Occurrence of normal and anomalous diffusion in polygonal billiard channels

From extensive numerical simulations, we find that periodic polygonal billiard channels with angles which are irrational multiples of pi generically exhibit normal diffusion (linear growth of the mean squared displacement) when they have a finite horizon, i.e. when no particle can travel arbitrarily far without colliding. For the infinite horizon case we present numerical tests showing that the mean squared displacement instead grows asymptotically as t log t. When the unit cell contains accessible parallel scatterers, however, we always find anomalous super-diffusion, i.e. power-law growth with an exponent larger than 1. This behavior cannot be accounted for quantitatively by a simple continuous-time random walk model. Instead, we argue that anomalous diffusion correlates with the existence of families of propagating periodic orbits. Finally we show that when a configuration with parallel scatterers is approached there is a crossover from normal to anomalous diffusion, with the diffusion coefficient exhibiting a power-law divergence.Comment: 9 pages, 15 figures. Revised after referee reports: redrawn figures, additional comments. Some higher quality figures available at http://www.fis.unam.mx/~dsander

The Reaction-Diffusion Front for A+B→∅A+B \to\emptyset in One Dimension

We study theoretically and numerically the steady state diffusion controlled reaction $A+B\rightarrow\emptyset$, where currents $J$ of $A$ and $B$ particles are applied at opposite boundaries. For a reaction rate $\lambda$, and equal diffusion constants $D$, we find that when $\lambda J^{-1/2} D^{-1/2}\ll 1$ the reaction front is well described by mean field theory. However, for $\lambda J^{-1/2} D^{-1/2}\gg 1$, the front acquires a Gaussian profile - a result of noise induced wandering of the reaction front center. We make a theoretical prediction for this profile which is in good agreement with simulation. Finally, we investigate the intrinsic (non-wandering) front width and find results consistent with scaling and field theoretic predictions.Comment: 11 pages, revtex, 4 separate PostScript figure

Scaling exponents for fracture surfaces in homogenous glass and glassy ceramics

We investigate the scaling properties of post-mortem fracture surfaces in silica glass and glassy ceramics. In both cases, the 2D height-height correlation function is found to obey Family-Viseck scaling properties, but with two sets of critical exponents, in particular a roughness exponent $\zeta\simeq 0.75$ in homogeneous glass and $\zeta\simeq 0.4$ in glassy ceramics. The ranges of length-scales over which these two scalings are observed are shown to be below and above the size of process zone respectively. A model derived from Linear Elastic Fracture Mechanics (LEFM) in the quasistatic approximation succeeds to reproduce the scaling exponents observed in glassy ceramics. The critical exponents observed in homogeneous glass are conjectured to reflect damage screening occurring for length-scales below the size of the process zone

Diffusion-Limited Annihilation with Initially Separated Reactants

A diffusion-limited annihilation process, A+B->0, with species initially separated in space is investigated. A heuristic argument suggests the form of the reaction rate in dimensions less or equal to the upper critical dimension $d_c=2$. Using this reaction rate we find that the width of the reaction front grows as $t^{1/4}$ in one dimension and as $t^{1/6}(\ln t)^{1/3}$ in two dimensions.Comment: 9 pages, Plain Te

Kinetics of A+B--->0 with Driven Diffusive Motion

We study the kinetics of two-species annihilation, A+B--->0, when all particles undergo strictly biased motion in the same direction and with an excluded volume repulsion between same species particles. It was recently shown that the density in this system decays as t^{-1/3}, compared to t^{-1/4} density decay in A+B--->0 with isotropic diffusion and either with or without the hard-core repulsion. We suggest a relatively simple explanation for this t^{-1/3} decay based on the Burgers equation. Related properties associated with the asymptotic distribution of reactants can also be accounted for within this Burgers equation description.Comment: 11 pages, plain Tex, 8 figures. Hardcopy of figures available on request from S

Localisation Transition of A Dynamic Reaction Front

We study the reaction-diffusion process $A+B\to \emptyset$ with injection of each species at opposite boundaries of a one-dimensional lattice and bulk driving of each species in opposing directions with a hardcore interaction. The system shows the novel feature of phase transitions between localised and delocalised reaction zones as the injection rate or reaction rate is varied. An approximate analytical form for the phase diagram is derived by relating both the domain of reactants $A$ and the domain of reactants $B$ to asymmetric exclusion processes with open boundaries, a system for which the phase diagram is known exactly, giving rise to three phases. The reaction zone width $w$ is described by a finite size scaling form relating the early time growth, relaxation time and saturation width exponents. In each phase the exponents are distinct from the previously studied case where the reactants diffuse isotropically.Comment: 13 pages, latex, uses eps

Order statistics for d-dimensional diffusion processes

We present results for the ordered sequence of first passage times of arrival of N random walkers at a boundary in Euclidean spaces of d dimensions

Order statistics of the trapping problem

When a large number N of independent diffusing particles are placed upon a site of a d-dimensional Euclidean lattice randomly occupied by a concentration c of traps, what is the m-th moment of the time t_{j,N} elapsed until the first j are trapped? An exact answer is given in terms of the probability Phi_M(t) that no particle of an initial set of M=N, N-1,..., N-j particles is trapped by time t. The Rosenstock approximation is used to evaluate Phi_M(t), and it is found that for a large range of trap concentracions the m-th moment of t_{j,N} goes as x^{-m} and its variance as x^{-2}, x being ln^{2/d} (1-c) ln N. A rigorous asymptotic expression (dominant and two corrective terms) is given for for the one-dimensional lattice.Comment: 11 pages, 7 figures, to be published in Phys. Rev.

A Cellular Automata Model for Citrus Variagated Chlorosis

A cellular automata model is proposed to analyze the progress of Citrus Variegated Chlorosis epidemics in S\~ao Paulo oranges plantation. In this model epidemiological and environmental features, such as motility of sharpshooter vectors which perform L\'evy flights, hydric and nutritional level of plant stress and seasonal climatic effects, are included. The observed epidemics data were quantitatively reproduced by the proposed model varying the parameters controlling vectors motility, plant stress and initial population of diseased plants.Comment: 10 pages, 10 figures, Scheduled tentatively for the issue of: 01Nov0

The effect of monomer evaporation on a simple model of submonolayer growth

We present a model for thin film growth by particle deposition that takes into account the possible evaporation of the particles deposited on the surface. Our model focuses on the formation of two-dimensional structures. We find that the presence of evaporation can dramatically affect the growth kinetics of the film, and can give rise to regimes characterized by different ``growth'' exponents and island size distributions. Our results are obtained by extensive computer simulations as well as through a simple scaling approach and the analysis of rate equations describing the system. We carefully discuss the relationship of our model with previous studies by Venables and Stoyanov of the same physical situation, and we show that our analysis is more general.Comment: 41 pages including figures, Revtex, to be published in Physical Review