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Register a new userSuperdiffusion and Transport in 2d-systems with Levy Like Quenched Disorder
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We present an extensive analysis of transport properties in superdiffusive two dimensional quenched random media, obtained by packing disks with radii distributed according to a Levy law. We consider transport and scaling properties in samples packed with two different procedures, at fixed filling fraction and at self-similar packing, and we clarify the role of the two procedures in the superdiffusive effects. Using the behavior of the filling fraction in finite size systems as the main geometrical parameter, we define an effective Levy exponents that correctly estimate the finite size effects. The effective Levy exponent rules the dynamical scaling of the main transport properties and identify the region where superdiffusive effects can be detected.
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After a short excursion from discovery of Brownian motion to the Richardson law of four thirds in turbulent diffusion, the article introduces the L{e}vy flight superdiffusion as a self-similar L{e}vy process. The condition of self-similarity converts the infinitely divisible characteristic function of the L{e}vy process into a stable characteristic function of the L{e}vy motion. The L{e}vy motion generalizes the Brownian motion on the base of the $alpha$-stable distributions theory and fractional order derivatives. The further development of the idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorovs equation for arbitrary Markovian processes. As particular case we obtain the fractional Fokker-Planck equation for L{e}vy flights. Some results concerning stationary probability distributions of L{e}vy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally we discuss results on the same characteristics and barrier crossing problems with L{e}vy flights, recently obtained with different approaches.
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We study Levy walks in quenched disordered one-dimensional media, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling relations for the random-walk probability and for the resistivity in the equivalent electric problem, we obtain the asymptotic behavior of the mean square displacement as a function of the exponent characterizing the scatterers distribution. We demonstrate that in quenched media different average procedures can display different asymptotic behavior. In particular, we estimate the moments of the displacement averaged over processes starting from scattering sites, in analogy with recent experiments. Our results are compared with numerical simulations, with excellent agreement.
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