Correlated continuous-time random walks in external force fields

Magdziarz, Marcin; Metzler, Ralf; Szczotka, Władysław; Żebrowski, Piotr

doi:10.1103/physreve.85.051103

Cited by 74 publications

(73 citation statements)

References 45 publications

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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: 68%

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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: 68%

“…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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“…The introduced model generalizes those studied in Chechkin et al [12] and Tejedor & Metzler [13]. Recent progress in the field of CTRWs with correlated temporal and spatial structure can be found in Meerschaert et al [14] and Magdziarz et al [15,16]. The Langevin description of some classes of anomalous diffusions has been recently studied in Magdziarz [17], Magdziarz et al [18] and Teuerle et al [19].…”

Section: Introductionmentioning

confidence: 76%

Asymptotic behaviour of random walks with correlated temporal structure

2013

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We introduce a continuous-time random walk process with correlated temporal structure. The dependence between consecutive waiting times is generated by weighted sums of independent random variables combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. We derive the corresponding diffusion limit and prove its subdiffusive character. Analysing the set of corresponding coupled Langevin equations, we verify the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking.

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“…intrinsically correlated), was also developed by using stochastic dynamic (Langevin) equations. This led to anomalous diffusion under the influence of an external force field [102]. A very helpful effective theoretical criterion for subdiffusion was found in the context of the reference fractional Brownian motion [103].…”

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confidence: 99%

The continuous time random walk, still trendy: fifty-year history, state of art and outlook

2017

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Abstract.In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Inspiring properties and achievementsIn their pioneering work published in year 1965 [1], physicists Eliott W. Montroll and George H. Weiss introduced the concept of continuous-time random walk (CTRW) as a way to make the interevent-time continuous and fluctuating. It is characterized by some distribution associated with a stochastic process, giving an insight into the process activity. This distribution, called pausing-or waiting-time one (WTD), permitted the description of both Debye (exponential) and, what is most significant, non-Debye (slowly-decaying) relaxations as well as normal and anomalous transport and diffusion [2,3] -thus the model involves fundamental aspects of the stochastic world -a real, complex world. Notably, ancestors of this concept are presented by Michael Shlesinger in [4].Let us incidentally comment that term "walk" in the name "continuous-time random walk" is commonly used in the generic sense comprising two concepts: namely, both the walk (associated with finite displacement velocity of the process) and flight (associated with an instantaneous displacement of the process). Thus we have to specify in a detailed way with what kind of process we are considering.The CTRW formalism was most conveniently developed by physicists Scher and Lax in terms of recursion relations [5][6][7][8][9][10]. In this context the distinction between Contribution to the Topical Issue "Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook", edited by Ryszard Kutner and Jaume Masoliver. a e-mail: ryszard.kutner@fuw.edu.pl discrete and continuous times [11] and also between separable and non-separable WTDs were introduced [12]. A thorough analysis of the latter, called also the nonindependent CTRW, was performed a decade ago [13] although, in the context of the concentrated lattice gases, it was performed much earlier [14,15]. These analyses took into account dependences over many correlated consecutive particle displacements and waiting (or interevent) times. The Scher and Lax formulation of the CTRW formalism is particularly convenient to study as well anomalous transport and diffusion as the non-Debye relaxation and their anomalous scaling properties (e.g., the nonlinear time growth of the process variance). Examples are the span of the walk, the first-passage times, survival probabilities, the number of distinct sites visited and, of course, mean and mean-square displacement if they exist. It is very interesting...

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Correlated continuous-time random walks in external force fields

Cited by 74 publications

References 45 publications

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Asymptotic behaviour of random walks with correlated temporal structure

The continuous time random walk, still trendy: fifty-year history, state of art and outlook

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