Transport properties and ageing for the averaged Lévy–Lorentz gas

Radice, Mattia; Onofri, Manuele; Artuso, Roberto; Cristadoro, Giampaolo

doi:10.1088/1751-8121/ab5990

Cited by 14 publications

(14 citation statements)

References 41 publications

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“…It follows that the exponential conspiracy in which distribution of diffusion constants is exponential is not a necessary condition for a cusp like behavior of P(x, t). We further remark that the non-analytical behavior is found also in the context of normal diffusion in [35][36][37] and within the anomalous one at [31][32][33][34]36,[38][39][40].…”

Section: Super -Statisticssupporting

confidence: 53%

“…The latter model consists of a “fast” and a “slow” phases, each one with a diffusion coefficient

and

, respectively [ 13 , 30 ]. Furthermore, the appearance of a cusp at small displacements also has been reported in different diffusive approaches like the Sinai model [ 31 ], employing the quenched trap model [ 32 , 33 , 34 , 35 , 36 , 37 ] or spatial dependence in the diffusivity [ 38 ], within the Lévy–Lorentz gas model [ 39 ] and using the fractional Fokker–Planck equation [ 40 ]. It is important to notice that the cusp found in [ 31 , 32 , 33 , 34 , 36 , 38 , 39 , 40 ] is within the context of anomalous diffusion in the MSD sense, and those presented in [ 35 , 36 , 37 ] are for normal diffusive systems.…”

Section: Introductionmentioning

confidence: 94%

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Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

Hidalgo-Soria

Barkai

Burov

2021

Entropy

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We study a two state “jumping diffusivity” model for a Brownian process alternating between two different diffusion constants, D+>D−, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient D−⟶0 is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements P(x,t) for x⟶0. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state D+ to a δ-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in P(x,t). We demonstrate how super-statistical framework is a zeroth order short time expansion of P(x,t), in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

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Section: Super -Statisticssupporting

confidence: 53%

“…The latter model consists of a “fast” and a “slow” phases, each one with a diffusion coefficient

and

Section: Introductionmentioning

confidence: 94%

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

Hidalgo-Soria

Barkai

Burov

2021

Entropy

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“…This fact, that can be easily observed in fig. 5, contrasts with the result obtained in the interval − 1 2 , 1 2 or in other stochastic processes where the two quantities M n and |x| n have the same asymptotic growth, see for example [46,47]. Data are obtained by considering n = 10 5 numbers of steps and 10 6 walks.…”

Section: Statistics Of Records and Maximummentioning

confidence: 79%

Exploring the Gillis model: a discrete approach to diffusion in logarithmic potentials

Onofri,

Pozzoli,

Radice

et al. 2020

Preprint

Self Cite

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Gillis model, introduced more than 60 years ago, is a nonhomogeneous random walk with a position dependent drift.Though parsimoniously cited both in the physical and mathematical literature, it provides one of the very few examples of a stochastic system allowing for a number of exact result, although lacking translational invariance. We present old and novel results for such model, which moreover we show represents a discrete version of a diffusive particle in the presence of a logarithmic potential.

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“…In particular, they are used as supports for various kinds of random walks, in order to study phenomena of anomalous transport and anomalous diffusion. An incomplete list of general or recent references on this topic includes [22,14,11,26,1,18,19].…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for Lévy flights on a 1D Lévy random medium

Stivanello

Bet

Bianchi

et al. 2021

Electron. J. Probab.

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We study a random walk on a point process given by an ordered array of points (ω k , k ∈ Z) on the real line. The distances ω k+1 − ω k are i.i.d. random variables in the domain of attraction of a β-stable law, with β ∈ (0, 1) ∪ (1, 2). The random walk has i.i.d. jumps such that the transition probabilities between ω k and ω depend on − k and are given by the distribution of a Z-valued random variable in the domain of attraction of an α-stable law, with α ∈ (0, 1) ∪ (1, 2). Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters α and β, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.

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Transport properties and ageing for the averaged Lévy–Lorentz gas

Cited by 14 publications

References 41 publications

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

Exploring the Gillis model: a discrete approach to diffusion in logarithmic potentials

Limit theorems for Lévy flights on a 1D Lévy random medium

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