First-passage times, mobile traps, and Hopf bifurcations

First-encounter time of two diffusing particles in two- and three-dimensional confinement

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 t_{B} 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=D_{1}+D_{2}, 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 D_{1}/D_{2}. We also investigate the slow-diffusion limit when D_{2}≪D_{1}, and we 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〉.

Brownian dynamics simulations for the narrow escape problem in the unit sphere

The narrow escape problem is a first-passage problem that concerns the calculation of the time needed for a Brownian particle to leave a domain with localized absorbing boundary traps, such that the measure of these traps is asymptotically small compared to the domain size. A common model for the mean first-passage time (MFPT) as a function of particle's starting location in a given domain with constant diffusivity is given by a Poisson partial differential equation subject to mixed Dirichlet-Neumann boundary conditions. The primary objective of this work is to perform direct numerical simulations of multiple particles undergoing Brownian motion in a three-dimensional spherical domain with boundary traps, compute MFPT values by averaging Brownian escape times, and compare these with explicit asymptotic results obtained previously by approximate solution of the Poisson problem. A close agreement of MFPT values is observed already at 10^{4} particle runs from a single starting point, providing a computational validation of the Poisson equation-based continuum model. Direct Brownian dynamics simulations are also used to study additional features of particle dynamics in narrow escape problems that cannot be captured in a continuum approach, such as average times spent by particles in a thin layer near the domain boundary, and effects of isotropic vs anisotropic near-boundary diffusion.

First-encounter time of two diffusing particles in confinement

We investigate how confinement may drastically change both the probability density of the first-encounter time and the associated survival probability in the case of two diffusing particles. To obtain analytical insights into this problem, we focus on two one-dimensional settings: a half-line and an interval. We first consider the case with equal particle diffusivities, for which exact results can be obtained for the survival probability and the associated first-encounter time density valid over the full time domain. We also evaluate the moments of the first-encounter time when they exist. We then turn to the case with unequal diffusivities and focus on the long-time behavior of the survival probability. Our results highlight the great impact of boundary effects in diffusion-controlled kinetics even for simple one-dimensional settings, as well as the difficulty of obtaining analytic results as soon as the translational invariance of such systems is broken.

Globally optimal volume-trap arrangements for the narrow-capture problem inside a unit sphere

The determination of statistical characteristics for particles undergoing Brownian motion in constrained domains has multiple applications in various areas of research. This work presents an attempt to systematically compute globally optimal configurations of traps inside a three-dimensional domain that minimize the average of the mean first passage time (MFPT) for the narrow capture problem, the average time it takes a particle to be captured by any trap. For a given domain, the mean first passage time satisfies a linear Poisson problem with Dirichlet-Neumann boundary conditions. While no closed-form general solution of such problems is known, approximate asymptotic MFPT expressions for small traps in a unit sphere have been found. These solutions explicitly depend on trap parameters, including locations, through a pairwise potential function. After probing the applicability limits of asymptotic formulas through comparisons with numerical and available exact solutions of the narrow capture problem, full three-dimensional global optimization was performed to find optimal trap positions in the unit sphere for 2≤N≤100 identical traps. The interaction energy values and geometrical features of the putative optimal trap arrangements are presented.

Hybrid asymptotic-numerical approach for estimating first-passage-time densities of the two-dimensional narrow capture problem

A hybrid asymptotic-numerical method is presented for obtaining an asymptotic estimate for the full probability distribution of capture times of a random walker by multiple small traps located inside a bounded two-dimensional domain with a reflecting boundary. As motivation for this study, we calculate the variance in the capture time of a random walker by a single interior trap and determine this quantity to be comparable in magnitude to the mean. This implies that the mean is not necessarily reflective of typical capture times and that the full density must be determined. To solve the underlying diffusion equation, the method of Laplace transforms is used to obtain an elliptic problem of modified Helmholtz type. In the limit of vanishing trap sizes, each trap is represented as a Dirac point source that permits the solution of the transform equation to be represented as a superposition of Helmholtz Green's functions. Using this solution, we construct asymptotic short-time solutions of the first-passage-time density, which captures peaks associated with rapid capture by the absorbing traps. When numerical evaluation of the Helmholtz Green's function is employed followed by numerical inversion of the Laplace transform, the method reproduces the density for larger times. We demonstrate the accuracy of our solution technique with a comparison to statistics obtained from a time-dependent solution of the diffusion equation and discrete particle simulations. In particular, we demonstrate that the method is capable of capturing the multimodal behavior in the capture time density that arises when the traps are strategically arranged. The hybrid method presented can be applied to scenarios involving both arbitrary domains and trap shapes.