Everlasting effect of initial conditions on single-file diffusion

Leibovich, N.; Barkai, Eli

doi:10.1103/physreve.88.032107

Cited by 70 publications

(106 citation statements)

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“…[36][37][38][39], confirm that different kind of memory processes may lead to weak ergodicity breaking, in particular that characterized by random time-averaged moments (inhomogeneous diffusion). It is expected that the same kind of results arise in continuous (time and space) random walk models with finite residence times and finite average jump lengths.…”

Section: Discussionmentioning

confidence: 69%

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Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini

2016

Phys. Rev. E

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We introduce a class of discrete random walk model driven by global memory effects. At any time the right-left transitions depend on the whole previous history of the walker, being defined by an urn-like memory mechanism. The characteristic function is calculated in an exact way, which allows us to demonstrate that the ensemble of realizations is ballistic. Asymptotically each realization is equivalent to that of a biased Markovian diffusion process with transition rates that strongly differs from one trajectory to another. Using this "inhomogeneous diffusion" feature the ergodic properties of the dynamics are analytically studied through the time-averaged moments. Even in the long time regime they remain random objects. While their average over realizations recover the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second time-averaged moment an ergodic limit and the limit of infinite lag times do not commutate. All these effects are induced by the memory effects. A generalized Einstein fluctuation-dissipation relation is also obtained for the time-averaged moments.

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Section: Discussionmentioning

confidence: 69%

“…On the other hand, we remark that the interplay between memory effects and weak ergodicity breaking was study previously such as for example in correlated continuous-time random walk models [36,37], single-file diffusion [38], and fractional Brownian-Langevin motion [39]. Here, we consider a different kind of memory processes.…”

Section: Introductionmentioning

confidence: 96%

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini

2016

Phys. Rev. E

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“…Recently, the dependence of the diffusion coefficient on the initial condition has been quantified for single-file diffusion, which is subdiffusive and can be related to fractional Brownian motion [70], indicating that this effect may be of general importance for long-range correlated systems. Another interesting question is how the sensitivity to initial conditions translates to the time-averaged diffusivity, where, instead of averaging over an ensemble of trajectories at a given time, the mean-square displacement is computed from a time average over a single trajectory.…”

Section: Discussionmentioning

confidence: 99%

Scaling Green-Kubo Relation and Application to Three Aging Systems

et al. 2014

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The Green-Kubo formula relates the spatial diffusion coefficient to the stationary velocity autocorrelation function. We derive a generalization of the Green-Kubo formula that is valid for systems with longrange or nonstationary correlations for which the standard approach is no longer valid. For the systems under consideration, the velocity autocorrelation function hvðt þ τÞvðtÞi asymptotically exhibits a certain scaling behavior and the diffusion is anomalous, hx 2 ðtÞi ≃ 2D ν t ν . We show how both the anomalous diffusion coefficient D ν and the exponent ν can be extracted from this scaling form. Our scaling GreenKubo relation thus extends an important relation between transport properties and correlation functions to generic systems with scale-invariant dynamics. This includes stationary systems with slowly decaying power-law correlations, as well as aging systems, systems whose properties depend on the age of the system. Even for systems that are stationary in the long-time limit, we find that the long-time diffusive behavior can strongly depend on the initial preparation of the system. In these cases, the diffusivity D ν is not unique, and we determine its values, respectively, for a stationary or nonstationary initial state. We discuss three applications of the scaling Green-Kubo relation: free diffusion with nonlinear friction corresponding to cold atoms diffusing in optical lattices, the fractional Langevin equation with external noise recently suggested to model active transport in cells, and the Lévy walk with numerous applications, in particular, blinking quantum dots. These examples underline the wide applicability of our approach, which is able to treat very different mechanisms of anomalous diffusion.

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“…In dimension d ≥ 2, tracer diffusion has been shown to remain normal, with a non trivial diffusion coefficient resulting from many-body interactions and well approximated by the Nakazato-Kitahara approach [21]. In a "single-file" geometry, where particles cannot bypass each other, the impact of EVIs is stronger and results in a subdiffusive behavior x 2 (t) ∝ t β with β = 1/2 [22][23][24][25][26][27]. In this context, determining the effect of EVIs on systems with geometrical constraints appears an important question which does not seem to have received much attention.…”

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confidence: 99%

Diffusion and Subdiffusion of Interacting Particles on Comblike Structures

et al. 2015

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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 ∼ 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.Subdiffusive motion of tracer particles in crowded media, e.g. biological cells, is widespread. Among the possible microscopic scenarios leading to this sublinear growth with time of the mean square displacement (MSD), the existence of geometric constraints related to the complexity of the environment plays an important role [1,2]. In this context, the comb model (see Fig. 1), in which a single particle moves on a two-dimensional space with the constraint that steps in the x direction are only allowed when the y coordinate of the particle positions is zero, has attracted considerable attention because of its simplicity and ability to reproduce subdiffusive behaviors of disordered systems [3].Comb-like structures have indeed been introduced as a first step to model diffusion in more complicated fractal structures like percolation clusters, the backbone and teeth of the comb representing the quasilinear structure and dangling ends of percolation clusters [4]. The particle can spend a long time exploring a tooth, which results in a subdiffusive motion along the backbone with x 2 (t) ∝ t α with α = 1/2. Since, numerous results have been obtained for this model [5][6][7][8][9][10][11][12], including the determination of the occupation time statistics [13], of mean first-passage times between two nodes of a finite comb [14] or the case of fractional Brownian walks on comb-like structures [15].In parallel, the comb model has been invoked to account for transport in real systems like spiny dendrites [10], diffusion of cold atoms [16] and mainly diffusion in crowded media like cells [17]. However, all existing studies have focused on single-particle diffusion, and interactions between particles have up to now been completely left aside. As an elementary model for diffusion of particles under short-range repulsive forces, we consider here excluded-volume interactions (EVIs) and focus on their impact on tracer dynamics on comb-like structures.From a theoretical point of view, lattice systems of interacting particles represent a protot...

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Everlasting effect of initial conditions on single-file diffusion

Cited by 70 publications

References 50 publications

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Scaling Green-Kubo Relation and Application to Three Aging Systems

Diffusion and Subdiffusion of Interacting Particles on Comblike Structures

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