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Register a new userLocal and average behavior in inhomogeneous superdiffusive media
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We consider a random walk on one-dimensional inhomogeneous graphs built from Cantor fractals. Our study is motivated by recent experiments that demonstrated superdiffusion of light in complex disordered materials, thereby termed Levy glasses. We introduce a geometric parameter $alpha$ which plays a role analogous to the exponent characterizing the step length distribution in random systems. We study the large-time behavior of both local and average observables; for the latter case, we distinguish two different types of averages, respectively over the set of all initial sites and over the scattering sites only. The single long jump approximation is applied to analytically determine the different asymptotic behaviours as a function of $alpha$ and to understand their origin. We also discuss the possibility that the root of the mean square displacement and the characteristic length of the walker distribution may grow according to different power laws; this anomalous behaviour is typical of processes characterized by Levy statistics and here, in particular, it is shown to influence average quantities.
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We study the effects of scattering lengths on Levy walks in quenched one-dimensional random and fractal quasi-lattices, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling properties of the random-walk probability distribution, we show that the effect of the varying scattering length can be reabsorbed in the multiplicative coefficient of the scaling length. This leads to a superscaling behavior, where the dynamical exponents and also the scaling functions do not depend on the value of the scattering length. Within the scaling framework, we obtain an exact expression for the multiplicative coefficient as a function of the scattering length both in the annealed and in the quenched random and fractal cases. Our analytic results are compared with numerical simulations, with excellent agreement, and are supposed to hold also in higher dimensions
Thermal conductivities are routinely calculated in molecular dynamics simulations by keeping the boundaries at different temperatures and measuring the slope of the temperature profile in the bulk of the material, explicitly using Fouriers law of heat conduction. Substantiated by the observation of a distinct linear profile at the center of the material, this approach has also been frequently used in superdiffusive materials, such as nanotubes or polymer chains, which do not satisfy Fouriers law at the system sizes considered. It has been recently argued that this temperature gradient procedure yields worse results when compared with a method based on the temperature difference at the boundaries -- thus taking into account the regions near the boundaries where the temperature profile is not linear. We study a realistic example, nanocomposites formed by adding boron nitride nanotubes to a polymer matrix of amorphous polyethylene, to show that in superdiffusive materials, despite the appearance of a central region with a linear profile, the temperature gradient method is actually inconsistent with a conductivity that depends on the system size, and, thus, it should be only used in normal diffusive systems.
We analyze the mean squared displacement of a Brownian particle in a medium with a spatially varying local diffusivity which is assumed to be periodic. When the system is asymptotically diffusive the mean squared displacement, characterizing the dispersion in the system, is, at late times, a linear function of time. A Kubo type formula is given for the mean squared displacement which allows the recovery of some known results for the effective diffusion constant $D_e$ in a direct way, but also allows an understanding of the asymptotic approach to the diffusive limit. In particular, as well as as computing the slope of a linear fit to the late time mean squared displacement, we find a formula for the constant where the fit intersects the y axis.
We study numerically the coarsening dynamics of the Ising model on a regular lattice with random bonds and on deterministic fractal substrates. We propose a unifying interpretation of the phase-ordering processes based on two classes of dynamical behaviors characterized by different growth-laws of the ordered domains size - logarithmic or power-law respectively. It is conjectured that the interplay between these dynamical classes is regulated by the same topological feature which governs the presence or the absence of a finite-temperature phase-transition.
We consider the transport of conserved charges in spatially inhomogeneous quantum systems with a discrete lattice symmetry. We analyse the retarded two point functions involving the charge and the associated currents at long wavelengths, compared to the scale of the lattice, and, when the DC conductivity is finite, extract the hydrodynamic modes associated with charge diffusion. We show that the dispersion relations of these modes are related to the eigenvalues of a specific matrix constructed from the DC conductivity and certain thermodynamic susceptibilities, thus obtaining generalised Einstein relations. We illustrate these general results in the specific context of relativistic hydrodynamics where translation invariance is broken using spatially inhomogeneous and periodic deformations of the stress tensor and the conserved $U(1)$ currents. Equivalently, this corresponds to considering hydrodynamics on a curved manifold, with a spatially periodic metric and chemical potential.
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