Short-time dynamics of an ising system on fractal structures

Critical short-time dynamics in a system with interacting static and diffusive populations

We study the critical short-time dynamical behavior of a one-dimensional model where diffusive individuals can infect a static population upon contact. The model presents an absorbing phase transition from an active to an inactive state. Previous calculations of the critical exponents based on quasistationary quantities have indicated an unusual crossover from the directed percolation to the diffusive contact process universality classes. Here we show that the critical exponents governing the slow short-time dynamic evolution of several relevant quantities, including the order parameter, its relative fluctuations, and correlation function, reinforce the lack of universality in this model. Accurate estimates show that the critical exponents are distinct in the regimes of low and high recovery rates.

Short-time critical dynamics of damage spreading in the two-dimensional Ising model

The short-time critical dynamics of propagation of damage in the Ising ferromagnet in two dimensions is studied by means of Monte Carlo simulations. Starting with equilibrium configurations at T=∞ and magnetization M=0 , an initial damage is created by flipping a small amount of spins in one of the two replicas studied. In this way, the initial damage is proportional to the initial magnetization M0 in one of the configurations upon quenching the system at T C, the Onsager critical temperature of the ferromagnetic-paramagnetic transition. It is found that, at short times, the damage increases with an exponent θ D=1.915(3) , which is much larger than the exponent θ=0.197 characteristic of the initial increase of the magnetization M(t). Also, an epidemic study was performed. It is found that the average distance from the origin of the epidemic (R2(t)) grows with an exponent z∗ ≈ η ≈ 1.9, which is the same, within error bars, as the exponent θ D. However, the survival probability of the epidemics reaches a plateau so that δ=0. On the other hand, by quenching the system to lower temperatures one observes the critical spreading of the damage at T D ≃ 0.51TC, where all the measured observables exhibit power laws with exponents θ D=1.026(3), δ=0.133(1), and z∗=1.74(3).

Criticality of a contact process with coupled diffusive and non-diffusive fields

We investigate the critical behavior of a model with two coupled critical densities, one of which is diffusive. The model simulates the propagation of an epidemic process in a population, which uses the underlying lattice to leave a track of the recent disease history. We determine the critical density of the population above which the system reaches an active stationary state with a finite density of active particles. We also perform a scaling analysis to determine the order parameter, the correlation length, and critical relaxation exponents. We show that the model does not belong to the usual directed percolation universality class and is compatible with the class of directed percolation with diffusive and conserved fields.

Phase-ordering kinetics on graphs

We study numerically the phase-ordering kinetics following a temperature quench of the Ising model with single spin-flip dynamics on a class of graphs, including geometrical fractals and random fractals, such as the percolation cluster. For each structure we discuss the scaling properties and compute the dynamical exponents. We show that the exponent a_{chi} for the integrated response function, at variance with all the other exponents, is independent of temperature and of the presence of pinning. This universal character suggests a strict relation between a_{chi} and the topological properties of the networks, in analogy to what is observed on regular lattices.

Critical behavior of an Ising system on the Sierpinski carpet: a short-time dynamics study

The short-time dynamic evolution of an Ising model embedded in an infinitely ramified fractal structure with noninteger Hausdorff dimension was studied using Monte Carlo simulations. Completely ordered and disordered spin configurations were used as initial states for the dynamic simulations. In both cases, the evolution of the physical observables follows a power-law behavior. Based on this fact, the complete set of critical exponents characteristic of a second-order phase transition was evaluated. Also, the dynamic exponent theta of the critical initial increase in magnetization, as well as the critical temperature, were computed. The exponent theta exhibits a weak dependence on the initial (small) magnetization. On the other hand, the dynamic exponent z shows a systematic decrease when the segmentation step is increased, i.e., when the system size becomes larger. Our results suggest that the effective noninteger dimension for the second-order phase transition is noticeably smaller than the Hausdorff dimension. Even when the behavior of the magnetization (in the case of the ordered initial state) and the autocorrelation (in the case of the disordered initial state) with time are very well fitted by power laws, the precision of our simulations allows us to detect the presence of a soft oscillation of the same type in both magnitudes that we attribute to the topological details of the generating cell at any scale.

Nearly space-filling fractal networks of carbon nanopores

Small-angle x-ray scattering, nitrogen adsorption, and scanning tunneling microscopy show that a series of activated carbons host an extended fractal network of channels with dimension D(p) = 2.8-3.0 (pore fractal), channel width 15-20 A (lower end of scaling), network diameter 3000-3400 A (upper end of scaling), and porosity of 0.3-0.6. We interpret the network as a stack of quasiplanar invasion percolation clusters, formed by oxidative removal of walls between closed voids of diameter of approximately 10 A and held in registry by fibrils of the biological precursor, and point out unique applications.