First passage time properties of diffusion with a broad class of stochastic diffusion coefficients

TL;DR

This work analyzes first passage time (FPT) statistics for diffusion with a broad class of positive, non-zero stochastic diffusion coefficients (DCs) in one dimension. By deriving diffusion equations for each sample path and averaging, the authors obtain a FPT distribution that is a superposition over the time-averaged diffusivity $S(t)$, expressed as $g(t,x_0)=\int_{\Omega}\frac{x_0 D(t;\omega)}{\sqrt{4\pi S^3(t;\omega) t^3}}\exp\left(-\frac{x_0^2}{4S(t;\omega)t}\right)P(d\omega)$. They show that absorption is certain in 1D, and stochastic DCs yield an enhanced likelihood of early arrivals compared to diffusion with the ensemble-averaged diffusivity, with the excess early-arrival probability governed by the distribution of $S(t)$ and the convergence of the normalized diffusivity to its mean. For ergodic DCs, the mean FPT is infinite and, at long times, the FPT distribution tends to the Lévy-Smirnov form; the speed of convergence depends on the autocovariance of the diffusivity fluctuations. The results extend to 3D diffusion outside a spherical absorbing boundary, where absorption is not guaranteed and the distance to the boundary modulates the excess early-arrival probability. Overall, the study highlights that non-ergodic or non-Markov DCs may be advantageous for efficient transport to distant targets in diffusion-limited processes, with potential implications for single-molecule reactions and molecular search in biology and chemistry.

Abstract

Diffusion in a heterogeneous environment or diffusion of a particle that shows conformational fluctuations can be described by Brownian motions with stochastic diffusion coefficients (sDCs). In this study, we investigate first passage time (FPT) properties of diffusion with a broad class of non-zero sDCs. We show that for diffusion in one-dimensional semi-infinite domain with an absorbing boundary, particles will eventually reach the boundary with probability one, and that diffusion with a sDC exhibits higher transport efficiency in an early arrival of particles at the boundary than would be expected under diffusion whose DC is the ensemble average (EA) of the sDC. When particles begin to reach an absorbing boundary before the change in a sDC occurs, diffusion with a sDC with a larger supremum exhibits a more efficient transport in an early arrival of particles at the boundary even if the EAs of sDCs are the same. For ergodic DCs, the mean FPT is infinite. In addition, if particles take a long time to reach an absorbing boundary, higher transport efficiency in an early arrival at the boundary almost disappears and the FPT distribution can be approximated by the Lévy-Smirnov distribution. We show that these three properties result from the convergence of the time average of the DC to the EA and the convergence speed is determined by the time scale of fluctuations in the DC. We finally discuss the similarities and differences of FPT properties between three-dimensional diffusion outside a spherical absorbing boundary and the one-dimensional diffusion. Our results indicate that fluctuations in DCs may need to be non-Markov and/or non-ergodic to allow efficient transport of particles to distant targets. Our results also suggest that fluctuations in a DC play an important role, for example, in diffusion-limited reactions triggered by single molecules in physics, chemistry, or biology.

Figures (4)

• Figure 1: Time dependence of the variance of $H(t)$. The solid line shows the time dependence estimated from Eq. (\ref{['eq:Hvarn']}). The dashed line shows the value for short times. The value is given by $\left( h_{+} - h_{-} \right)^{2} / 4$. Time is normalized by $t_{c}$.
• Figure 2: First passage time distributions (FPTDs) at different values of $t_{s}$. The solid lines show the distributions estimated from Eq. (\ref{['eq:rhogfgttdl']}). The open circles show the distributions estimated from simulations. The dashed lines show the distributions for the corresponding diffusion with ensemble-averaged diffusivity and estimated from Eq. (\ref{['eq:gbm2']}). (a) $t_{s} = 0.1$. (b) $t_{s} = 1$. (c) $t_{s} = 10$. (d) $t_{s} = 100$.
• Figure 3: The dependence of the excess cumulative probability (ECP) of early arriving particles (EAP) on $t_{s}$. The solid line shows the dependence estimated from Eq. (\ref{['eq:pr']}).
• Figure 4: Comparison of the first passage time distributions for different combinations of $h_{+}$ and $h_{-}$ values. The solid line shows the distribution for $h_{+} = 1.2$ and $h_{-} = 0.8$. The thick dashed line shows the distribution for $h_{+} = 1.4$ and $h_{-} = 0.6$. The dotted line shows the distribution for $h_{+} = 1.6$ and $h_{-} = 0.4$. The dash-dotted line shows the distribution for $h_{+} = 1.8$ and $h_{-} = 0.2$. The distributions are estimated from Eq. (\ref{['eq:rhogfgttdl']}). The thin dashed line shows the distribution for the corresponding diffusion with ensemble-averaged diffusivity and estimated from Eq. (\ref{['eq:gbm2']}). $t_{s} = 0.1$ and $D_{m} = 0.5$.