Extreme value statistics and arcsine laws for heterogeneous diffusion processes

TL;DR

This paper analyzes the extreme value statistics of heterogeneous diffusion processes with power-law spatially varying diffusion coefficients, deriving exact distributions and revealing contrasting features compared to standard Brownian motion, including modified arcsine laws.

Contribution

It provides exact probability distributions for maximum displacement, arg-maximum, residence time, and last-passage time in heterogeneous diffusion, highlighting differences from classical Brownian motion.

Findings

Distributions of key times differ from arcsine laws for non-zero alpha.

Identifies a critical alpha where residence time distribution peaks at half the duration.

Reveals rich contrasting features in heterogeneous diffusion compared to standard Brownian motion.

Abstract

Heterogeneous diffusion with spatially changing diffusion coefficient arises in many experimental systems like protein dynamics in the cell cytoplasm, mobility of cajal bodies and confined hard-sphere fluids. Here, we showcase a simple model of heterogeneous diffusion where the diffusion coefficient $D(x)$ varies in power-law way, i.e. $D(x) \sim |x|^{-\alpha}$ with the exponent $\alpha >-1$. This model is known to exhibit anomalous scaling of the mean squared displacement (MSD) of the form $\sim t^{\frac{2}{2+\alpha}}$ and weak ergodicity breaking in the sense that ensemble averaged and time averaged MSDs do not converge. In this paper, we look at the extreme value statistics of this model and derive, for all $\alpha$, the exact probability distributions of the maximum spatial displacement $M(t)$ and arg-maximum $t_m(t)$ (i.e. the time at which this maximum is reached) till duration $t$. In the second part of our paper, we analyze the statistical properties of the residence time $t_r(t)$ and the last-passage time $t_{\ell}(t)$ and compute their distributions exactly for all values of $\alpha$. Our study unravels that the heterogeneous version $(\alpha \neq 0)$ displays many rich and contrasting features compared to that of the standard Brownian motion (BM). For example, while for BM $(\alpha =0)$, the distributions of $t_m(t),~t_r(t)$ and $t_{\ell}(t)$ are all identical (\textit{\'{a} la} "arcsine laws" due to L\'{e}vy), they turn out to be significantly different for non-zero $\alpha$. Another interesting property of $t_r(t)$ is the existence of a critical $\alpha$ (which we denote by $\alpha _c=-0.3182$) such that the distribution exhibits a local maximum at $t_r = t/2$ for $\alpha < \alpha _c$