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 $\langle \cal{T}\rangle$, 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 $D_2 \ll D_1$ 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 $\langle \cal{T}\rangle$.