Spectral design of anomalous diffusion

Eliazar, Iddo

doi:10.1016/j.physa.2023.129066

Cited by 2 publications

(14 citation statements)

References 69 publications

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“…Evidently, when the Fourier variable is zero ω = 0 then: ´∞ 0 y (β−1)/2 dy = ∞ and hence Q ω (∞) = ∞. When the Fourier variable is non-zero ω = 0 then [3]:…”

Section: A5 Asymptotic Analysis Of the Increments' Gap Functionmentioning

confidence: 96%

Beta Brownian motion

Eliazar

2024

J. Phys. A: Math. Theor.

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Brownian Motion (BM) is the paradigmatic model of diffusion. Transcending from diffusion to anomalous diffusion, the prominent Gaussian generalizations of BM are Scaled BM (SBM) and Fractional BM (FBM). In the sub/super diffusivity regimes: SBM is characterized by aging/anti-aging, and FBM is characterized by anti-persistence/persistence. BM is neither aging/anti-aging, nor persistent/anti-persistent. Within the realm of diffusion, a recent Gaussian generalization of BM, Weird BM (WBM), was shown to display aging/anti-aging and persistence /anti-persistence. This paper introduces and explores the anomalous-diffusion counterpart of WBM, Beta BM (BBM), and shows that: the weird behaviors of WBM become even weirder when elevating to BBM. Indeed, BBM displays a rich assortment of anomalous behaviors, and an even richer assortment of combinations of anomalous behaviors. In particular, the BBM anomalous behaviors include aging/anti-aging and persistence/anti-persistence -- which BBM displays in both the sub and super diffusivity regimes. So, anomalous behaviors that are unattainable by the prominent models of SBM and FBM are well attainable by the BBM model.

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“…Evidently, when the Fourier variable is zero ω = 0 then: ´∞ 0 y (β−1)/2 dy = ∞ and hence Q ω (∞) = ∞. When the Fourier variable is non-zero ω = 0 then [3]:…”

Section: A5 Asymptotic Analysis Of the Increments' Gap Functionmentioning

confidence: 96%

Beta Brownian motion

Eliazar

2024

J. Phys. A: Math. Theor.

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“…So, {X (t; ω) ; t ⩾ 0} is the trajectory-in the complex plane-of a complex version of the random motion [149]. As argued in section 3, the temporal variance function of equation ( 1) is a quantitative measure of the random-motion's diffusivity.…”

Section: Spectral Preludementioning

confidence: 98%

“…The random motion 'runs' over the real line. Per the frequency ω, 'elevate' the random motion from the real line to the complex plane as follows [149]. Firstly, shift from the randommotion's real velocities Ẋ (t) to the 'spiraling' complex velocities Ẋ (t; ω) := exp (iωt) Ẋ (t).…”

Section: Spectral Preludementioning

confidence: 99%

“…A general temporal scaling is 1/g (t), where g (t) is a general gauge function: a function that is monotone increasing from zero g (0) = 0 to infinity lim t→∞ g (t) = ∞. Cases where a limit Ω (ω) = lim t→∞ [V (t; ω) /g (t)] exists and is positive-with a gauge function that is not asymptotically linear, and without the stationarity condition of the Wiener-Khinchin theoremmanifest generalized Wiener-Khinchin theorems [149][150][151][152][153][154][155][156][157] 7 . And, in such cases the limit Ω (ω) manifests a generalized spectral density.…”

Section: Wiener-khinchin Theorems and Spectral Densitiesmentioning

confidence: 99%

“…In the statistical-physics community generalized Wiener-Khinchin theorems are often referred to as 'aging Wiener-Khinchin theorems'[152][153][154][155][156][157]. In order to avoid confusion with the notions of aging and anti-aging (noted in section 6.2), here we maintain the term 'generalized Wiener-Khinchin theorems'[149][150][151].…”

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confidence: 99%

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Regular and anomalous diffusion: I. Foundations

Eliazar

2024

J. Phys. A: Math. Theor.

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Diffusion is a generic term for random motions whose positions become more and more diffuse with time. Diffusion is of major importance in numerous areas of science and engineering, and the research of diffusion is vast and profound. This paper is the first in a stochastic `intro series' to the multidisciplinary field of diffusion. The paper sets off from a basic question: how to quantitatively measure diffusivity? Having answered the basic question, the paper carries on to a follow-up question regarding statistical behaviors of diffusion: what further knowledge can the diffusivity measure provide, and when can it do so? The answers to the follow-up question lead to an assortment of notions and topics including: persistence and anti-persistence; aging and anti-aging; spectral densities, white noise, and their generalizations; the Wiener-Khinchin theorem and its generalizations; and colored noises. Then, observing diffusion from a macro level, the paper culminates with: the universal emergence of power-law diffusivity; the three universal diffusion regimes -- one regular, and two anomalous; and the universal emergence of 1/f noise. The paper is entirely self-contained, and its prerequisites are undergraduate mathematics and statistics.

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Spectral design of anomalous diffusion

Cited by 2 publications

References 69 publications

Beta Brownian motion

Beta Brownian motion

Regular and anomalous diffusion: I. Foundations

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