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FPTOPT Report Summary

Project ID: 277998
Funded under: FP7-IDEAS-ERC
Country: France

Final Report Summary - FPTOPT (First-passage times and optimization of target search strategies)

How long does it take a random walker to reach a given target? This quantity, known as a first-passage time (FPT), has been the subject of a growing number of theoretical studies over the past decade. The importance of FPTs originates from the crucial role played by properties related to first encounters in various real situations, including transport in disordered media, diffusion limited reactions, or more generally target search processes. However, determining the FPT in confined geometries remains a largely open problem. (i) First-passage times in confinement and (ii) their optimization were at the heart of this project.

The main class of results obtained regarding point (i) concerns the determination of FPTS of non-Markovian processes (random walks with memory).

An important advance concerns the determination of mean first-passage times (MFPTs) of extended searchers like polymers. Relying on a description of the dynamics in terms of a Rouse model (the simplest model of polymer dynamics), we have determined the kinetics of both inter and intra molecular reactions. In particular, our analysis reveals a strongly non-Markovian regime in one dimension, where the Markovian and non-Markovian dependences of the reaction time on the initial distance are different.
At this occasion, a new method of calculation has been developed that we have generalized to arbitrary Gaussian non Markovian processes.

Concerning the question (ii) of the potential optimisation of the FPT, 3 main types of results have been obtained :
a) We have derived exact expressions of the mean FPT to a bulk target for a random searcher that performs boundary-mediated diffusion (alternance of surface and bulk diffusions) in a circular domain. Although nonintuitive for bulk targets, it is found that boundary excursions, if fast enough, can minimize the search time. A scaling analysis generalizes these findings to domains of arbitrary shapes and underlines their robustness. Overall, these results provide a generic mechanism of optimization of search kinetics in interfacial systems, which could have important implications in chemical physics. In the context of animal behavior sciences, it shows that following the boundaries of a domain can accelerate a search process, and therefore suggests that thigmotactism could be a kinetically efficient behavior.

b) We have also considered a minimal model of persistent random searcher with a short range memory. We calculated exactly for such a searcher the mean FPT to a target in a bounded domain and find that it admits a nontrivial minimum as function of the persistence length. This reveals an optimal search strategy which differs markedly from the simple ballistic motion obtained in the case of Poisson distributed targets. Our results show that the distribution of targets plays a crucial role in the random search problem. In particular, in the biologically relevant cases of either a single target or regular patterns of targets, we find that, in strong contrast to repeated statements in the literature, persistent random walks with exponential distribution of excursion lengths can minimize the search time, and in that sense perform better than any Levy walk.

c) Finally, we have been interested in the time taken a random searcher to visit all sites of a given domain? This time, known as the cover time, is a key observable to quantify the efficiency of exhaustive searches, which require a complete exploration of an area and not only the discovery of a single target. Examples range from immune-system cells chasing pathogens to animals harvesting resources, from robotic exploration for cleaning or demining to the task of improving search algorithms. Despite its broad relevance, the cover time has remained elusive and so far explicit results have been scarce and mostly limited to regular random walks. We have determined the full distribution of the cover time for a broad range of random search processes, including Lévy strategies, intermittent strategies, persistent random walks and random walks on complex networks, and reveal its universal features. We showed that for all these examples the mean cover time can be minimized, and that the corresponding optimal strategies also minimize the mean search time for a single target, unambiguously pointing towards their robustness.

The developments on first-passage times performed in the frame of this ERC project have also allowed us to investigate a problem that had not been envisaged at the beginning of the project. It concerns the key question of the determination of transport properties of a driven tracer in a hard-core lattice gas. This question is crucial both in the context of active microrheology experiments and in the fundamental field of non equilibrium statistical physics. Technically, the developments we have performed have been made possible thanks to the determination of the first-passage properties that were at the heart of the project. We have obtained an unexpected number of results in this field of driven tracer diffusion in a hard-core lattice gas, such as a spectacular superdiffusion induced by the very geometry of the system in crowded systems or the existence of a velocity anomaly.

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