Renewal theory with fat-tailed distributed sojourn times: Typical versus rare

Wang, Wanli; Schulz, Johannes H. P.; Deng, Weihua; Barkai, Eli

doi:10.1103/physreve.98.042139

Cited by 36 publications

(48 citation statements)

References 56 publications

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“…For the SM tagging technique, the number of monomers z of the complex on which the single light emitter is found is random. Its distribution is related with the distribution of sizes P (N ) by an auxiliary technique from renewal theory [46][47][48]. Placing on a stretched line all the different size complexes found in the system at some measurement time, we find the straddling interval around some auxiliary large size Ñ .…”

Section: Tracking In a Many Body Scenariomentioning

confidence: 97%

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Hitchhiker model for Laplace diffusion processes

Hidalgo-Soria

Barkai

2020

Phys. Rev. E

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Aggregation and fragmentation of single molecules in the cell environment lead to a spectrum of diffusivities and to statistical laws of movement very different from typical Brownian motion. Current models of intracellular transport do not explain at a microscopical level the emergence of theses deviations. Employing a many body approach, which we call the Hitchhiker model, we elucidate how the widely observed exponential tails in the particle spreading, i.e. the Laplace distribution and the modulations of the diffusivities, are controlled by size fluctuations of single molecules. By means of numerical simulations Laplace distributions are obtained whether we track one molecule or many molecules in parallel. However, we show that the diffusivity varies significantly depending on which tracking protocol is applied. Using a renewal process in the space of sizes, we quantify to what extent the average diffusivity in the single molecule technique is decreased compared with the ensemble average. I.

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Section: Tracking In a Many Body Scenariomentioning

confidence: 97%

“…To quantify the difference in the diffusivity arising from the usage of distinct tagging methods we use tools from renewal theory [46][47][48]. Typical phenomena described by this framework are: arrival times of particles to a detector, or a bus arriving to a station.…”

Section: Tracking In a Many Body Scenariomentioning

confidence: 99%

Hitchhiker model for Laplace diffusion processes

Hidalgo-Soria

Barkai

2020

Phys. Rev. E

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“…The interval [0, T ] consists in M + n events τ + n and M − n of τ − n , plus uncompleted initial and final events of duration I n and E n respectively [56]. Let us observe that both I n and E n are not distributed according to the distribution P ± n of excursion times τ ± n .…”

Section: Appendix D: Ensemble Of Initial Conditionsmentioning

confidence: 99%

Dynamical glass in weakly nonintegrable Klein-Gordon chains

et al. 2019

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Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a non-integrable perturbation creates a coupling network in action space which can be short-or long-ranged. We analyze the dynamics of observables which become the conserved actions in the integrable limit. We compute distributions of their finite time averages and obtain the ergodization time scale TE on which these distributions converge to δ-distributions. We relate TE to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations σ + τ dominating the means µ + τ and establish that TE ∼ (σ + τ ) 2 /µ + τ . The Lyapunov time TΛ (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long-and short-range coupling networks by tuning its energy density. For long-range coupling networks TΛ ≈ σ + τ , which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a dynamical glass, where TE grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which satisfies TΛ µ + τ . This effect arises from the formation of highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of non-chaotic regions. These structures persist up to the ergodization time TE.

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“…However, in the long-time limit it is easy to calculate MSDs based on the non-normalized density. This method is also valid for high-order moments [49]. From Eq.…”

Section: Appendix A: Calculation Of Eq (32)mentioning

confidence: 97%

Transport in disordered systems: The single big jump approach

Wang

Vezzani

Burioni

et al. 2019

Phys. Rev. Research

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In a growing number of strongly disordered and dense systems, the dynamics of a particle pulled by an external force field exhibits superdiffusion. In the context of glass-forming systems, supercooled glasses, and contamination spreading in porous media, it was suggested that this behavior be modeled with a biased continuous-time random walk. Here we analyze the plume of particles lagging far behind the mean, with the single big jump principle. Revealing the mechanism of the anomaly, we show how a single trapping time, the largest one, is responsible for the rare fluctuations in the system. These nontypical fluctuations still control the behavior of the mean square displacement, which is the most basic quantifier of the dynamics in many experimental setups. We show how the initial conditions, describing either the stationary state or nonequilibrium case, persist forever in the sense that the rare fluctuations are sensitive to the initial preparation. To describe the fluctuations of the largest trapping time, we modify Fréchet's law from extreme value statistics, taking into consideration the fact that the large fluctuations are very different from those observed for independent and identically distributed random variables.

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Renewal theory with fat-tailed distributed sojourn times: Typical versus rare

Cited by 36 publications

References 56 publications

Hitchhiker model for Laplace diffusion processes

Hitchhiker model for Laplace diffusion processes

Dynamical glass in weakly nonintegrable Klein-Gordon chains

Transport in disordered systems: The single big jump approach

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