We present a simple criterion based on the Einstein relation for determining whether diffusion in systems governed by a generalized Langevin equation with long-range memory is normal, superdiffusive, or subdiffusive. We support our analysis with numerical simulations.
We study anomalous diffusion for one-dimensional systems described by a generalized Langevin equation. We show that superdiffusive systems can be divided into two classes: normal and fast. For fast superdiffusion we prove that the Fluctuation-Dissipation Theorem does not hold. As a result, the system acquires an effective temperature. This effective temperature is a signature of metastability found in many complex systems such as spin-glass and granular material.
This paper is concerned with the topological nature of the arithmetic sum of two Cantor sets. For the class of homogeneous Cantor sets, there are frve possible structures for the sum: a Cantor set, a closed interval, or three mixed models called L, R and M-Cantorvals. Examples are given showing that all these possible structures actually occur. I n the case of symmetric homogeneous Cantor sets. there are in fact only three possible topological types for the sum: a Cantor set, a closed interval or an M-Cantomal. For homogeneous Cantor sets generated by two intervals stronger results are given. Finally, an example is given showing that the result for homogeneous Cantar sets is not true in the more general class of the affine Cantor sets.
A recent Letter [M. H. Lee, Phys. Rev. Lett. 98, 190601 (2007)] has called attention to the fact that irreversibility is a broader concept than ergodicity, and that therefore the Khinchin theorem [A. I. Khinchin, (Dover, New York, 1949)] may fail in some systems. In this Letter we show that for all ranges of normal and anomalous diffusion described by a generalized Langevin equation the Khinchin theorem holds.
A discrete atomistic solid-on-solid model is proposed to describe dissolution of a crystalline solid by a liquid. The model is based on the simple assumption that the probability per unit time of a unit cell being removed is proportional to its exposed area. Numerical simulations in one dimension demonstrate that the model has very good scaling properties. After removal of only about 10(2) monolayers, independently of the substrate size, the etched surface shows almost time-independent short-range correlations and the receding surface presents the Family-Vicsek scaling behavior. The scaling parameters alpha=0.491+/-0.002 and beta=0.330+/-0.001 indicate that the system belongs to the Kardar-Parisi-Zhang universality class. The imposition of periodic boundary conditions on the simulations reduces the effective system size by a factor of 0.68 without changing the exponents alpha and beta. Surprisingly, the periodic condition changes drastically the statistics of the surface height fluctuations and the short-range correlations. Without periodic conditions, that statistics is, up to 3 standard deviations, an asymmetric Lévy distribution with mu=1.82+/-0.01, and outside this region the statistics is Gaussian. With periodic conditions, that statistics is Gaussian, except for large negative fluctuations.
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