Extreme value statistics and arcsine laws for heterogeneous diffusion processes

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 $(\alpha \ne 0)$ 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{á la} "arcsine laws" due to Lévy), 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 $\alpha <{\alpha }_c$

arXiv comment18 pages, 9 figures

References in corpus (59)

• Universality Classes for Extreme Value Statistics
• State-dependent diffusion: thermodynamic consistency and its path integral formulation
• Stochastic Ergodicity Breaking: a Random Walk Approach
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme value statistics of correlated random variables: a pedagogical review
• Layering and position-dependent diffusive dynamics of confined fluids
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Top eigenvalue of a random matrix: large deviations and third order phase transition
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• Drift without flux: Brownian walker with a space dependent diffusion coefficient
• Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes
• The Local and the Occupation Time of a Particle Diffusing in a Random Medium
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity
• Maximal Height Scaling of Kinetically Growing Surfaces
• Exact distribution of the maximal height of p vicious walkers
• Infinite Ergodic Theory for Heterogeneous Diffusion Processes
• Statistics at the tip of a branching random walk and the delay of traveling waves
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
• On distributions of functionals of anomalous diffusion paths
• Spatial extent of an outbreak in animal epidemics
• On the time to reach maximum for a variety of constrained Brownian motions
• Extreme Value Statistics and Traveling Fronts: An Application to Computer Science
• Generalized arcsine laws for fractional Brownian motion
• Properties of additive functionals of Brownian motion with resetting
• Hitting probability for anomalous diffusion processes
• Traveling Waves, Front Selection, and Exact Nontrivial Exponents in a Random Fragmentation Problem
• Generalised `Arcsine' laws for run-and-tumble particle in one dimension
• Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
• Time to reach the maximum for a random acceleration process
• Condensation and Extreme Value Statistics
• Distribution of the time at which N vicious walkers reach their maximal height
• Extremal statistics of curved growing interfaces in 1+1 dimensions
• Residence Time Statistics for Normal and Fractional Diffusion in a Force Field
• Statistical Properties of Functionals of the Paths of a Particle Diffusing in a One-Dimensional Random Potential
• Extremal statistics for stochastic resetting systems
• Residence Time Distribution for a Class of Gaussian Markov Processes
• Generalized persistence exponents: an exactly soluble model
• Occupation time of a run-and-tumble particle with resetting
• Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries
• Random acceleration process under stochastic resetting
• The longest excursion of stochastic processes in nonequilibrium systems
• Distribution of the time of the maximum for stationary processes
• Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes
• Time between the maximum and the minimum of a stochastic process
• Distribution of the Time Between Maximum and Minimum of Random Walks
• Traveling Front Solutions to Directed Diffusion Limited Aggregation, Digital Search Trees and the Lempel-Ziv Data Compression Algorithm
• Extremal Properties of Random Systems
• Exact extreme value statistics at mixed order transitions
• Ergodic properties of heterogeneous diffusion processes in a potential well
• Arcsine Laws in Stochastic Thermodynamics
• Run-and-Tumble particle in inhomogeneous media in one dimension
• Residence time statistics for $N$ blinking quantum dots and other stochastic processes
• Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models
• Brownian Motion in wedges, last passage time and the second arc-sine law
• Extreme value theory for constrained physical systems
• Last-passage time for linear diffusions and application to the emptying time of a box

Cited by in corpus (17)

• First-passage Brownian functionals with stochastic resetting
• Time to reach the maximum for a stationary stochastic process
• First-passage functionals for Ornstein Uhlenbeck process with stochastic resetting
• Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
• First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion
• Efficiency of random search with space-dependent diffusivity
• Extremal statistics for a resetting Brownian motion before its first-passage time
• Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier
• Occupation time of a system of Brownian particles on the line with steplike initial condition
• Extreme value statistics of positive recurrent centrally biased random walks
• Optimal conditions for first passage of jump processes with resetting
• Generalized arcsine laws for a sluggish random walker with subdiffusive growth
• Exact joint distributions of three global characteristic times for Brownian motion
• Full Record Statistics of 1d Random Walks
• Ergodic properties of functionals of Gaussian processes
• Short-time large deviations of first-passage functionals for high-order stochastic processes
• Records and occupation time statistics for area-preserving maps