Non-Markoffian diffusion in a one-dimensional disordered lattice

Recent treatments of diffusion in a one-dimensional disordered lattice by Machta using a renormalization-group approach, and by Alexander and Orbach using an effective medium approach, lead to a frequency-dependent (or non-Markoffian) diffusion coefficient. Their results are confirmed by a direct calculation of the diffusion coefficient.Research supported by NSF Grant No. CHE 77-16308.     

Numerical simulation of diffusion in a three-dimensional disordered lattice

The conductivity in the Anderson's model of a disordered diamond lattice is studied numerically via a direct simulation of the particle diffusion. In the vicinity of mobility edges the results reveal the continuous variation of diffusivity and the divergence of the amplitude fluctuations. The critical exponent for diffusivity t_w = 1.45 ± 0.1, obtained at fixed energy E = 0, seems to support the analogy with the classical percolation.     

Conductivity in a two-dimensional disordered model with anisotropic long-range hopping

We consider a two-dimensional system of particles localized on randomly distributed sites of a square lattice with anisotropic transition matrix elements between localized sites. The diagram and replica methods are used. The conductivity of a system in different limits of local sites and particles densities is calculated. The model is relevant to the problem of strong nonmagnetic impurities in superconductors with symmetry of the order parameter. Zh. éksp. Teor. Fiz. 115, 1093–1106 (March 1999) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Depinning of a metastable disordered vortex lattice

We report on experiments investigating the depinning dynamics of a strongly pinned vortex lattice in 2H-NbSe2. We find that the depinning process starts at currents that are well below the critical current of the entire lattice and that it is governed by the formation of contiguous channels of mobile vortices connecting the sample edges. We obtain the formation time of the first channel by monitoring the delayed voltage response to a driving current step and by measuring the ramping rate dependence of the critical current. The subsequent increase in the number of moving vortices is determined from the temporal evolution of the voltage response and the critical current.     

Perfect imaging through a disordered waveguide lattice

It is experimentally demonstrated that perfect imaging is possible in disordered wave guiding media, provided that the disorder is off-diagonal, i.e., that only the spacing varies randomly between the otherwise identical lattice sites. On-diagonal disorder or Kerr nonlinearity destroys the imaging.     

An anomalous non-self-similar infiltration and fractional diffusion equation

Problems of anomalous infiltration in porous media are considered. As follows from the analysis of experimental data, modification of the infiltration equation is necessary. A fractional diffusion equation with variable order of the time-derivative operator for describing the liquid infiltration in porous media is proposed. The physical meaning of this fractional equation is explained. This equation provides good agreement with existing experimental data for both the subdiffusion and the superdiffusion. The treatment of experimental data for the absorption of water in a fired-clay brick and for water infiltration in cement mortar using this fractional equation of diffusion is presented. Various formulae, which can be useful for applications, have been developed.     

Bloch-like oscillations in a one-dimensional lattice with long-range correlated disorder

We study the dynamics of an electron subjected to a uniform electric field within a tight-binding model with long-range-correlated diagonal disorder. The random distribution of site energies is assumed to have a power spectrum S(k) approximately 1/k(alpha) with alpha>0. de Moura and Lyra [Phys. Rev. Lett. 81, 3735 (1998)]] predicted that this model supports a phase of delocalized states at the band center, separated from localized states by two mobility edges, provided alpha>2. We find clear signatures of Bloch-like oscillations of an initial Gaussian wave packet between the two mobility edges and determine the bandwidth of extended states, in perfect agreement with the zero-field prediction.     

Transport and trapping on a one-dimensional disordered lattice

Exact solutions are obtained for the incoherent tranport and trapping of particles moving through a one-dimensional disordered solid, containing a fraction p of broken bonds and a concentration x of deep traps.     

Example of diffusion in a disordered Lorentz gas

We prove a diffusion law for a disordered Lorentz gas obtained by modification of a model of Gates, Gerst, Kac in Ref. 1, even though the motion is not a Markovian one in the technical sense of the word.     

Random walk on a disordered directed Bethe lattice

The random walk of a particle on a directed Bethe lattice of constant coordinanceZ is examined in the case of random hopping rates. As a result, the higher the coordinance, the narrower the regions of anomalous drift and diffusion. The annealed and quenched mean square dispersions are calculated in all dynamical phases. In opposition to the one-dimensional (Z=2) case, the annealed and quenched mean quadratic dispersions are shown to be identical in all phases.We shall employ indifferently the expressions Bethe lattice or infinite Cayley tree to denote an infinite ramified lattice of constant coordinanceZ.^{(4, 5)}     

Biased random walks on a disordered one-dimensional lattice

We employ the spectral properties of cyclic random matrices to understand the approach to equilibrium of a biased random walk on a disordered lattice. The simplicity of the calculations and the flexibility with which randomness can be modelled makes it a worthwhile alternative.     

Quantum-state transfer through long-range correlated disordered channels

We study quantum-state transfer in XX spin-1/2 chains where both communicating spins are weakly coupled to a channel featuring disordered on-site magnetic fields. Fluctuations are modeled by long-range correlated sequences with self-similar profile obeying a power-law spectrum. We show that the channel is able to perform almost perfect quantum-state transmissions even in the presence of significant amounts of disorder provided the degree of those correlations is strong enough, with the cost of having long transfer times and unavoidable timing errors. Still, we show that the lack of mirror symmetry in the channel does not affect much the likelihood of having high-quality outcomes. Our results suggest that coexistence between localized and delocalized states can diminish effects of static perturbations in solid-state devices for quantum communication.     

Collective diffusion in a lattice gas automaton

We report on non-mean-field and ring-kinetic-theory calculations of both the momentum autocorrelation function and the collective diffusion coefficient in a diffusive lattice gas automaton. For both quantities the ring approximation is calculated exactly. A saddle point method yields a leadingt ^–2 and a subleadingt ^–5/2 long-time tail in the momentum autocorrelation function. The ring kinetic corrections to the mean field value of the diffusion coefficient are in good agreement with computer simulations.     

Anderson localization in a periodic photonic lattice with a disordered boundary

We investigate experimentally the light evolution inside a two-dimensional finite periodic array of weakly coupled optical waveguides with a disordered boundary. For a completely localized initial condition away from the surface, we find that the disordered boundary induces an asymptotic localization in the bulk, centered around the initial position of the input beam.     

The statistics of self-avoiding walks on a disordered lattice

The relevance of lattice disorder on the critical behaviour of self-avoiding walks is discussed. A crossover from nonclassical to classical behaviour seems to take place.Supported by Special Research Area SFB 125Supported by German Academic Exchange Service (DAAD).