A. Shehter

Anomalous Size Dependence of Relaxational Processes

We consider relaxation processes that exhibit a stretched exponential behavior. We find that in those systems, where the relaxation arises from two competing exponential processes, the size of the system may play a dominant role. Above a crossover time tx that depends logarithmically on the size of the system the relaxation changes from a stretched exponential to a simple exponential decay, where the decay rate also depends logarithmically on the size of the system. This result is relevant to large-scale Monte-Carlo simulations and should be amenable to experimental verification in low-dimensional and mesoscopic systems.

“Logistic map”: an analytical solution

An analytical solution for the well-known quadratic recursion, the logistic map, is presented. Our derivation is based on the analogy between this recursion and a probabilistic problem that can be solved analytically. The solution is represented as a power of a transfer matrix. The proposed method allows to solve a more general quadratic mapping.

Complex dynamics in initially separated reaction-diffusion systems

We review recent developments in the study of the diffusion reaction systems of the type A + B → C in which the reactants are initially separated. We consider the case where the A and B particles are initially placed uniformly in Euclidean space at x > 0 and x < 0, respectively. We find that whereas for d ⩾ 2 the mean field exponent characterizes the width of the reaction zone, fluctuations are relevant in the one-dimensional system. We present analytical and numerical results for the reaction rate on fractals and percolation systems at criticality.We also study the case where the particles are Levy flights in d = 1. Finally, we consider experimentally, analytically, and numerically the reaction A + Bstatic → C, where species A diffuses from a localized source.

Stretched-exponential relaxation : the role of system size

Abstract We investigate the role of system size on stretched-exponential relaxation which arises from a convolution of two competing exponential processes. We find that above a cross-over time t x that depends logarithmically on the size of the system the relaxation changes from a stretched exponential to a single-exponential decay. The rate of the exponential also depends logarithmically on the system size. This anomalous size dependence is exemplified by the trapping problem and by the model of hierarchically constrained dynamics.

Anomalous dynamics in reaction-diffusion systems

SummaryWe review recent developments in the study of the diffusion reaction systems of the type A+B→C in which the reactants are initially separated. We consider the case where the A and B particles are initially placed uniformly in Euclidean space atx>0 andx<0, respectively. We find that whereas ford≥2 the mean-field exponent characterizes the width of the reaction zone, fluctuations are relevant in the one-dimensional system. We also presented analytical and numerical results for the reaction rate on fractals and percolation systems.

REACTION-FRONT DYNAMICS IN A+B→C WITH INITIALLY-SEPARATED REACTANTS

We review recent developments in the study of the diffusion reaction system of the type A+B→C in which the reactants are initially separated. We consider the case where the A and B particles are initially placed uniformly in Euclidean space at x>0 and x<0 respectively. We find that whereas for d≥2 a single scaling exponent characterizes the width of the reaction zone, a multiscaling approach is needed to describe the one-dimensional system. We also present analytical and numerical results for the reaction rate on fractals and percolation systems.

SURFACE ROUGHENING WITH QUENCHED DISORDER IN d-DIMENSIONS

We review recent numerical simulations of several models of interface growth in d- dimensional media with quenched disorder. These models belong to the universality class of anisotropic diode-resistor percolation networks. The values of the roughness exponent α=0.63±0.01 (d=1+1) and α=0.48±0.02 (d=2+1) are in good agreement with our recent experiments. We study also the diode-resistor percolation on a Cayley tree. We find that thus suggesting that the critical exponent for βp=∞ and that the upper critical dimension in this problem is d=dc=∞. Other critical exponents on the Cayley tree are: τ=3,ν||=ν⊥=γ=σ=0. The exponents related to roughness are: α=β=0, z=2.

Analytical Solution of the Logistic Equation

An analytical solution for the well-known quadratic recursion, also known as the logistic map, is presented. Our derivation makes use of the analogy between this equation and Galton–Watson processes. The latter can be treated in a linear way using the transfer matrix technique. The solution is represented as a power of an appropriate transfer matrix.

Anomalous kinetics in A + B → C with initially-separated reactants

We review recent developments in the study of the diffusion reaction system of the type A+B --~ C in which the reactants are initially separated. We consider the case where the A and/3 particles are initially placed uniformly in Euclidean space at x > 0 and x 2 a single scaling exponent characterizes the width of the reaction zone, a multiscaling approach is needed to describe the one-dimensional system. We also present analytical and numerical results for the reaction rate on fractals and percolation systems.

Mercury spreading on silver films: Interface characteristics

Abstract We have studied the spreading of a liquid Hg drop on a Ag film. Geometrical characteristics of the propagating Hg spot front, its roughness and hull exponents have been investigated. The mechanism of Hg spot front formation has been analysed by optical microscopy, transmission electron microscopy, scanning electron microscopy and X-ray diffraction. We found that at short scales (up to about 20 μm) the roughness exponent of the Hg steady state interface is α ≈ 1 which is characteristic for self-similar objects. We also studied the hull exponent ν at these scales and found that ν = 1.3–1.7. We suggest that on these scales a self-similar interface is generated by the process of invasion percolation. The invasion of Hg through the grain boundaries, which is observed at a high magnification, might be the origin of the roughening of the front interface. On larger scales the roughness saturates and the corresponding exponents are α = 0 and ν = 1.