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

• Published in: Nature (London) 2016-06, Vol.534 (7607), p.356-359
• Main Authors: Guérin, T., Levernier, N., Bénichou, O., Voituriez, R.
• Format: Article
• Language: English
• Subjects: 639/766/530/2804; 639/766/94; Condensed Matter; Disease transmission; Humanities and Social Sciences; Kinetics; letter; Mathematical research; multidisciplinary; Physics; Science; Statistical Mechanics; Stochastic models; Stochastic processes
• ISSN: 0028-0836; 1476-4687
• DOI: 10.1038/nature18272

An analytical method of determining the mean first-passage time (the time taken by a random walker in confinement to reach a target point) is presented for a Gaussian non-Markovian random walker, thus revealing the importance of memory effects in first-passage statistics. The statistics of a random walker with memory Stochastic processes that fulfil the Markov property — that is, processes that are memoryless in the sense that future states depend only on present and not on past states — are relatively well understood. Exact formulas for various important properties, for example, to describe diffusion of reactants, have been developed under this assumption. However, many processes are not well described by a Markov process and a non-Markovian theory has proved hard to formulate. Here Olivier Bénichou and colleagues develop a theoretical framework to calculate a key property of non-Markovian processes, namely the mean first-passage time (the first-passage time is the time a random walker takes to reach a target point in a confining domain). Comparing with numerical simulations, they confirm that this framework provides a good description of many important processes, such as telomere motion or tracer diffusion. Although many extensions of the theory, such as getting a grip on the full statistics of the first-passage time, are yet to be conceived, this result is a big step towards a rigorous theoretical description of non-Markovian processes. The first-passage time, 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 1 . Its importance comes from its crucial role in quantifying the efficiency of processes as varied as diffusion-limited reactions 2 , 3 , target search processes 4 or the spread of diseases 5 . Most methods of determining the properties of first-passage time in confined domains have been limited to Markovian (memoryless) processes 3 , 6 , 7 . However, as soon as the random walker interacts with its environment, memory effects cannot be neglected: that is, the future motion of the random walker does not depend only on its current position, but also on its past trajectory. Examples of non-Markovian dynamics include single-file diffusion in narrow channels 8 , or the motion of a tracer particle either attached to a polymeric chain 9 or diffusing in simple 10 or complex fluids such as nematics 11 , dense soft colloids 12 or viscoelastic solutions 13 , 14 . H