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      A Continuous-Time Random Walk Extension of the Gillis Model

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          Abstract

          We consider a continuous-time random walk which is the generalization, by means of the introduction of waiting periods on sites, of the one-dimensional non-homogeneous random walk with a position-dependent drift known in the mathematical literature as Gillis random walk. This modified stochastic process allows to significantly change local, non-local and transport properties in the presence of heavy-tailed waiting-time distributions lacking the first moment: we provide here exact results concerning hitting times, first-time events, survival probabilities, occupation times, the moments spectrum and the statistics of records. Specifically, normal diffusion gives way to subdiffusion and we are witnessing the breaking of ergodicity. Furthermore we also test our theoretical predictions with numerical simulations.

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          Most cited references53

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          Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

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            Modeling non-Fickian transport in geological formations as a continuous time random walk

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              Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.

              Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.
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                Author and article information

                Journal
                Entropy (Basel)
                Entropy (Basel)
                entropy
                Entropy
                MDPI
                1099-4300
                18 December 2020
                December 2020
                : 22
                : 12
                : 1431
                Affiliations
                [1 ]Center for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy; gpozzoli@ 123456uninsubria.it (G.P.); m.radice1@ 123456uninsubria.it (M.R.); m.onofri1@ 123456uninsubria.it (M.O.)
                [2 ]Istituto Nazionale di Fisica Nucleare—Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
                Author notes
                Author information
                https://orcid.org/0000-0001-9510-6657
                https://orcid.org/0000-0001-9887-1047
                https://orcid.org/0000-0002-4353-4251
                https://orcid.org/0000-0002-6463-1093
                Article
                entropy-22-01431
                10.3390/e22121431
                7766702
                33353053
                1fa4d31d-f7aa-47de-b320-ad8fda93c236
                © 2020 by the authors.

                Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

                History
                : 24 November 2020
                : 15 December 2020
                Categories
                Article

                gillis model,ctrw,biased processes,anomalous diffusion,ergodicity,first-time events

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