First-Passage Times for Random Walks in Bounded Domains

We present a novel computational method of first-passage times between a starting site and a target site of regular bounded lattices. We derive accurate expressions for all the moments of this first-passage time, validated by numerical simulations. Their range of validity is discussed. We also consider the case of a starting site and two targets. In addition, we present the extension to continuous Brownian motion. These results are of great relevance to any system involving diffusion in confined media.

Figure 1

Modifications of the original lattice: Arrows denote one-way links.

Figure 2

3D—Influence of the distance between the source and the target. Simulations (red crosses) vs theory (plain line). The domain is a cube of side 31, the target being in the middle of it. The source takes all the positions in a cube of side 15 centered on the target.

Figure 3

3D—Influence of the distance between the target and a flat boundary. Simulations (red crosses) vs theory (plain line). The domain is a cube of side 41 whose center is at $(0,0,0)$; the source is at $(0,0,x-15)$, and the target is at $(0,0,x-20)$.

Figure 4

2D—Influence of the distance between the source and the target. Simulations (red crosses) vs theory: blue dotted line, without the uniform correction; black plain line (in the middle of the set of points), with this correction. The domain is a square of side 61, the target being in the middle of it, and the source takes all possible positions.

Figure 5

3D—Higher-order moments: Theory (black curve) vs simulation (red crosses). The blue dotted curve is the moments of the exponential distribution whose average time is the MFPT. The $n$th moment is normalized by $N^n$; the domain is a cube of side 51 centered on the target at $(0,0,0)$ and the source at $(2,2,1)$.

Figure 6

2D—Two-target simulations. Simulations (red crosses) vs theory (plain line). One target is fixed at $(-5,0)$; the source is fixed at $(5,0)$; the other target is at $(x,3)$. The domain is a square of side 201; the middle is the point $(0,0)$.