Equivalence of position–position auto-correlations in the Slicer Map and the Lévy–Lorentz gas

The big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory and quenched disorder. Doing so we are able to predict rare events using the extended big jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure and in a class of Lévy walks with memory. We argue that the generalized big jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy tailed distributions, ranging from contamination spreading to active transport in the cell. PACS numbers:

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“…In particular, the competition of time scales in Eqs. (27)(28)(29)(30)(31) provides the full behavior of R q (T ) as a function of time [27,53]:…”

Section: A the Lévy Lorentz Gasmentioning

confidence: 99%

Single-big-jump principle in physical modeling

2019

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“…According to the moments scaling (1.1) for q = 2, the Lévy-Lorentz gas should exhibit normal diffusion with exponent 1 if α > 3/2 and superdiffusion with exponent larger than 1, and equal respectively to 5/2 − α or 2 − α 2 /(α + 1), if 1 < α < 3/2 or 0 < α < 1. Very recently, the asymptotic behavior (1.1) has been corroborated through simplified models, both deterministic dynamical systems [20,21] and random walks [22,23], which were argued to approximate the Lévy-Lorentz gas to some extent. The present work was stimulated by the quest for providing rigorous insights into the large fluctuations of the Lévy-Lorentz gas and for establishing (1.1) on a solid mathematical ground.…”

Section: Transport Properties and Heuristicsmentioning

confidence: 90%

Large fluctuations and transport properties of the Lévy-Lorentz gas

Zamparo¹

2020

Preprint

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The Lévy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d. random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this paper we study the large fluctuations of the continuoustime process and the resulting transport properties of the model, both annealed and quenched, confirming and extending previous work by physicists that pertain to the annealed framework. Specifically, focusing on the particle displacement, and under the assumption that the tail distribution of the interdistances between scatterers is regularly varying at infinity, we prove a uniform large deviation principle for the annealed fluctuations and present the asymptotics of annealed moments, demonstrating annealed superdiffusion. Then, we provide an upper large deviation estimate for the quenched fluctuations and the asymptotics of quenched moments, showing that, unexpectedly, the asymptotically stable diffusive regime conditional on a typical arrangement of the scatterers is normal diffusion. Although the Lévy-Lorentz gas seems to be accepted as a model for anomalous diffusion, our findings lead to the conclusion that superdiffusion is a metastable behavior, which develops into normal diffusion on long timescales.

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Introduction to Nonequilibrium Statistical Physics and Its Foundations

Rondoni

2020

Frontiers and Progress of Current Soft Matter Research

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Equivalence of position–position auto-correlations in the Slicer Map and the Lévy–Lorentz gas

Cited by 7 publications

References 30 publications

Single-big-jump principle in physical modeling

Single-big-jump principle in physical modeling

Large fluctuations and transport properties of the Lévy-Lorentz gas

Introduction to Nonequilibrium Statistical Physics and Its Foundations

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