Zeev Olami

Variation of the Gutenberg‐Richter b values and nontrivial temporal correlations in a Spring‐Block Model for earthquakes

We show that a two-dimensional spring-block model for earthquakes is equivalent to a continuous, nonconservative cellular automaton model. The level of conservation is a function of the relevant elastic parameters describing the model. The model exhibits power law distributions for the energy released during an earthquake. The corresponding exponent is not universal. It is a function of the level of conservation. Thus the observed variation in the b value in the Gutenberg-Richter law could be explained by a variation in the elastic parameters. We address the problem of the boundary conditions and display results for two extreme possibilities. Furthermore, we discuss the correlation in the interoccurrence time of earthquakes. The model exhibits the features of real earthquakes: the occurrence of small earthquakes is random, while the larger earthquakes seem to be bunched. Primarily, the results of our work indicate that the dynamic of earthquakes is intimately related to the nonconservative nature of the model, which gives birth to both the change in the exponent and the correlations in interoccurrence time.

Flame front propagation VI: Dynamics and Wrinkling of Radially Propagating Fronts Inferred from Scaling Laws in Channel Geometries

Flame Propagation is used as a prototypical example of expanding fronts that wrinkle without limit in radial geometries but reach a simple shape in channel geometry. We show that the relevant scaling laws that govern the radial growth can be inferred once the simpler channel geometry is understood in detail. In radial geometries (in contrast to channel geometries) the effect of external noise is crucial in accelerating and wrinkling the fronts. Nevertheless, once the interrelations between system size, velocity of propagation and noise level are understood in channel geometry, the scaling laws for radial growth follow.

Flame front propagation IV: Random Noise and Pole-Dynamics in Unstable Front Propagation II

The current paper is a corrected version of our previous pape r (Olami et al., PRE 55 (3),(1997)). Similarly to previous version we investigate th problem of flame propagation. This problem is studied as an example of unstab le fronts that wrinkle on many scales. The analytic tool of pole expansion in the com plex plane is employed to address the interaction of the unstable growth pro cess with random initial conditions and perturbations. We argue that the effect of ra ndom noise is immense and that it can never be neglected in sufficiently large syste m . We present simulations that lead to scaling laws for the velocity and accele ration of the front as a function of the system size and the level of noise, and analyt ic arguments that explain these results in terms of the noisy pole dynamics.This ver ion corrects some very critical errors made in (Olami et al., PRE 55 (3),(1997)) and makes more detailed description of excess number of poles in system , nu mber of poles that appear in the system in unit of time, life time of pole. It allo ws us to understand more correctly dependence of the system parameters on noise than in (Olami et al., PRE55 (3),(1997))The problem of flame propagation is studied as an example of unstable fronts that wrinkle on many scales. The analytic tool of pole expansion in the complex plane is employed to address the interaction of the unstable growth process with random initial conditions and perturbations. We argue that the effect of random noise is immense and that it can never be neglected in sufficiently large systems. We present simulations that lead to scaling laws for the velocity and acceleration of the front as a function of the system size and the level of noise and also analytic arguments that explain these results in terms of the noisy pole dynamics.