Broken scaling in the Forest Fire Model

We investigate the scaling behavior of the cluster size distribution in the Drossel-Schwabl Forest Fire model (DS-FFM) by means of large scale numerical simulations, partly on (massively) parallel machines. It turns out that simple scaling is clearly violated, as already pointed out by Grassberger [P. Grassberger, J. Phys. A: Math. Gen. 26, 2081 (1993)], but largely ignored in the literature. Most surprisingly the statistics not seems to be described by a universal scaling function, and the scale of the physically relevant region seems to be a constant. Our results strongly suggest that the DS-FFM is not critical in the sense of being free of characteristic scales.Comment: 9 pages in RevTEX4 format (9 figures), submitted to PR

Synchronization and Coarsening (without SOC) in a Forest-Fire Model

We study the long-time dynamics of a forest-fire model with deterministic tree growth and instantaneous burning of entire forests by stochastic lightning strikes. Asymptotically the system organizes into a coarsening self-similar mosaic of synchronized patches within which trees regrow and burn simultaneously. We show that the average patch length grows linearly with time as t-->oo. The number density of patches of length L, N(L,t), scales as ^{-2}M(L/), and within a mean-field rate equation description we find that this scaling function decays as e^{-1/x} for x-->0, and as e^{-x} for x-->oo. In one dimension, we develop an event-driven cluster algorithm to study the asymptotic behavior of large systems. Our numerical results are consistent with mean-field predictions for patch coarsening.Comment: 5 pages, 4 figures, 2-column revtex format. To be submitted to PR

A Cellular Automaton Model for Diffusive and Dissipative Systems

We study a cellular automaton model, which allows diffusion of energy (or equivalently any other physical quantities such as mass of a particular compound) at every lattice site after each timestep. Unit amount of energy is randomly added onto a site. Whenever the local energy content of a site reaches a fixed threshold $E_{c1}$, energy will be dissipated. Dissipation of energy propagates to the neighboring sites provided that the energy contents of those sites are greater than or equal to another fixed threshold $E_{c2} (\leq E_{c1})$. Under such dynamics, the system evolves into three different types of states depending on the values of $E_{c1}$ and $E_{c2}$ as reflected in their dissipation size distributions, namely: localized peaks, power laws, or exponential laws. This model is able to describe the behaviors of various physical systems including the statistics of burst sizes and burst rates in type-I X-ray bursters. Comparisons between our model and the famous forest-fire model (FFM) are made.Comment: in REVTEX 3.0. Figures available on request. Extensively revised. Accepted by Phys.Rev.

Universal Behavior of the Coefficients of the Continuous Equation in Competitive Growth Models

The competitive growth models involving only one kind of particles (CGM), are a mixture of two processes one with probability $p$ and the other with probability $1-p$. The $p-$dependance produce crossovers between two different regimes. We demonstrate that the coefficients of the continuous equation, describing their universality classes, are quadratic in $p$ (or $1-p$). We show that the origin of such dependance is the existence of two different average time rates. Thus, the quadratic $p-$dependance is an universal behavior of all the CGM. We derive analytically the continuous equations for two CGM, in 1+1 dimensions, from the microscopic rules using a regularization procedure. We propose generalized scalings that reproduce the scaling behavior in each regime. In order to verify the analytic results and the scalings, we perform numerical integrations of the derived analytical equations. The results are in excellent agreement with those of the microscopic CGM presented here and with the proposed scalings.Comment: 9 pages, 3 figure

Order Parameter and Scaling Fields in Self-Organized Criticality

We present a unified dynamical mean-field theory for stochastic self-organized critical models. We use a single site approximation and we include the details of different models by using effective parameters and constraints. We identify the order parameter and the relevant scaling fields in order to describe the critical behavior in terms of usual concepts of non equilibrium lattice models with steady-states. We point out the inconsistencies of previous mean-field approaches, which lead to different predictions. Numerical simulations confirm the validity of our results beyond mean-field theory.Comment: 4 RevTex pages and 2 postscript figure

How self-organized criticality works: A unified mean-field picture

We present a unified mean-field theory, based on the single site approximation to the master-equation, for stochastic self-organized critical models. In particular, we analyze in detail the properties of sandpile and forest-fire (FF) models. In analogy with other non-equilibrium critical phenomena, we identify the order parameter with the density of ``active'' sites and the control parameters with the driving rates. Depending on the values of the control parameters, the system is shown to reach a subcritical (absorbing) or super-critical (active) stationary state. Criticality is analyzed in terms of the singularities of the zero-field susceptibility. In the limit of vanishing control parameters, the stationary state displays scaling characteristic of self-organized criticality (SOC). We show that this limit corresponds to the breakdown of space-time locality in the dynamical rules of the models. We define a complete set of critical exponents, describing the scaling of order parameter, response functions, susceptibility and correlation length in the subcritical and supercritical states. In the subcritical state, the response of the system to small perturbations takes place in avalanches. We analyze their scaling behavior in relation with branching processes. In sandpile models because of conservation laws, a critical exponents subset displays mean-field values ($\nu=1/2$ and $\gamma = 1$) in any dimensions. We treat bulk and boundary dissipation and introduce a new critical exponent relating dissipation and finite size effects. We present numerical simulations that confirm our results. In the case of the forest-fire model, our approach can distinguish between different regimes (SOC-FF and deterministic FF) studied in the literature and determine the full spectrum of critical exponents.Comment: 21 RevTex pages, 3 figures, submitted to Phys. Rev.

Dynamics of driven interfaces near isotropic percolation transition

We consider the dynamics and kinetic roughening of interfaces embedded in uniformly random media near percolation treshold. In particular, we study simple discrete ``forest fire'' lattice models through Monte Carlo simulations in two and three spatial dimensions. An interface generated in the models is found to display complex behavior. Away from the percolation transition, the interface is self-affine with asymptotic dynamics consistent with the Kardar-Parisi-Zhang universality class. However, in the vicinity of the percolation transition, there is a different behavior at earlier times. By scaling arguments we show that the global scaling exponents associated with the kinetic roughening of the interface can be obtained from the properties of the underlying percolation cluster. Our numerical results are in good agreement with theory. However, we demonstrate that at the depinning transition, the interface as defined in the models is no longer self-affine. Finally, we compare these results to those obtained from a more realistic reaction-diffusion model of slow combustion.Comment: 7 pages, 9 figures, to appear in Phys. Rev. E (1998

Structural Instability in Polyacene : A Projector Quantum Monte Carlo Study

We have studied polyacene within the Hubbard model to explore the effect of electron correlations on the Peierls' instability in a system marginally away from one-dimension. We employ the projector quantum Monte Carlo method to obtain ground state estimates of the energy and various correlation functions. We find strong similarities between polyacene and polyacetylene which can be rationalized from the real-space valence-bond arguments of Mazumdar and Dixit. Electron correlations tend to enhance the Peierls' instability in polyacene. This enhancement appears to attain a maximum at $U/t \sim 3.0$ and the maximum shifts to larger values when the alternation parameter is increased. The system shows no tendency to destroy the imposed bond-alternation pattern, as evidenced by the bond-bond correlations. The cis- distortion is seen to be favoured over the trans- distortion. The spin-spin correlations show that undistorted polyacene is susceptible to a SDW distortion for large interaction strength. The charge-charge correlations indicate the absence of a CDW distortion for the parameters studied.Comment: 13 pages, 10 figures available on reques

Non-Lorentzian single-molecule line shape: Pseudolocal phonons and coherence transfer

The excitation line shape of a single terrylene molecule in a naphthalene crystal has been investigated. In addition to the conventional Lorentzian, it consists of a dispersive component in the core region and a sideband. This is due to a pseudolocal phonon caused by the substitution of a host molecule with the chromophore. When the pseudolocal phonon is excited, the resonance frequency of the chromophore slightly changes, resulting in the appearance of a second, quasiresonant transition. Coherence transfer between these two optical transitions causes the deviation from the purely Lorentzian line shape

Clar's Theory, STM Images, and Geometry of Graphene Nanoribbons

We show that Clar's theory of the aromatic sextet is a simple and powerful tool to predict the stability, the \pi-electron distribution, the geometry, the electronic/magnetic structure of graphene nanoribbons with different hydrogen edge terminations. We use density functional theory to obtain the equilibrium atomic positions, simulated scanning tunneling microscopy (STM) images, edge energies, band gaps, and edge-induced strains of graphene ribbons that we analyze in terms of Clar formulas. Based on their Clar representation, we propose a classification scheme for graphene ribbons that groups configurations with similar bond length alternations, STM patterns, and Raman spectra. Our simulations show how STM images and Raman spectra can be used to identify the type of edge termination