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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.
The Binder cumulant (BC) has been widely used for locating the phase transition point accurately in systems with thermal noise. In systems with quenched disorder, the BC may show subtle finite-size effects due to large sample-to-sample fluctuations. We study the globally coupled Kuramoto model of interacting limit-cycle oscillators with random natural frequencies and find an anomalous dip in the BC near the transition. We show that the dip is related to non-self-averageness of the order parameter at the transition. Alternative definitions of the BC, which do not show any anomalous behavior regardless of the existence of non-self-averageness, are proposed.
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.
We study by Monte Carlo simulations the effect of quenched orientational disorder in systems of interacting classical dipoles on a square lattice. Each dipole can lie along any of two perpendicular axes that form an angle psi with the principal axes of the lattice. We choose psi at random and without bias from the interval [-Delta, Delta] for each site of the lattice. For 0<Delta <~ pi/4 we find a thermally driven second order transition between a paramagnetic and a dipolar antiferromagnetic order phase and critical exponents that change continously with Delta. Near the case of maximum disorder Delta ~ pi/4 we still find a second order transition at a finite temperature T_c but our results point to weak instead of {it strong} long-ranged dipolar order for temperatures below T_c.
The effect of quenched (frozen) disorder on the collective motion of active particles is analyzed. We find that active polar systems are far more robust against quenched disorder than equilibrium ferromagnets. Long ranged order (a non-zero average velocity $langle{bf v}rangle$) persists in the presence of quenched disorder even in spatial dimensions $d=3$; in $d=2$, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs. In equilibrium systems, only quasi-long-ranged order in $d=3$ and short ranged order in $d=2$ are possible. Our theoretical predictions for two dimensions are borne out by simulations.
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