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

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

💡 Research Summary

This paper addresses a previously unexplored statistical relationship between the first‑reaction time (τ) of a diffusing particle and the boundary local time (ℓ_τ) accumulated up to the reaction event on a partially reactive surface. While classical first‑passage‑time (FPT) theory assumes a perfectly absorbing (Dirichlet) boundary, many realistic systems feature a Robin boundary with finite reactivity κ, so that a particle may collide with the surface many times before a successful reaction occurs. The authors adopt the encounter‑based framework, defining the boundary local time ℓ_t as the limit of the rescaled number of crossings of a thin layer adjacent to the reactive region Γ. Reaction is assumed to happen when ℓ_t exceeds a random threshold ˆℓ, whose distribution ψ(ℓ) models the microscopic chemistry; the most common choice is an exponential law ψ(ℓ)=q e^{−qℓ} with q=κ/D.

Under this construction τ = inf{t>0 : ℓ_t>ˆℓ}=T_{ˆℓ} is the first‑crossing time of a random threshold. Consequently the joint density of (ℓ,τ) factorises as
P(ℓ,t|x₀)=ψ(ℓ) U(ℓ,t|x₀),
where U(ℓ,t|x₀) is the first‑crossing‑time density for a fixed deterministic threshold ℓ. The authors obtain U via a Laplace transform in time and a spectral expansion of a generalized Steklov eigenvalue problem:

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