Mean first-passage times of non-Markovian random walkers in confinement

Abstract

The first-passage time (FPT), defined as the time a random walker takes to reach a target point in a confining domain, is a key quantity in the theory of stochastic processes. Its importance comes from its crucial role to quantify the efficiency of processes as varied as diffusion-limited reactions, target search processes or spreading of diseases. Most methods to determine the FPT properties in confined domains have been limited to Markovian (memoryless) processes. However, as soon as the random walker interacts with its environment, memory effects can not be neglected. Examples of non Markovian dynamics include single-file diffusion in narrow channels or the motion of a tracer particle either attached to a polymeric chain or diffusing in simple or complex fluids such as nematics \cite{turiv2013effect}, dense soft colloids or viscoelastic solution. Here, we introduce an analytical approach to calculate, in the limit of a large confining volume, the mean FPT of a Gaussian non-Markovian random walker to a target point. The non-Markovian features of the dynamics are encompassed by determining the statistical properties of the trajectory of the random walker in the future of the first-passage event, which are shown to govern the FPT kinetics.This analysis is applicable to a broad range of stochastic processes, possibly correlated at long-times. Our theoretical predictions are confirmed by numerical simulations for several examples of non-Markovian processes including the emblematic case of the Fractional Brownian Motion in one or higher dimensions. These results show, on the basis of Gaussian processes, the importance of memory effects in first-passage statistics of non-Markovian random walkers in confinement.

Submitted version. Supplementary Information can be found on the Nature website : http://www.nature.com/nature/journal/v534/n7607/full/nature18272.html

References in corpus (3)

• First-passage times in complex scale-invariant media
• Non-Markovian polymer reaction kinetics
• Anomalous Polymer Dynamics Is Non-Markovian: Memory Effects and The Generalized Langevin Equation Formulation

Cited by in corpus (15)

• Generalized arcsine laws for fractional Brownian motion
• Inverse square Lévy walks are not optimal search strategies for $d\ge 2$
• Emergence of molecular friction in liquids: bridging between the atomistic and hydrodynamic pictures
• Survival probability of stochastic processes beyond persistence exponents
• First-encounter time of two diffusing particles in confinement
• Exact enumeration approach to first-passage time distribution of non-Markov random walks
• Search efficiency of biased migration towards stationary or moving targets in heterogeneously structured environments
• Target finding in fibrous biological environments
• Non-stationary Markovian Replication Process causing Diverse Diffusions
• Memory in random bouncing ball dynamics
• Sign-changes as a universal concept in first-passage time calculations
• Mean first-passage time of an anisotropic diffusive searcher
• Interface Collisions
• Brownian motion: from kinetics to hydrodynamics
• Trapping problem on star-type graphs with applications