Reaction-diffusion system with self-organized critical behavior

We describe the construction of a conserved reaction-diffusion system that exhibits self-organized critical (avalanche-like) behavior under the action of a slow addition of particles. The model provides an illustration of the general mechanism to generate self-organized criticality in conserving systems. Extensive simulations in d = 2 and 3 reveal critical exponents compatible with the universality class of the stochastic Manna sandpile model. Received 16 November 2000     

Damage-spreading in the parallel Bak-Sneppen model

We study the behavior under perturbations of the Parallel Bak-Sneppen model (PBS) in 1+1 dimension, which has been shown to belong to the universality class of Directed Percolation (DP) in 1+1 dimensions [#!SD96!#]. We focus our attention on the damage-spreading features of the PBS model with both random and deterministic updating, which are studied and compared to the known results for the extremal Bak-Sneppen model (BS) and for DP. For both random and deterministic updating, we observe a power law growth of the Hamming distance. In addition, we compute analytically the asymptotic plateau reached by the distance after the growing phase. Received: 24 September 1998 / Revised: 17 November 1998 / Accepted: 19 November 1998     

Damage-spreading in the Bak-Sneppen model without noise

We study the behavior under perturbations in the, recently introduced, Bak-Sneppen model with deterministic updating. We focus our attention on the damage-spreading features and show that the value of the growth exponent for the distance, , coincides with that of the random updating Bak-Sneppen model. Moreover, we generalize this analysis by considering a broader set of initial perturbations for which the value of is preserved. Received: 24 June 1998 / Accepted: 9 July 1998     

Sensitivity to initial conditions in the Bak-Sneppen model of biological evolution

We consider biological evolution as described within the Bak and Sneppen 1993 model. We exhibit, at the self-organized critical state, a power-law sensitivity to the initial conditions, calculate the associated exponent, and relate it to the recently introduced nonextensive thermostatistics. The scenario which here emerges without tuning strongly reminds of that of the tuned onset of chaos in say logistic-like one-dimensional maps. We also calculate the dynamical exponent z. Received: 5 November 1997 / Received in final form: 11 November 1997 / Accepted: 19 November 1997     

Detonation type waves in phase (chemical) transformation processes in condensed matter

Fast self sustained waves (autowaves) associated with chemical or phase transformations are observed in many situations in condensed matter. They are governed neither by diffusion of matter or heat (as in combustion processes) nor by a travelling shock wave (as in gaseous detonation). Instead, they result from a coupling between phase transformation and the stress field, and may be classified as gasless detonation autowaves in solids. We propose a simple model to describe these regimes. The model rests on the classical equations of elastic deformations in a 1-dimensional solid bar, with the extra assumption that the phase (chemical) transformation induces a change of the sound velocity. The transformations are assumed to occur through a chain branched mechanism, which starts when the mechanical stress exceeds a given threshold. Our investigation shows that supersonic autowaves exist in this model. In the absence of diffusion (dissipation factor, losses), a continuum of travelling wave solutions is found. In the presence of diffusion, a steady state supersonic wave solution is found, along with a slower wave controlled by diffusion. Received 15 October 1998     

Propagation and ignition of fast gasless detonation waves of phase or chemical transformation in condensed matter

Fast self sustained waves of chemical or phase transformations, observed in several contexts in condensed matter effectively result in “gasless detonation". The phenomenon is modelled by coupling the reaction diffusion equation, describing chemical or phase transformations, and the wave equation, describing elastic perturbations. The coupling considered in this work involves (i) a dependence of the sound velocity on the chemical (phase) field, and (ii) the destruction of the initial chemical equilibrium when the strain exceeds a critical value (strain induced phase transition). Both the case of an initially unstable state (first order kinetics) and metastable state (second order kinetics) are considered. An exhaustive analytic and numerical study of travelling waves reveals the existence of supersonic modes of transformations. The practically important problem of ignition of fast waves by mechanical perturbation is investigated. With the present model, the critical strain necessary to ignite gasless detonation by local perturbations is determined. Received 18 November 1999     

The short-time critical behaviour of the Ginzburg-Landau model with long-range interaction

The renormalisation group approach is applied to the study of the short-time critical behaviour of the d-dimensional Ginzburg-Landau model with long-range interaction of the form in momentum space. Firstly the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents and of the order parameter and the response function respectively, are calculated to the second order in . Received 9 June 2000 and Received in final form 2 August 2000     

Persistence distributions in a conserved lattice gas with absorbing states

Extensive simulations are performed to study the persistence behavior of a conserved lattice gas model exhibiting an absorbing phase transition from an active phase into an inactive phase. Both the global and the local persistence exponents are determined in two and higher dimensions. The local persistence exponent obeys a scaling relation involving the order parameter exponent of the absorbing phase transition. Furthermore we observe that the global persistence exponent exceeds its local counterpart in all dimensions in contrast to the known persistence behavior in reversible phase transitions. Received 27 August 2001 and Received in final form 15 November 2001     

Temperature scaling, glassiness and stationarity in the Bak-Sneppen model

We show that the emergence of criticality in the locally-defined Bak-Sneppen model corresponds to separation over a hierarchy of timescales. Near to the critical point the model obeys scaling relations, with exponents which we derive numerically for a one-dimensional system. We further describe how the model can be related to the glass model of Bouchaud (J. Phys. I France 2, 1705 (1992)), and we use this insight to comment on the usual assumption of stationarity in the Bak-Sneppen model. Finally, we propose a general definition of self-organised criticality which is in partial agreement with other recent definitions. Received 14 January 2000 and Received in final form 18 April 2000     

Non-commutative geometry and irreversibility

A kinetics built upon q-calculus, the calculus of discrete dilatations, is shown to describe diffusion on a hierarchical lattice. The only observable on this ultrametric space is the “quasi-position” whose eigenvalues are the levels of the hierarchy, corresponding to the volume of phase space available to the system at any given time. Motion along the lattice of quasi-positions is irreversible. Received: 24 June 1997 / Revised: 15 September 1997 / Accepted: 6 October 1997     

Shear-driven heat flow in absence of a temperature gradient

Jaynesian statistical inference is used to predict that steady, non-uniform Couette flow in a simple liquid will generate a heat flux proportional to the gradient of the square of the strain-rate when the temperature gradient is negligible. The heat flux is divided into phonon and self-diffusion components, with the latter coupling to the elastic strain and inelastic strain-rate. Operators for all these are substituted into the information-theoretic phase-space distribution. By taking moments of an exact equation for this distribution derived by Robertson, one obtains an evolution equation for the self-diffusion component of the heat flux which, in a steady state, predicts shear-driven heat flow. Received 10 September 1998     

Finite-size effects in the self-organized critical forest-fire model

We study finite-size effects in the self-organized critical forest-fire model by numerically evaluating the tree density and the fire size distribution. The results show that this model does not display the finite-size scaling seen in conventional critical systems. Rather, the system is composed of relatively homogeneous patches of different tree densities, leading to two qualitatively different types of fires: those that span an entire patch and those that do not. As the system size becomes smaller, the system contains less patches, and finally becomes homogeneous, with large density fluctuations in time. Received 24 April 1999 and Received in final form 26 October 1999     

Short-time behavior of the kinetic spherical model with long-ranged interactions

The kinetic spherical model with long-ranged interactions and an arbitrary initial order m₀ quenched from a very high temperature to T is solved. In the short-time regime, the bulk order increases with a power law in both the critical and phase-ordering dynamics. To the latter dynamics, a power law for the relative order is found in the intermediate time-regime. The short-time scaling relations of small m₀ are generalized to an arbitrary m₀ and all the time larger than . The characteristic functions for the scaling of m₀ and for are obtained. The crossover between scaling regimes is discussed in detail. Received 17 September 1999     

Randomness and a step-like distribution of pile heights in avalanche models

This paper considers a one-parameter family of sand-piles. The family exhibits the crossover between the models with deterministic and stochastic relaxation. The mean pile height is used to describe the crossover. The height densities corresponding to the models with relaxation of both types approach one another as the parameter increases. Relaxation is supposed to deal with the local losses of grains by a fixed amount. In that case the densities show a step-like behaviour in contrast to the peaked shape found in the models with the local loss of grains down to a fixed level [S. Lübeck, Phys. Rev. E 62, 6149 (2000)]. A spectral approach based on the long-run properties of the pile height considers the models with deterministic and random relaxation more accurately and distinguishes between the two cases for admissible parameter values.     

Spinodal decomposition of an ABv model alloy: Patterns at unstable surfaces

We develop mean-field kinetic equations for a lattice gas model of a binary alloy with vacancies (ABv model) in which diffusion takes place by a vacancy mechanism. These equations are applied to the study of phase separation of finite portions of an unstable mixture immersed in a stable vapor. Due to a larger mobility of surface atoms, the most unstable modes of spinodal decomposition are localized at the vapor-mixture interface. Simulations show checkerboard-like structures at the surface or surface-directed spinodal waves. We determine the growth rates of bulk and surface modes by a linear stability analysis and deduce the relation between the parameters of the model and the structure and length scale of the surface patterns. The thickness of the surface patterns is related to the concentration fluctuations in the initial state. Received 28 October 1998     

Revisiting the nonequilibrium phase transition of the triplet-creation model

The nonequilibrium phase transition in the triplet-creation model is investigated using critical spreading and the conservative diffusive contact process. The results support the claim that at high enough diffusion the phase transition becomes discontinuous. As the diffusion probability increases the critical exponents change continuously from the ordinary directed percolation (DP) class to the compact directed percolation (CDP). The fractal dimension of the critical cluster, however, switches abruptly between those two universality classes. Strong crossover effects in both methods make it difficult, if not impossible, to establish the exact location of the tricritical point.     

Depinning of a domain wall in the 2d random-field Ising model

We report studies of the behaviour of a single driven domain wall in the 2-dimensional non-equilibrium zero temperature random-field Ising model, closely above the depinning threshold. It is found that even for very weak disorder, the domain wall moves through the system in percolative fashion. At depinning, the fraction of spins that are flipped by the proceeding avalanche vanishes with the same exponent as the infinite percolation cluster in percolation theory. With decreasing disorder strength, however, the size of the critical region decreases. Our numerical simulation data appear to reflect a crossover behaviour to an exponent at zero disorder strength. The conclusions of this paper strongly rely on analytical arguments. A scaling theory in terms of the disorder strength and the magnetic field is presented that gives the values of all critical exponent except for one, the value of which is estimated from scaling arguments. Received: 13 February 1998 / Accepted: 30 March 1998     

Non-linear reciprocity in extended thermodynamics from the Robertson formalism

Robertson has found a projection operator which, applied to the Liouville equation, yields an exact equation for , the information-theoretic phase-space distribution. If the Robertson equation is multiplied by a set [0pt]{} of functions representing physical fluxes, odd under momentum reversal and even under configuration inversion, a set of evolution equations is obtained for time-dependent ensemble averages which are variables of extended thermodynamics. In earlier work, a perturbation calculation was developed, assuming just one variable , for an operator [0pt] occurring in the Robertson equation. This calculation is extended here to the case where there are variables. The coefficients in the evolution equations depend on {} and explicitly on time t at short times. It is shown here that these coefficients exhibit Onsager symmetry at long times, after the transient explicit t-dependence has disappeared, to . Received 13 September 1999 and Received in final form 4 April 2000     

Two-variable macroscopic model for spin-crossover solids: Static and dynamic effects of the correlations

In order to investigate the role of nearest neighbors correlations in the relaxation of the High Spin fraction in spin crossover compounds, we have developed a two macro-variable dynamical model based on Kubo's treatement of the master equation. This is compared to the local equilibrium approach, where short-range correlations are assumed to follow adiabatically the long range-order parameter. The sigmoidal shape of the relaxation, previously associated with the effects of interactions, and the so-called “tail effect”, i.e. the extra-slowing down at long times due to the correlations are obtained. The accurate comparison to experimental relaxation data confirms the coexistence of short-range and long-range interactions in spin-crossover solids. Received 20 April 2000     

Intermittent criticality in the long-range connective sandpile (LRCS) model

We here propose a long-range connective sandpile model with variable connection probability Pc which has an important impact on the slope of the power-law frequency-size distribution of avalanches. The long-range connection probability Pc is changed according to an explicit function of the size of the latest event, although the evolution rule of Pc may be different in various physical systems. Such version of the sandpile model demonstrates large fluctuations in the dynamical variable 〈Z〉(t) (the spatially averaged amount of grains retained within the grid at each time step), indicating the state of intermittent criticality in the system. Many researches have suggested that the earthquake fault system is an intermittent criticality system, which would imply some level of statistical predictability of great events. Our modified sandpile model thus provides a testing ground for many proposed precursory measures related to great earthquakes.