renormalization, is shown to be expressed through a scaling function of a single variable R/√t. We provide general integral formulas for scaling functions for arbitrary dimensionality d<4. In contrast, we show that no scaling function exists for higher dimensionalities d≥4.

Single-molecule diffusion in a periodic potential at a solid-liquid interface

We used single-molecule tracking experiments to observe the motion of small hydrophobic fluorescent molecules at the interface between water and a solid surface that exhibited periodic chemical patterns. The dynamics were characterized by non-ergodic, continuous time random walk statistics. The step-size distributions displayed enhanced probability of steps to periodic distances, consistent with theoretical predictions for diffusion in an atomic/molecular scale periodic potential. Surprisingly, this general behavior was observed here for surfaces exhibiting characteristic length scales three orders of magnitude larger than atomic/molecular dimensions, and may provide a new way to understand and control solid-liquid interfacial diffusion for molecular targeting applications.

Splitting probabilities and interfacial territory covered by two-dimensional and three-dimensional surface-mediated diffusion

We consider the mean territory covered by a particle that performs surface-mediated diffusion inside a spherical confining domain (in two and three dimensions) before exit through an opening on the surface. This quantity can be expressed in terms of the splitting probability between two targets on the surface. We derive a general formula that relates this splitting probability to the mean first passage time to a single target that has been recently calculated for such a surface-mediated diffusion process. This formula is exact for pointlike targets and is shown to be accurate for extended targets. The mean covered territory is then found and analyzed for an arbitrary extension of the exit region in both two- and three-dimensional spherical domains.

Enhanced transport through desorption-mediated diffusion

We present a master equation approach to the study of the bulk-mediated surface diffusion mechanism in a three-dimensional confined domain. The proposed scheme allowed us to evaluate analytically a number of magnitudes that were used to characterize the efficiency of the bulk-mediated surface transport mechanism, for instance, the mean escape time from the domain, and the mean number of distinct visited sites on the confined domain boundary.

Exact mean exit time for surface-mediated diffusion

We present an exact expression for the mean exit time through the cap of a confining sphere for particles alternating phases of surface and of bulk diffusion. The present approach is based on an integral equation which can be solved analytically. In contrast to the statement of Berezhkovskii and Barzykin [J. Chem. Phys. 136, 54115 (2012)], we show that the mean exit time can be optimized with respect to the desorption rate, under analytically determined criteria.

Narrow-escape-time problem: the imperfect trapping case

We present a master equation approach to the narrow escape time (NET) problem, i.e., the time needed for a particle contained in a confining domain with a single narrow opening to exit the domain for the first time. We introduce a finite transition probability, ν, at the narrow escape window, allowing the study of the imperfect trapping case. Ranging from 0 to ∞, ν allowed the study of both extremes of the trapping process: that of a highly deficient capture and situations where escape is certain ("perfect trapping" case). We have obtained analytic results for the basic quantity studied in the NET problem, the mean escape time, and we have studied its dependence in terms of the transition (desorption) probability over (from) the surface boundary, the confining domain dimensions, and the finite transition probability at the escape window. Particularly we show that the existence of a global minimum in the NET depends on the "imperfection" of the trapping process. In addition to our analytical approach, we have implemented Monte Carlo simulations, finding excellent agreement between the theoretical results and simulations.