Features of barrier crossing event for Lévy flights

The steady-state time characteristics of anomalous diffusion in the form of Lévy flights in a symmetric bistable quartic potential are examined. Based on a previously developed technique, we obtain the exact formula for the correlation time of particle displacement in the case of Cauchy stable noise. The dependence of the correlation time on the height of the potential barrier separating the two stable states and the position of the potential wells is analyzed in detail. As shown, for a sufficiently high potential barrier the correlation time does not depend on its height that can be interpreted as a revision of Kramers' law for Lévy flight superdiffusion

Spectral characteristics of steady-state Lévy flights in confinement potential profiles

The steady-state correlation characteristics of superdiffusion in the form of Levy flights in one-dimensional confinement potential profiles are investigated both theoretically and numerically. Specifically, for Cauchy stable noise we calculate the steady-state probability density function for an infinitely deep rectangular potential well and for a symmetric steep potential well of the type U(x) 1ex2m. For these potential profiles and arbitrary Levy index \u3b1, we obtain the asymptotic expression of the spectral power density

Statistics of residence time for Lévy flights in unstable parabolic potentials

We analyze the residence time problem for an arbitrary Markovian process describing nonlinear systems without a steady state. We obtain exact analytical results for the statistical characteristics of the residence time. For diffusion in a fully unstable potential profile in the presence of Lévy noise we get the conditional probability density of the particle position and the average residence time. The noise-enhanced stability phenomenon is observed in the system investigated. Results from numerical simulations are in very good agreement with analytical ones

Oligogermanes containing only electron-withdrawing substituents: Synthesis and properties

A series of germanes Ar3GeX, containing electron-withdrawing substituents [Ar = p-FC6H4, 1a–d, 1a (X = Cl), 1b (X = Br), 1c (X = H), 1d (X = NMe2); p-F3CC6H4, 2a–d, 2a (X = Cl), 2b (X = Br), 2c (X = H), 2d (X = NMe2)], was synthesized and used to prepare symmetrical digermanes Ar3Ge–GeAr3, (p-FC6H4)3GeGe(C6H4F-p)3 (3), and (p-F3CC6H4)3GeGe(C6H4CF3-p)3 (4) and trigermane [(p-F3CC6H4)3Ge]2Ge(C6F5)2 (5) by hydrogermolysis reaction. The properties of all compounds were investigated by multinuclear NMR and for oligogermanes by UV/vis and fluorescence spectroscopy, as well as by electrochemical methods. In addition, the molecular structures of 1a, 1b, 2b, 2c, and 3–5 were studied by X-ray diffraction analysis. Compound 5 showed a significantly shifted UV/visible absorption to the red field in comparison with previously described derivatives