A random telegraph signal of Mittag-Leffler type

Ferraro, Simone; Manzini, Michele; Masoero, A.; Scalas, Enrico

doi:10.1016/j.physa.2009.06.036

Cited by 11 publications

(9 citation statements)

References 30 publications

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“…It comes from the theory of a random walk in a random potential energy landscape and considers a random variation of barrier heights on the length scale of individual ion positions in the glass network [41,42]. The ColeCole spectral density is given by [43]: exponential ACF model. A non-exponentiality of the dynamical process is introduced and controlled via the width parameter α.…”

Section: The Power-law Waiting Times Modelmentioning

confidence: 99%

Lithium ion dynamics in Li2S+GeS2+GeO2 glasses studied using 7Li NMR field-cycling relaxometry and line-shape analysis

Gabriel

Petrov

Kim

et al. 2015

Solid State Nuclear Magnetic Resonance

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We use (7)Li NMR to study the ionic jump motion in ternary 0.5Li2S+0.5[(1-x)GeS2+xGeO2] glassy lithium ion conductors. Exploring the "mixed glass former effect" in this system led to the assumption of a homogeneous and random variation of diffusion barriers in this system. We exploit that combining traditional line-shape analysis with novel field-cycling relaxometry, it is possible to measure the spectral density of the ionic jump motion in broad frequency and temperature ranges and, thus, to determine the distribution of activation energies. Two models are employed to parameterize the (7)Li NMR data, namely, the multi-exponential autocorrelation function model and the power-law waiting times model. Careful evaluation of both of these models indicates a broadly inhomogeneous energy landscape for both the single (x=0.0) and the mixed (x=0.1) network former glasses. The multi-exponential autocorrelation function model can be well described by a Gaussian distribution of activation barriers. Applicability of the methods used and their sensitivity to microscopic details of ionic motion are discussed.

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Section: The Power-law Waiting Times Modelmentioning

confidence: 99%

Lithium ion dynamics in Li2S+GeS2+GeO2 glasses studied using 7Li NMR field-cycling relaxometry and line-shape analysis

Gabriel

Petrov

Kim

et al. 2015

Solid State Nuclear Magnetic Resonance

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“…Renewal theory is the branch of probability theory that generalizes Poisson processes for arbitrary holding times [16]. A far less general class of processes, called fractal renewal processes [27], was extensively studied by mathematicians [17,28], and the statistical physical community [18,21,42,49]. An early application is the work of Berger and Mandelbrot on communication networks [9].…”

Section: Introductionmentioning

confidence: 99%

Renewal Theory for a System with Internal States

Niemann

Barkai

Kantz

2016

Math. Model. Nat. Phenom.

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We investigate a stochastic signal described by a renewal process for a system with N states. Each state has an associated joint distribution for the signal's intensity and its holding time. We calculate multi-point distributions, correlation functions, and the powerspectrum of the signal. Focusing on fat tailed power-law distributed sojourn times in the states of the system, we investigate 1/f noise in this widely applicable model. When the mean waiting time is infinite, the averaged sample spectrum depends both on the age of the process, i.e. the time elapsing from start of the process and the start of observation, and on the total time of observation. Fluctuations of the periodogram estimator of the power-spectrum are investigated for aged systems and are found to be determined by the distribution of the number of renewals in the observation time window. These reduce to the Mittag-Leffler distribution when the start of observation is also the start of the process. When the average waiting time is finite we find a time independent Wienerian spectrum computed from the stationary correlation function of the signal.

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“…͑18͒, is known and widely studied including its generalizations. 31,32 The solution found for the density in Sec. III ͓Eq.…”

Section: The Telegrapher's Equationmentioning

confidence: 98%

Density approach to ballistic anomalous diffusion: An exact analytical treatment

Bologna

Ascolani

Grigolini

2010

Journal of Mathematical Physics

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This paper addresses the problem of deriving the probability distribution density of a diffusion process generated by a nonergodic dichotomous fluctuation using the Liouville equation (density method). The velocity of the diffusing particles fluctuates from the value of 1 to the value of −1, and back, with the distribution density of time durations τ of the two states proportional to 1/τμ in the asymptotic time limit. The adopted density method allows us to establish an exact analytical expression for the probability distribution density of the diffusion process generated by these fluctuations. Contrary to intuitive expectations, the central part of the diffusion distribution density is not left empty when moving from μ>2 (ergodic condition) to μ<2 (nonergodic condition). The intuitive expectation is realized for μ<μcr, with μcr≈1.6. For values of μ>μcr, the monomodal distribution density with a minimum at the origin is turned into a bimodal one, with a central bump whose intensity increases for μ→2. The exact theoretical treatment applies to the asymptotic time limit, which establishes for the diffusion process the ballistic scaling value δ=1. To assess the time evolution toward this asymptotic time condition, we use a numerical approach which relates the emergence of the central bump at μ=μcr with the generation of the ordinary scaling δ=0.5, which lasts for larger and larger times for μ coming closer and closer to the critical value μ=2. We assign to the waiting time distribution density two different analytical forms: one derived from the Manneville intermittence (MI) theory and one from the Mittag-Leffler (ML) survival probability. The adoption of the ML waiting time distribution density generates an exact analytical prediction, whereas the MI method allows us to get the same asymptotic time limit as the ML one for μ<2 as a result of an approximation. The joint adoption of these two waiting time distribution densities sheds light into the critical nature of the condition μ=2 and into why this is the critical point for the MI process, representing the phase transition from the nonergodic to the ergodic regime. Our main result can be interpreted as a new derivation of Lamperti distribution.

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A random telegraph signal of Mittag-Leffler type

Cited by 11 publications

References 30 publications

Lithium ion dynamics in Li2S+GeS2+GeO2 glasses studied using 7Li NMR field-cycling relaxometry and line-shape analysis

Lithium ion dynamics in Li2S+GeS2+GeO2 glasses studied using 7Li NMR field-cycling relaxometry and line-shape analysis

Renewal Theory for a System with Internal States

Density approach to ballistic anomalous diffusion: An exact analytical treatment

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