First-encounter time of two diffusing particles in confinement

Abstract

The statistics of the first-encounter time of diffusing particles changes drastically when they are placed under confinement. In the present work, we make use of Monte Carlo simulations to study the behavior of a two-particle system in two-and three-dimensional domains with reflecting boundaries. Based on the outcome of the simulations, we give a comprehensive overview of the behavior of the survival probability S(t) and the associated first-encounter time probability density H(t) over a broad time range spanning several decades. In addition, we provide numerical estimates and empirical formulas for the mean first-encounter time T , as well as for the decay time T characterizing the monoexponential long-time decay of the survival probability. Based on the distance between the boundary and the center of mass of two particles, we obtain an empirical lower bound tB for the time at which S(t) starts to significantly deviate from its counterpart for the no boundary case. Surprisingly, for small-sized particles, the dominant contribution to T depends only on the total diffusivity D = D1 + D2, in sharp contrast to the one-dimensional case. This contribution can be related to the Wiener sausage generated by a fictitious Brownian particle with diffusivity D. In two dimensions, the first subleading contribution to T is found to depend weakly on the ratio D1/D2. We also investigate the slow-diffusion limit when D2 ≪ D1 and discuss the transition to the limit when one particle is a fixed target. Finally, we give some indications to anticipate when T can be expected to be a good approximation for T .

Distribution of first-reaction times with target regions on boundaries of shell-like domains

“…(b) In some instances, the target may themselves perform a slow diffusive motion along the boundary of the domain. Recently, some analytical approaches have been developed to study the survival of a diffusive particle in the presence of diffusive mobile sites in unbounded domains [85][86][87][88][89][90], which ultimately gives an access to the first-reaction time PDF. An extension of these approaches to the case in which a particle diffuses within the inter-boundary region while the target perform diffusive motion along the boundaries represents an interesting problem in its own right.…”

Statistics of boundary encounters by a particle diffusing outside a compact planar domain

“…In other words, a single encounter is counted when two particles hit each other and then moved away and become separated by a distance exceeding a. In turn, T aN is the first moment when the number of such encounters exceeds N. As an encounter of two species is the necessary step for most bimolecular reactions, reproduction in animals, and virus spreading in humans, both quantities can be used in describing various encounters in chemistry, biology and ecology (see [50][51][52][53][54][55][56][57][58] and references therein).…”

Conditioning two diffusion processes with respect to their first-encounter properties

“…In the present paper, our goal is to apply the general framework described above to the conditioning of two diffusion processes with respect to their first-encounter properties. Indeed, the first-encounter problem is very important for many applications (see the recent works [85,86] and references therein) and belongs to the broad field of first-passage problems that has attracted a lot of interest [87][88][89][90][91].…”