Diffusivity in one-dimensional generalized Mott variable-range hopping models

We consider random walks in a random environment which are generalized versions of well-known effective models for Mott variable-range hopping. We study the homogenized diffusion constant of the random walk in the one-dimensional case. We prove various estimates on the low-temperature behavior which confirm and extend previous work by physicists.Comment: Published in at http://dx.doi.org/10.1214/08-AAP583 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Recurrence and transience for long-range reversible random walks on a random point process

We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.Comment: 34 page

Synchronization in fiber lasers arrays

We consider an array of fiber lasers coupled through the nearest neighbors. The model is a generalized nonlinear Schroedinger equation where the usual Laplacian is replaced by the graph Laplacian. For a graph with no symmetries, we show that there is no resonant transfer of energy between the different eigenmodes. We illustrate this and confirm our result on a simple graph. This shows that arrays of fiber ring lasers can be made temporally coherent

Invariance principle for Mott variable range hopping and other walks on point processes

We consider a random walk on a homogeneous Poisson point process with energy marks. The jump rates decay exponentially in the A-power of the jump length and depend on the energy marks via a Boltzmann--like factor. The case A=1 corresponds to the phonon-induced Mott variable range hopping in disordered solids in the regime of strong Anderson localization. We prove that for almost every realization of the marked process, the diffusively rescaled random walk, with arbitrary start point, converges to a Brownian motion whose diffusion matrix is positive definite, and independent of the environment. Finally, we extend the above result to other point processes including diluted lattices.Comment: 47 pages, minor corrections, submitte

Two tone response of radiofrequency signals using the voltage output of a Superconducting Quantum Interference Filter

In the presence of weak time harmonic electromagnetic fields, Superconducting Quantum Interference Filters (SQIFs) show the typical behavior of non linear mixers. The SQIFs are manufactured from high-T_c grain boundary Josephson junctions and operated in active microcooler. The dependence of dc voltage output V_dc vs. static external magnetic field B is non-periodic and consists of a well pronounced unique dip at zero field, with marginal side modulations at higher fields. We have successfully exploited the parabolic shape of the voltage dip around B=0 to mix quadratically two external time harmonic rf-signals, at frequencies f_1 and f_2 below the Josephson frequency f_J, and detect the corresponding mixing signal at f_1-f_2. When the mixing takes place on the SQIF current-voltage characteristics the component at 2f_2 - f_1 is present. The experiments suggest potential applications of a SQIF as a non-linear mixing device, capable to operate at frequencies from dc to few GHz with a large dynamic range.Comment: 10 pages, 3 Figures, submitted to J. Supercond. (as proceeding of the HTSHFF Symposium, June 2006, Cardiff

Fractional Fokker-Planck Equation for Ultraslow Kinetics

Several classes of physical systems exhibit ultraslow diffusion for which the mean squared displacement at long times grows as a power of the logarithm of time ("strong anomaly") and share the interesting property that the probability distribution of particle's position at long times is a double-sided exponential. We show that such behaviors can be adequately described by a distributed-order fractional Fokker-Planck equations with a power-law weighting-function. We discuss the equations and the properties of their solutions, and connect this description with a scheme based on continuous-time random walks