Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini, Adrián A.

doi:10.1103/physreve.94.052142

Cited by 10 publications

(25 citation statements)

References 59 publications

(93 reference statements)

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“…This feature here is related to the non-stationary power-law decay of the transition probabilities to their stationary values. In contrast with previous results [24,40], for the studied model we also show that time-averaged response to a bias dye out in the asymptotic regime.…”

Section: Introductioncontrasting

confidence: 99%

“…For µ = 1, the urn-like dynamics of Ref. [40] is recovered, while for λ = 0 the elephant model arises [29] (see Refs. [30,31] where this model is written in terms of the number of transitions t ± ).…”

Section: Random Walk Dynamicsmentioning

confidence: 97%

“…As in Refs. [29,40], both the time and position coordinates are discrete. Hence, in each discrete time step (t → t + δt) the walker perform a jump of length δx to the right or to the left.…”

Section: Random Walk Dynamicsmentioning

confidence: 99%

“…In Ref. [40] we introduced a globally correlated diffusive dynamics that leads to (ensemble) ballistic behaviors and also characterized its time-averaged moments. Interestingly, the memory effects also lead to weak ergodicity breaking.…”

Section: Introductionmentioning

confidence: 99%

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A memory-induced diffusive-superdiffusive transition: ensemble and time-averaged observables

Budini

2017

Preprint

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The ensemble properties and time-averaged observables of a memory-induced diffusivesuperdiffusive transition are studied. The model consists in a random walker whose transitions in a given direction depend on a weighted linear combination of the number of both right and left previous transitions. The diffusion process is nonstationary and its probability develops the phenomenon of aging. Depending on the characteristic memory parameters, the ensemble behavior may be normal, superdiffusive, or ballistic. In contrast, the time-averaged mean squared displacement is equal to that of a normal undriven random walk, which renders the process non-ergodic. In addition, and similarly to Levy walks [Godec and Metzler, Phys. Rev. Lett. 110, 020603 (2013)], for trajectories of finite duration the time-averaged displacement apparently become random with properties that depend on the measurement time and also on the memory properties. These features are related to the non-stationary power-law decay of the transition probabilities to their stationary values. Time-averaged response to a bias is also calculated. In contrast with Levy walks [Froemberg and Barkai, Phys. Rev. E 87, 030104(R) (2013)], the response always vanishes asymptotically.

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Section: Introductioncontrasting

confidence: 99%

Section: Random Walk Dynamicsmentioning

confidence: 97%

Section: Random Walk Dynamicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A memory-induced diffusive-superdiffusive transition: ensemble and time-averaged observables

Budini

2017

Preprint

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“…Recently, relevant physical phenomena of distributional behaviors have been experimentally unveiled, e.g., intensity of fluorescence in quantum dots [40,41], diffusion coefficient of a diffusing biomolecule in living cells [42][43][44][45], and interface fluctuations in Kardar-Parisi-Zhang universality class [46], where time averages of an observable, obtained from different realizations under the same experimental setup, do not converge to a constant but remain random. These distributional behaviors of time averages for some observ-ables have been investigated by several stochastic models describing anomalous diffusion processes [19,39,[47][48][49][50][51][52][53][54].…”

Section: Introductionmentioning

confidence: 99%

Infinite invariant density in a semi-Markov process with continuous state variables

Akimoto,

Barkai,

Radons

2019

Preprint

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We report on a fundamental role of a non-normalized invariant density, i.e., infinite invariant density, in a semi-Markov process with continuous state variables, where the state is determined by the inter-event time of successive changes of states. The state could be the velocity in the generalized Levy walk model or the energy of a particle in the trap model. We analytically show that the density for the state value accumulates in the vicinity of the zero and is composed of two parts: the density near the zero increases with its support becoming zero in the long-time limit and the other is a trace of a non-normalized formal steady state, which gradually disappears in the long-time limit. The first is normalized and serves as a description of the typical fluctuations, and the second is non-normalized and serves effectively as a steady state. Moreover, we demonstrate two distributional behaviors for time-averaged observables in the non-stationary processes, where the shape of the distribution is determined by whether the observable is integrable with respect to the infinite density. The infinite invariant density plays an important role in characterizing the non-stationary accumulation process.

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Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing

Hou

Cherstvy

Metzler

et al. 2018

Phys. Chem. Chem. Phys.

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We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.

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Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Cited by 10 publications

References 59 publications

A memory-induced diffusive-superdiffusive transition: ensemble and time-averaged observables

A memory-induced diffusive-superdiffusive transition: ensemble and time-averaged observables

Infinite invariant density in a semi-Markov process with continuous state variables

Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing

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