Mean first-passage time to a small absorbing target in three-dimensional elongated domains

We derive an approximate formula for the mean first-passage time (MFPT) to a small absorbing target of arbitrary shape inside an elongated domain of a slowly varying axisymmetric profile. For this purpose, the original Poisson equation in three dimensions is reduced to an effective one-dimensional problem on an interval with a semipermeable semiabsorbing membrane. The approximate formula captures correctly the dependence of the MFPT on the distance to the target, the radial profile of the domain, and the size and the shape of the target. This approximation is validated by Monte Carlo simulations.

Figure 1

Projection onto the $xz$ plane of three domains used for Monte Carlo simulations: truncated cylinder, with $r\left(z\right)=1$ (i); truncated cone, with $r\left(z\right)=1+z/\ell$ (ii); and a domain with an oscillating profile $r\left(z\right)=1+\frac{1}{2}sin(2\pi z/\ell )$ (iii), with $\ell =5$. The vertical dotted line shows the symmetry axis in the $z$ direction ($r=0$); the horizontal dashed line indicates the location of uniformly distributed starting points (at $z=2$). A target (in black) is located at $(x_T,0,\ell /2)$: a disk of radius $\rho =0.2$ with $x_T=0$ (i), a cube of edge $2\rho =0.4$ with $x_T=0.6$ (ii), and a sphere of radius $\rho =0.2$ with $x_T=-0.4$ (iii). On the right is an example of a simulated trajectory inside the domain with an oscillating profile, colored from dark blue to dark red according to elapsed time until the first passage to the target (black sphere) at the center.

Figure 2

Mean first-passage time as a function of $r_T$ for diffusion towards a target centered at ${\mathit{x}}_T=(r_T,0,\ell /2)$ with (a) $\rho =0.2$ or (b) $\rho =0.1$, $D=1$, $\ell =5$, the starting point ${\mathit{x}}_0$ uniform at the cross section at $z_0=2$, in three settings shown in Fig. 1: (i) a disk of radius $\rho$ inside a truncated cylinder of radius 1, (ii) a cube of edge $2\rho$ inside a truncated cone $r\left(z\right)=1+z/\ell$, and (iii) a sphere of radius $\rho$ inside an oscillating domain $r\left(z\right)=1+\frac{1}{2}sin(2\pi z/\ell )$. Lines show theoretical predictions (8); symbols present the mean values from 1000 realizations obtained via Monte Carlo simulations with the time step $\delta ={10}^{-6}$.