Correlation between the first-reaction time and the acquired boundary local time

TL;DR

This work develops an encounter-based framework to quantify the joint statistics of the first-reaction time $\tau$ and the boundary local time $\ell_\tau$ for diffusion with a partially reactive boundary. By linking the first-crossing time of $\ell$, $\mathcal{T}_\ell$, to the exponential threshold $\hat{\ell}$ with rate $q$, the authors derive the joint density $\mathcal{P}_q(\ell,t|\bm{x}_0)= q e^{-q\ell} U(\ell,t|\bm{x}_0)$ and obtain the correlation coefficient $C_q$ between $\tau$ and $\ell_\tau$. They provide exact results for basic geometries (ball, disk, annulus, spherical shell) and validate them with Monte Carlo simulations, extending the analysis to disordered media with interior obstacles via annulus and shell approximations. The study reveals universal limits $C_q\to1$ as $q\to0$ and $C_q\to0$ as $q\to\infty$, with geometry-dependent corrections in intermediate regimes, offering insights into diffusion-controlled reactions on surfaces and potential design principles for porous or catalytic media.

Abstract

We investigate the statistical correlation between the first-reaction time of a diffusing particle and its boundary local time accumulated until the reaction event. Since the reaction event occurs after multiple encounters of the particle with a partially reactive boundary, the boundary local time as a proxy for the number of such encounters is not independent of, but intrinsically linked to, the first-reaction time. We propose a universal theoretical framework to derive their joint probability density and, in particular, the correlation coefficient. To illustrate the dependence of these correlations on the boundary reactivity and shape, we obtain explicit analytical solutions for several basic domains. The analytical results are complemented by Monte Carlo simulations, which we employ to examine the role of interior obstacles on correlations in disordered media. Applications of these statistical results in chemical physics are discussed

Figures (7)

• Figure 1: Computational domains with interior obstacles of the same radius $R$: the central one (in blue) is the reactive target, while the others are inert. (a) Regular square-lattice packing of $m \times m$ identical disks inside the unit disk, with $m=5$. The shown configuration has the maximal radius $R_{\mathrm{max}}$ such that all disks touch each other compactly without overlapping. (b) Random packing of 25 disks with the maximal radius $R_{\mathrm{max}}$ that corresponds to a fixed area occupation of 50%. The coordinates of all disks are provided in Appendix \ref{['app:random25']}. (c) Regular cubic-lattice packing of $m \times m \times m$ identical balls inside the unit ball with $m=3$, and the maximal radius $R_{\mathrm{max}}$.
• Figure 2: Correlation coefficient $C_q(\circ)$ for (a) a unit ball and (b) a unit disk ($R=1$). Comparison between Monte Carlo results (MC, blue dots) and analytical expressions (\ref{['eq:Cq_sphere_circ']}, \ref{['eq:Cq_disk_circ']}) (orange curves). Numerical statistics are obtained from $N=10^6$ particles, with $D=1$. Green dashed lines indicate the asymptotic behaviors $C_q(\circ) \approx 5\sqrt{7/13}/(qR)$ for $d=3$ and $C_q(\circ) \approx 4 \sqrt{3/5} / (qR)$ for $d=2$ in the limit $q\to\infty$.
• Figure 3: Joint histograms of the FRT $\tau$ and the BLT $\ell_\tau$ for $q = 10^{-2}, 10^{-1}, 10^0, 10^1$ obtained by Monte Carlo simulations with $N=10^6$ particles, where the starting points are uniform inside the unit ball. Both $\tau$ and $\ell_\tau$ are normalized by their respective mean values. In each panel, the scatter of blue dots represents individual Monte Carlo results of the pair $(\tau,\ell_\tau)$, whose density approximated the underlying joint probability density $\mathcal{P}_q(\ell, t | \circ)$ from Eq. (\ref{['eq:Pjoint']}). The marginal distributions of $\tau / \langle \tau \rangle$ and $\ell_\tau / \langle \ell_\tau \rangle = q \ell_\tau$ are shown above and right of each joint plot.
• Figure 4: Mean FRT and correlation coefficient for the domain shown in Fig. \ref{['fig:dm5sq']} with $m = 5$. Numerical results are obtained by Monte Carlo simulations with $N=10^6$ particles that are initially uniformly distributed inside the domain. Symbols correspond to different reactivities: $q=10^3$ (squares), $q=10^2$ (downward triangles), $q=10^1$ (upward triangles), and $q=10^0$ (diamonds). (a) Mean FRT as a function of the scaled disk radius $R/R_{\mathrm{max}}$. The solid black curve represents the mean first-passage time (i.e., $q=\infty$), and colored dashed curves show the annulus approximation (\ref{['eq:tau1_annulus_circ']}). (b) Correlation coefficient $C_q(\circ)$ as a function of scaled disk radius. Colored dashed curves correspond to the annulus approximation (\ref{['eq:Cq_annulus_circ']}), in which $L$ is replaced by $L_{\mathrm{eff}}$ from Eq. (\ref{['eq:annulus_app']}).
• Figure 5: Role of the obstacle-lattice density $m$ at fixed reactivity $q=10^2$. Numerical results obtained by Monte Carlo simulations with $N=10^6$ particles that are initially uniformly distributed inside the domain shown in Fig. \ref{['fig:dm5sq']}. Numerical results are shown for: $m=3$ (squares), $m=5$ (downward triangles), $m=7$ (upward triangles), and $m=9$ (diamonds). (a) Mean FRT as a function of the scaled disk radius $R / R_{\mathrm{max}}$. Colored dashed curves exhibit the annulus approximation (\ref{['eq:tau1_annulus_circ']}) for each $m$. (b) Correlation coefficient $C_q(\circ)$ as a function of the scaled disk radius. Colored dashed curves exhibit the annulus approximation (\ref{['eq:Cq_annulus_circ']}), in which $L$ is replaced by $L_{\mathrm{eff}}$ from Eq. (\ref{['eq:annulus_app']}).