Proton and Deuteron Field-Cycling NMR Relaxometry of Liquids in Porous Glasses: Evidence for Lévy-Walk Statistics

Stapf, Siegfried; Kimmich, Rainer; Seitter, R.-O.

doi:10.1103/physrevlett.75.2855

Cited by 246 publications

(200 citation statements)

References 14 publications

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“…Another well-known case is molecular dynamics of fluids near surfaces: relative to bulk fluids, there is a strong effect on the rotational diffusion as probed by field-cycling NMR relaxometry. 21 On the other hand, the influence on translational diffusion is only minor. The latter finding was observed both by field-gradient NMR diffusometry 22 and neutron scattering.…”

mentioning

confidence: 98%

Comment on “Neutron scattering study of the dynamics of the polymer melt under nanoscopic confinement” [J. Chem. Phys. 131, 174901 (2009)]

Kimmich

Fatkullin

2011

The Journal of Chemical Physics

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confidence: 98%

Comment on “Neutron scattering study of the dynamics of the polymer melt under nanoscopic confinement” [J. Chem. Phys. 131, 174901 (2009)]

Kimmich

Fatkullin

2011

The Journal of Chemical Physics

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“…This leads to the important consequence that, except in the Gaussian case α = 2, a Lévy probability density has a divergent second moment [2]. Signatures of Lévy statistics have been experimentally observed in a variety of physical systems [3] ranging from micelle systems [4] to porous glasses [5] and subrecoil laser cooling [6]. Another non-Gaussian distribution, which naturally arises within the framework of nonextensive statistical mechanics [7,8], is the Tsallis distribution, P q (x) = Z −1 q [1 − β(1 − q)x 2 ] 1/(1−q) with 1 ≤ q < 3 [9].…”

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confidence: 99%

Anomalous diffusion and Tsallis statistics in an optical lattice

2003

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We point out a connection between anomalous quantum transport in an optical lattice and Tsallis' generalized thermostatistics. Specifically, we show that the momentum equation for the semiclassical Wigner function that describes atomic motion in the optical potential, belongs to a class of transport equations recently studied by Borland [PLA 245, 67 (1998)]. The important property of these ordinary linear Fokker-Planck equations is that their stationary solutions are exactly given by Tsallis distributions. Dissipative optical lattices are therefore new systems in which Tsallis statistics can be experimentally studied.

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“…In a broad variety of systems, however, it has been found that correlations in space or time give birth to anomalous transport whose pdf is non-Gaussian and/or whose mean squared displacement is non-linear in time [3]. These systems include charge carrier transport in amorphous semiconductors [4], tracer dispersion in convection rolls and rotating flows [5,6], capillary surface waves [7], the motion of bacteriae and the flight of an albatross [8], intracellular transport [9], transport in micelles [10], 2-D dusty plasmas [11], the dynamics in (bio)polymeric systems [12], and the NMR diffusometry in porous glasses and percolation clusters [13], among others.…”

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confidence: 99%

Superdiffusive Klein-Kramers equation: Normal and ano malous time evolution and Lévy walk moments

Metzler

Sokolov

2002

Europhys. Lett.

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We introduce a fractional Klein-Kramers equation which describes sub-ballistic superdiffusion in phase space in the presence of a space-dependent external force field. This equation defines the differential Lévy walk model whose solution is shown to be non-negative. In the velocity coordinate, the probability density relaxes in Mittag-Leffler fashion towards the Maxwell distribution whereas in the space coordinate, no stationary solution exists and the temporal evolution of moments exhibits a competition between Brownian and anomalous contributions. Classically, Brownian stochastic transport processes in the phase space spanned by velocity v and coordinate x are described by the deterministic Klein-Kramers equation (KKE). In the low and high friction limits, the KKE reduces to the Rayleigh equation which describes the relaxation of the velocity probability density function (pdf) towards the Maxwell distribution, and the FokkerPlanck-Smoluchowski equation controlling the temporal approach of the Gibbs-Boltzmann equilibrium, respectively [1, 2]. The KKE is therefore a fundamental equation in non-equilibrium systems dynamics.Brownian transport is characterised through the Gaussian pdf and the linear time dependence x 2 (t) = 2Kt of the mean squared displacement in the force-free diffusion limit, its universality being guaranteed by the central limit theorem [2]. In a broad variety of systems, however, it has been found that correlations in space or time give birth to anomalous transport whose pdf is non-Gaussian and/or whose mean squared displacement is non-linear in time [3]. These systems include charge carrier transport in amorphous semiconductors [4], tracer dispersion in convection rolls and rotating flows [5,6] The systems we are interested in fall into the broad class whose force-free diffusion behaviour is characterised through the power-law form x 2 (t) ∝ t κ , which separates into subdiffusion (0 < κ < 1) and superdiffusion (κ > 1). The continuous time random walk (CTRW) model has proved to be a well-suited framework which accounts for such anomalous diffusion for the entire spectrum of κ [14]. Especially in the sub-ballistic superdiffusive domain 1 < κ < 2, Lévy walks which couple long flight times with a time cost have been a successful tool, e.g., in fluid dynamics [6,15]. The space-time coupling of Lévy walks leads to finite moments and they are therefore fundamentally different from Lévy flights which exhibit a diverging variance [16].In the presence of external force fields, the CTRW approach is less flexible. It has been realised that fractional equations constitute a tailor-made framework to formulate the underlying dynamics equations in coordinate and phase space; see, for instance, [16,17,18,19,20] and references therein. In the subdiffusive domain 0 < κ < 1, a fractional KKE was derived from CTRW models [21] and from the Chapman-Kolmogorov equation [22]. A similar consistent generalisation to systems in the regime 1 < κ < 2 is still outstanding. In this note, we propose the fractional KKE

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Proton and Deuteron Field-Cycling NMR Relaxometry of Liquids in Porous Glasses: Evidence for Lévy-Walk Statistics

Cited by 246 publications

References 14 publications

Comment on “Neutron scattering study of the dynamics of the polymer melt under nanoscopic confinement” [J. Chem. Phys. 131, 174901 (2009)]

Comment on “Neutron scattering study of the dynamics of the polymer melt under nanoscopic confinement” [J. Chem. Phys. 131, 174901 (2009)]

Anomalous diffusion and Tsallis statistics in an optical lattice

Superdiffusive Klein-Kramers equation: Normal and ano malous time evolution and Lévy walk moments

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