On the Cole-Cole relaxation function and related Mittag-Leffler distribution

Weron, Karina; Kotulski, Marcin

doi:10.1016/0378-4371(96)00209-9

Cited by 130 publications

(100 citation statements)

References 20 publications

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“…In particular fractional approach to sub-diffusion naturally leads to anomalous response functions commonly found in many systems e.g. the ColeCole relaxation [24,38,39]. In the next three subsections we will explore the response of FLE with and without the Harmonic potential and also investigate the behavior of the imaginary part of the complex susceptibility, i.e.…”

Section: Response To An External Fieldmentioning

confidence: 99%

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

2008

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The dynamical phase diagram of the fractional Langevin equation is investigated for harmonically bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of the system. Four different critical exponents are found. (i) αc = 0.402 ± 0.002 marks a transition to a non-monotonic under-damped phase, (ii) αR = 0.441... marks a transition to a resonance phase when an external oscillating field drives the system, (iii) αχ 1 = 0.527... and (iv) αχ 2 = 0.707... marks transition to a double peak phase of the "loss" when such an oscillating field present. As a physical explanation we present a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing over-damped, under-damped regimes, motion and resonances, show behaviors different from normal.

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Section: Response To An External Fieldmentioning

confidence: 99%

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

2008

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“…The symbol "L" stands for the "Laplace transform of". These considerations may be easily extended to the fractional dynamics [29][30][31][32][33][34][35][36] by substituting the ordinary time derivative operator d/dt by a fractional (Riemann-Liouville) derivative operator of arbitrary order 0 D α t , such that…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Anomalous Dielectric Relaxation in Binary Mixtures of Mesogenic Solvent/Non-Mesogenic Solute

et al. 2005

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“…It corresponds to the Cole-Cole model of glassy dielectric media [27], whereas the limit α → 1 corresponds to an exponential relaxation with E 1 (x) = exp(x).…”

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confidence: 99%

Anomalous Escape Governed by Thermal1/fNoise

Goychuk

Hänggi

2007

Phys. Rev. Lett.

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We present an analytic study for subdiffusive escape of overdamped particles out of a cusp-shaped parabolic potential well which are driven by thermal, fractional Gaussian noise with a 1/ω 1−α power spectrum. This long-standing challenge becomes mathematically tractable by use of a generalized Langevin dynamics via its corresponding non-Markovian, time-convolutionless master equation: We find that the escape is governed asymptotically by a power law whose exponent depends exponentially on the ratio of barrier height and temperature. This result is in distinct contrast to a description with a corresponding subdiffusive fractional Fokker-Planck approach; thus providing experimentalists an amenable testbed to differentiate between the two escape scenarios. The theme of anomalous sub-diffusion and rate kinetics continuous to flourish over the last years. This topic is driven by the availability of a wealth of intriguing experimental data, ranging from anomalous diffusion in amorphous materials, quantum dots, protein dynamics, actin networks, and biological cells [1,2,3,4,5,6,7,8,9,10,11]. Suitable theoretical descriptions derive from continuous time random walks (CTRW) [1,12], the CTRWbased fractional Fokker-Planck (FFP)-approach [13,14], or the generalized Langevin equation (GLE) [15,16]. The GLE-subdiffusion implies power-law-correlated thermal forces (or fractional Gaussian noise (fGn) [17]) possessing infinite memory with a 1/ω 1−α power spectrum [7,18]. Such random forces emerge when coupling the system to sub-Ohmic thermal baths with spectral densities J(ω) ∝ ω α , 0 < α < 1 [16].

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On the Cole-Cole relaxation function and related Mittag-Leffler distribution

Cited by 130 publications

References 20 publications

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

Anomalous Dielectric Relaxation in Binary Mixtures of Mesogenic Solvent/Non-Mesogenic Solute

Anomalous Escape Governed by Thermal1/fNoise

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