First-passage times to anisotropic partially reactive targets

We investigate restricted diffusion in a bounded domain towards a small partially reactive target in three- and higher-dimensional spaces. We propose a simple explicit approximation for the principal eigenvalue of the Laplace operator with mixed Robin-Neumann boundary conditions. This approximation involves the harmonic capacity and the surface area of the target, the volume of the confining domain, the diffusion coefficient, and the reactivity. The accuracy of the approximation is checked by using a finite-elements method. The proposed approximation determines also the mean first-reaction time, the long-time decay of the survival probability, and the overall reaction rate on that target. We identify the relevant lengthscale of the target, which determines its trapping capacity, and we investigate its relation to the target shape. In particular, we study the effect of target anisotropy on the principal eigenvalue by computing the harmonic capacity of prolate and oblate spheroids in various space dimensions. Some implications of these results in chemical physics and biophysics are briefly discussed.

Figure 1

A confining domain $\mathrm{\Omega }$ with reflecting boundary $\partial {\mathrm{\Omega }}_0$ (in gray). A particle diffuses (blue trajectory) from a starting point $\mathit{x}$ (black filled circle) towards an anisotropic target $\mathrm{\Gamma }$ (in red).

Figure 2

The eigenfunction $u_1^{\left(\infty \right)}\left(r\right)$ of the Laplace operator in a three-dimensional shell-like domain between two concentric spheres of radii $b=0.1$ and $R=1$, with the Dirichlet boundary condition on the target, $u_1^{\left(\infty \right)}\left(b\right)=0$, and the Neumann boundary condition on the outer sphere, $\left({\partial }_r{u}_1^{\left(\infty \right)}\right)\left(R\right)=0$. This eigenfunction is known explicitly (see [29] for details) and depends only on the radial coordinate $r=\left|\mathit{x}\right|$. For comparison, the harmonic function $v\left(r\right)=(1/b-1/r)/(1/b-1/R)$ satisfying $v\left(b\right)=0$ and $v\left(R\right)=1$ is shown by a dashed line.

Figure 3

(a), (b) The principal eigenvalue ${\lambda }_1^{\left(\infty \right)}$ of the Laplace operator for a perfectly reactive prolate (a) and oblate (b) spheroidal target with semiaxes $a\le b=0.2$ surrounded by a concentric reflecting spherical surface of radius $R=1$. Symbols present the numerical computation by a finite-elements method (see Appendix pp4), whereas thick lines show the approximate relation (14). In three dimensions, a thick blue line presents the improved approximation (15) with the “corrected” capacity $C^{\prime }$ from Eq. (16), whereas a thin blue line indicates the leading-order approximation (14). (c) The relative error of the above approximations shown by filled symbols for prolate spheroids and by empty symbols for oblate spheroids in ${\mathbb{R}}^d$ with $d=3,4,5,6$ (see the legend). Note that a minor increase of the relative error for $d=6$ at small $a$ can be a numerical artifact due to an insufficient mesh size.

Figure 4

The trapping length $L$ from Eqs. (49) and (59) of prolate (lines) and oblate (symbols) spheroids for several dimensions $d$. Note that $L/b=1/(d-2)$ at $a/b=1$.

Figure 5

The principal eigenvalue ${\lambda }_1^{\left(q\right)}$ as a function $q$ for prolate (a) and oblate (b) spheroidal targets with semiaxes $a=0.1$ and $b=0.2$ surrounded by a reflecting concentric spherical surface of radius $R=1$. Symbols present the numerical computation by a finite-elements method (see Appendix pp4), whereas thick lines show the approximate relation (5). In three dimensions, a thick blue line presents Eq. (5) with the “corrected” capacity $C^{\prime }$ from Eq. (16), whereas a thin blue line corresponds to the capacity $C$. The trapping length $L$ given by Eqs. (49) and (59) is 0.1300, 0.0596, 0.0379, and 0.0276 for prolate spheroids, and 0.1669, 0.0864, 0.0588, and 0.0447 for oblate spheroids, with $d=3,4,5,6$, respectively.

Figure 6

(a) A prolate spheroidal target (in red) is enclosed by an outer reflecting sphere (in gray). (b) An equivalent planar domain with an elliptic target (in red) and an outer circular reflecting boundary (in gray).