Average volume of the domain visited by randomly injected spherical Brownian particles in d dimensions

Diffusion-limited encounter rate in a three-dimensional lattice of connected compartments studied by Brownian-dynamics simulations

We considered the rate at which a diffusing particle encounters a target in a three-dimensional lattice of compartments with semipermeable walls. This work expands a previous theory [Li et al., Phys. Rev. Lett. 113, 028303 (2014)] for the encounter rate in the dilute limit of targets to the general case of any density of targets. We also used Brownian dynamics simulations to evaluate the approximations in the analytical theory. We find that the largest errors in the analytical theory are on the order of 10%. This work therefore demonstrates an analytical theory capable of describing the encounter rates in compartmentalized environments for any level of confinement and any target density.

Calculated rates of diffusion-limited reactions in a three-dimensional network of connected compartments: application to porous catalysts and biological systems

We describe the diffusion limit for reaction rates in a three-dimensional system of connected compartments. This model exhibits the length-scale dependent diffusion that can be observed in many heterogeneous environments, such as porous catalysts and biological environments. We obtain a simple analytical expression for the diffusion limit applicable to any scale of the compartment confinement. This diffusion limit exceeds the classic Smoluchowski diffusion limit that was derived for homogeneous environments but is often applied to biological reactions in heterogeneous environments. We expect our new diffusion limit to provide a more appropriate upper bound on reaction rates in biological systems, porous structures, and other heterogeneous environments where obstacles create local confinement.

Diffusion-controlled annihilation A+B-->0: the growth of an A-particle island from a localized A source in the B-particle sea

We present the growth dynamics of an island of particles A injected from a localized A source into a sea of particles B and dying in the course of diffusion-controlled annihilation A+B-->0. We show that in the one-dimensional (1D) case the island grows unlimitedly at any source strength Lambda, and the dynamics of its growth does not depend asymptotically on the diffusivity of B particles. In the 3D case the island grows only at Lambda> Lambda(c), achieving asymptotically a stationary state (static island). In the marginal 2D case the island unlimitedly grows at any Lambda but at Lambda< Lambda(*) the time of its formation becomes exponentially large. For all cases the numbers of surviving and dying A particles are calculated, and the scaling of the reaction zone is derived.

Occupancy of a single site by many random walkers

We consider an infinite number of noninteracting lattice random walkers with the goal of determining statistical properties of the time, out of a total time T, that a single site has been occupied by n random walkers. Initially the random walkers are assumed uniformly distributed on the lattice except for the target site at the origin, which is unoccupied. The random-walk model is taken to be a continuous-time random walk and the pausing-time density at the target site is allowed to differ from the pausing-time density at other sites. We calculate the dependence of the mean time of occupancy by n random walkers as a function of n and the observation time T. We also find the variance for the cumulative time during which the site is unoccupied. The large-T behavior of the variance differs according as the random walk is transient or recurrent. It is shown that the variance is proportional to T at large T in three or more dimensions, it is proportional to T(3/2) in one dimension and to T ln T in two dimensions.

Territory covered by N random walkers

The problem of evaluating the number of distinct sites S(N)(t) covered up to time t by N random walkers is revisited. For the nontrivial time regime and for N>>1 we show how to get the asymptotic behavior of S(N)(t) and we calculate the main and first two corrective terms. The mth corrective term decays mildly as 1/ln(m) N. For d-dimensional (d=1,2,3) simple cubic lattices, the main term is the volume of the hypersphere of radius [(ln N(2))2Dt/d](1/2), D being the diffusion constant, and the corrective terms account for the roughening of the surface of the set of visited sites.