arXiv subjects

R. Voituriez

Publications and source records attributed to R. Voituriez.

At least 19 recordsLinked to original sources

arXiv2026

Aging Record Statistics in Saturating Self-Interacting Random Walks

The record age tau_k, defined as the time between the k-th and k+1-st record-breaking events, is a central observable of extreme-value statistics. In Markovian processes, the absence of memory makes tau_k independent of k. How memory breaks this invariance and induces aging, meaning a dependence of tau_k on k, remains a fundamental question, closely connected to widely observed aging phenomena in non-Markovian dynamics. In this Letter, we derive the exact asymptotic distribution of tau_k for saturating self-interacting random walks, a broad class of non-Markovian processes. We uncover two asymptotic regimes, in agreement with recent scaling predictions: at short times (tau much smaller than k squared), record statistics are governed by the geometry of the explored region, while at long times (tau much larger than k squared), memory effects become subdominant and reduce to nontrivial prefactor corrections. Our exact result provides a rare analytic window beyond scaling theory and extends to a framework that fully quantifies aging dynamics in the presence of saturating self-interaction.

cond-mat.stat-mech↗

arXiv2024

Long-term memory induced correction to Arrhenius law

The Kramers escape problem is a paradigmatic model for the kinetics of rare events, which are usually characterized by Arrhenius law. So far, analytical approaches have failed to capture the kinetics of rare events in the important case of non-Markovian processes with long-term memory, as occurs in the context of reactions involving proteins, long polymers, or strongly viscoelastic fluids. Here, based on a minimal model of non-Markovian Gaussian process with long-term memory, we determine quantitatively the mean FPT to a rare configuration and provide its asymptotics in the limit of a large energy barrier $E$. Our analysis unveils a correction to Arrhenius law, induced by long-term memory, which we determine analytically. This correction, which we show can be quantitatively significant, takes the form of a second effective energy barrier $E'<E$ and captures the dependence of rare event kinetics on initial conditions, which is a hallmark of long-term memory. Altogether, our results quantify the impact of long-term memory on rare event kinetics, beyond Arrhenius law.

cond-mat.stat-mech↗

arXiv2024

Forming 1D Periodic J-aggregates by Mechanical Bending of BNNTs: Evidence of Activated Molecular Diffusion

Driving molecular assembly into micrometer-scale patterns is key for defining advanced materials of interest in various fields, including life sciences, photovoltaics, and quantum photonics. However, the driving process competes with other forces, such as Brownian motion, ripening phenomena, capillary forces, and non-specific adsorption. Here we report on a guided diffusion mechanism of luminescent dye molecules encapsulated inside boron nitride nanotubes (BNNTs). Correlative measurements between BNNT bending and molecular position along the BNNT axis reveal an efficient and long-range migration of dyes from curved to straight regions of the nanotube. This curvature activated diffusion forms clusters of bright J-aggregates in periodic patterns of well-defined spacing and length. A phenomenological model of guided molecular transport in bended BNNTs is used to describe this directed 1D diffusion inside BNNT. It is shown to accurately predict the position and morphologies of a J-aggregate as a function of nanotube length. Coupling topological stimuli to 1D molecular diffusion at the nanoscale is here presented as an interesting tool capable of reconfiguring various emissive patterns of functional molecules at the mesoscopic scale.

cond-mat.mes-hall↗

arXiv2024

Evidence and quantification of memory effects in competitive first passage events

Splitting probabilities quantify the likelihood of a given outcome out of competitive events. This key observable of random walk theory, historically introduced as the gambler's ruin problem, is well understood for memoryless (Markovian) processes. However, in complex systems such as polymer fluids, the motion of a particle should typically be described as a process with memory, for which splitting probabilities are much less characterized analytically. Here, we introduce an analytical approach that provides the splitting probabilities for one-dimensional isotropic non-Markovian Gaussian processes with stationary increments, in the case of two targets. This analysis shows that splitting probabilities are controlled by the out of equilibrium trajectories observed after the first passage. This is directly evidenced in a prototypical experimental reaction scheme in viscoelastic fluids. These results are extended to $d$-dimensional processes in large confining volumes, opening a path towards the study of competitive events in complex media.

cond-mat.stat-mech↗

arXiv2023

Imperfect Narrow Escape problem

We consider the kinetics of the imperfect narrow escape problem, i.e. the time it takes for a particle diffusing in a confined medium of generic shape to reach and to be adsorbed by a small, imperfectly reactive patch embedded in the boundary of the domain, in two or three dimensions. Imperfect reactivity is modeled by an intrinsic surface reactivity $κ$ of the patch, giving rise to Robin boundary conditions. We present a formalism to calculate the exact asymptotics of the mean reaction time in the limit of large volume of the confining domain. We obtain exact explicit results in the two limits of large and small reactivities of the reactive patch, and a semi-analytical expression in the general case. Our approach reveals an anomalous scaling of the mean reaction time as the inverse square root of the reactivity in the large reactivity limit, valid for an initial position near the extremity of the reactive patch. We compare our exact results with those obtained within the ``constant flux approximation''; we show that this approximation turns out to give exactly the next-to-leading order term of the small reactivity limit, and provides a good approximation of the reaction time far from the reactive patch for all reactivities, but not in the vicinity of the boundary of the reactive patch due to the above mentioned anomalous scaling. These results thus provide a general framework to quantify the mean reaction times for the imperfect narrow escape problem.

cond-mat.stat-mech↗

arXiv2022

Everlasting impact of initial perturbations on first-passage times of non-Markovian random walks

Persistence, defined as the probability that a fluctuating signal has not reached a threshold up to a given observation time, plays a crucial role in the theory of random processes. It quantifies the kinetics of processes as varied as phase ordering, reaction diffusion or interface relaxation dynamics. The fact that persistence can decay algebraically with time with non trivial exponents has triggered a number of experimental and theoretical studies. However, general analytical methods to calculate persistence exponents cannot be applied to the ubiquitous case of non-Markovian systems relaxing transiently after an imposed initial perturbation. Here, we introduce a theoretical framework that enables the non perturbative determination of persistence exponents of $d$-dimensional Gaussian non-Markovian processes with general non stationary dynamics relaxing to a steady state after an initial perturbation. Two prototypical classes of situations are analyzed: either the system is subjected to a temperature quench at initial time, or its past trajectory is assumed to have been observed and thus known. Altogether, our results reveal and quantify, on the basis of Gaussian processes, the deep impact of initial perturbations on first-passage statistics of non-Markovian processes. Our theory covers the case of spatial dimension higher than one, opening the way to characterize non-trivial reaction kinetics for complex systems with non-equilibrium initial conditions.

cond-mat.stat-mech↗

arXiv2021

Joint statistics of space and time exploration of $1d$ random walks

The statistics of first-passage times of random walks to target sites has proved to play a key role in determining the kinetics of space exploration in various contexts. In parallel, the number of distinct sites visited by a random walker and related observables have been introduced to characterize the geometry of space exploration. Here, we address the question of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies the coupling between kinetics and geometry of search trajectories. Focusing on 1-dimensional systems, we present a general method and derive explicit expressions of this joint distribution for several representative examples of Markovian search processes. In addition, we obtain a general scaling form, which holds also for non Markovian processes and captures the general dependence of the joint distribution on its space and time variables. We argue that the joint distribution has important applications to various problems, such as a conditional form of the Rosenstock trapping model, and the persistence properties of self-interacting random walks.

cond-mat.stat-mech↗

arXiv2019

Survival probability of stochastic processes beyond persistence exponents

For many stochastic processes, the probability $S(t)$ of not-having reached a target in unbounded space up to time $t$ follows a slow algebraic decay at long times, $S(t)\sim S_0/t^θ$. This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent $θ$ has been studied at length, the prefactor $S_0$, which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for $S_0$ for a compact random walk in unbounded space by establishing an analytic relation with the mean first-passage time of the same random walk in a large confining volume. Our analytical results for $S_0$ are in good agreement with numerical simulations, even for strongly correlated processes such as Fractional Brownian Motion, and thus provide a refined understanding of the statistics of longest first-passage events in unbounded space.

cond-mat.stat-mech↗

arXiv2018

Tracer diffusion in crowded narrow channels. Topical review

We summarise different results on the diffusion of a tracer particle in lattice gases of hard-core particles with stochastic dynamics, which are confined to narrow channels -- single-files, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters. We show that in such geometries a surprisingly rich, sometimes even counter-intuitive, behaviour emerges, which is absent in unbounded systems. We also present a survey of different results obtained for a tracer particle diffusion in unbounded systems, which will permit a reader to have an exhaustively broad picture of the tracer diffusion in crowded environments.

cond-mat.stat-mech↗

arXiv2017

Universal first-passage statistics of aging processes

Many out of equilibrium phenomena, such as diffusion-limited reactions or target search processes, are controlled by first-passage events. So far the general determination of the mean first-passage time (FPT) to a target in confinement has left aside aging processes, involved in contexts as varied as glassy dynamics, tracer diffusion in biological membranes or transport of cold atoms in optical lattices. Here we consider general non-Markovian scale-invariant processes in arbitrary dimension, displaying aging, and demonstrate that all the moments of the FPT obey universal scalings with the confining volume with non trivial exponents. Our analysis shows that a nonlinear scaling of the mean FPT with the volume is the hallmark of aging and provides a general tool to quantify its impact on first-passage kinetics in confinement.

cond-mat.stat-mech↗

arXiv2017

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

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.

cond-mat.stat-mech↗

arXiv2015

Cover times of random searches

How long does it take 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, robotized exploration by e.g. automated cleaners or deminers, or algorithmics. 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. Here we determine the full distribution of the cover time for a broad range of random search processes, which includes the prominent examples of Lévy strategies, intermittent strategies, persistent random walks and random walks on complex networks, and reveal its universal features. We show 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.

cond-mat.stat-mech↗

arXiv2015

Nonlinear response and emerging nonequilibrium micro-structures for biased diffusion in confined crowding environments

We study analytically the dynamics and the micro-structural changes of a host medium caused by a driven tracer particle moving in a confined, quiescent molecular crowding environment. Imitating typical settings of active micro-rheology experiments, we consider here a minimal model comprising a geometrically confined lattice system -- a two-dimensional strip-like or a three-dimensional capillary-like -- populated by two types of hard-core particles with stochastic dynamics -- a tracer particle driven by a constant external force and bath particles moving completely at random. Resorting to a decoupling scheme, which permits us to go beyond the linear-response approximation (Stokes regime) for arbitrary densities of the lattice gas particles, we determine the force-velocity relation for the tracer particle and the stationary density profiles of the host medium particles around it. These results are validated a posteriori by extensive numerical simulations for a wide range of parameters. Our theoretical analysis reveals two striking features: a) We show that, under certain conditions, the terminal velocity of the driven tracer particle is a nonmonotonic function of the force, so that in some parameter range the differential mobility becomes negative, and b) the biased particle drives the whole system into a nonequilibrium steady-state with a stationary particle density profile past the tracer, which decays exponentially, in sharp contrast with the behavior observed for unbounded lattices, where an algebraic decay is known to take place.

cond-mat.stat-mech↗

arXiv2015

Diffusion and subdiffusion of interacting particles on comb-like structures

We study the dynamics of a tracer particle (TP) on a comb lattice populated by randomly moving hard-core particles in the dense limit. We first consider the case where the TP is constrained to move on the backbone of the comb only, and, in the limit of high density of particles, we present exact analytical results for the cumulants of the TP position, showing a subdiffusive behavior $\sim t^{3/4}$. At longer times, a second regime is observed, where standard diffusion is recovered, with a surprising non analytical dependence of the diffusion coefficient on the particle density. When the TP is allowed to visit the teeth of the comb, based on a mean-field-like Continuous Time Random Walk description, we unveil a rich and complex scenario, with several successive subdiffusive regimes, resulting from the coupling between the inhomogeneous comb geometry and particle interactions. Remarkably, the presence of hard-core interactions speeds up the TP motion along the backbone of the structure in all regimes.

cond-mat.stat-mech↗

arXiv2014

Microscopic theory for negative differential mobility in crowded environments

We study the behavior of the stationary velocity of a driven particle in an environment of mobile hard-core obstacles. Based on a lattice gas model, we demonstrate analytically that the drift velocity can exhibit a nonmonotonic dependence on the applied force, and show quantitatively that such negative differential mobility (NDM), observed in various physical contexts, is controlled by both the density and diffusion time scale of obstacles. Our study unifies recent numerical and analytical results obtained in specific regimes, and makes it possible to determine analytically the region of the full parameter space where NDM occurs. These results suggest that NDM could be a generic feature of biased (or active) transport in crowded environments.

cond-mat.stat-mech↗

arXiv2014

Exit time distribution in spherically symmetric two-dimensional domains

The distribution of exit times is computed for a Brownian particle in spherically symmetric two- dimensional domains (disks, angular sectors, annuli) and in rectangles that contain an exit on their boundary. The governing partial differential equation of Helmholtz type with mixed Dirichlet- Neumann boundary conditions is solved analytically. We propose both an exact solution relying on a matrix inversion, and an approximate explicit solution. The approximate solution is shown to be exact for an exit of vanishing size and to be accurate even for large exits. For angular sectors, we also derive exact explicit formulas for the moments of the exit time. For annuli and rectangles, the approximate expression of the mean exit time is shown to be very accurate even for large exits. The analysis is also extended to biased diffusion. Since the Helmholtz equation with mixed boundary conditions is encountered in microfluidics, heat propagation, quantum billiards, and acoustics, the developed method can find numerous applications beyond exit processes.

physics.comp-ph↗

arXiv2014

Mean exit time for surface-mediated diffusion: spectral analysis and asymptotic behavior

We consider a model of surface-mediated diffusion with alternating phases of pure bulk and surface diffusion. For this process, we compute the mean exit time from a disk through a hole on the circle. We develop a spectral approach to this escape problem in which the mean exit time is explicitly expressed through the eigenvalues of the related self-adjoint operator. This representation is particularly well suited to investigate the asymptotic behavior of the mean exit time in the limit of large desorption rate $λ$. For a point-like target, we show that the mean exit time diverges as $\sqrtλ$. For extended targets, we establish the asymptotic approach to a finite limit. In both cases, the mean exit time is shown to asymptotically increase as $λ$ tends to infinity. We also revise the optimality regime of surface-mediated diffusion. Although the presentation is limited to the unit disk, the spectral approach can be extended to other domains such as rectangles or spheres.

math-ph↗

arXiv2013

Geometric friction directs cell migration

In the absence of environmental cues, a migrating cell performs an isotropic random motion. Recently, the breaking of this isotropy has been observed when cells move in the presence of asymmetric adhesive patterns. However, up to now the mechanisms at work to direct cell migration in such environments remain unknown. Here, we show that a non-adhesive surface with asymmetric micro-geometry consisting of dense arrays of tilted micro-pillars can direct cell motion. Our analysis reveals that most features of cell trajectories, including the bias, can be reproduced by a simple model of active Brownian particle in a ratchet potential, which we suggest originates from a generic elastic interaction of the cell body with the environment. The observed guiding effect, independent of adhesion, is therefore robust and could be used to direct cell migration both in vitro and in vivo.

cond-mat.soft↗