The branching structure of diffusion-limited aggregates

I analyze the topological structures generated by diffusion-limited aggregation (DLA), using the recently developed "branched growth model". The computed bifurcation number B for DLA in two dimensions is B ~ 4.9, in good agreement with the numerically obtained result of B ~ 5.2. In high dimensions, B -> 3.12; the bifurcation ratio is thus a decreasing function of dimensionality. This analysis also determines the scaling properties of the ramification matrix, which describes the hierarchy of branches.Comment: 6 pages, 1 figure, Euro-LaTeX styl

On the predictive power of Local Scale Invariance

Local Scale Invariance (LSI) is a theory for anisotropic critical phenomena designed in the spirit of conformal invariance. For a given representation of its generators it makes non-trivial predictions about the form of universal scaling functions. In the past decade several representations have been identified and the corresponding predictions were confirmed for various anisotropic critical systems. Such tests are usually based on a comparison of two-point quantities such as autocorrelation and response functions. The present work highlights a potential problem of the theory in the sense that it may predict any type of two-point function. More specifically, it is argued that for a given two-point correlator it is possible to construct a representation of the generators which exactly reproduces this particular correlator. This observation calls for a critical examination of the predictive content of the theory.Comment: 17 pages, 2 eps figure

Influence of pore-scale disorder on viscous fingering during drainage

We study viscous fingering during drainage experiments in linear Hele-Shaw cells filled with a random porous medium. The central zone of the cell is found to be statistically more occupied than the average, and to have a lateral width of 40% of the system width, irrespectively of the capillary number $Ca$. A crossover length $w_f \propto Ca^{-1}$ separates lower scales where the invader's fractal dimension $D\simeq1.83$ is identical to capillary fingering, and larger scales where the dimension is found to be $D\simeq1.53$. The lateral width and the large scale dimension are lower than the results for Diffusion Limited Aggregation, but can be explained in terms of Dielectric Breakdown Model. Indeed, we show that when averaging over the quenched disorder in capillary thresholds, an effective law $v\propto (\nabla P)^2$ relates the average interface growth rate and the local pressure gradient.Comment: 4 pages, 4 figures, submitted to Phys Rev Letter

Absorbing state phase transitions with quenched disorder

Quenched disorder - in the sense of the Harris criterion - is generally a relevant perturbation at an absorbing state phase transition point. Here using a strong disorder renormalization group framework and effective numerical methods we study the properties of random fixed points for systems in the directed percolation universality class. For strong enough disorder the critical behavior is found to be controlled by a strong disorder fixed point, which is isomorph with the fixed point of random quantum Ising systems. In this fixed point dynamical correlations are logarithmically slow and the static critical exponents are conjecturedly exact for one-dimensional systems. The renormalization group scenario is confronted with numerical results on the random contact process in one and two dimensions and satisfactory agreement is found. For weaker disorder the numerical results indicate static critical exponents which vary with the strength of disorder, whereas the dynamical correlations are compatible with two possible scenarios. Either they follow a power-law decay with a varying dynamical exponent, like in random quantum systems, or the dynamical correlations are logarithmically slow even for weak disorder. For models in the parity conserving universality class there is no strong disorder fixed point according to our renormalization group analysis.Comment: 17 pages, 8 figure

Multicomponent binary spreading process

I investigate numerically the phase transitions of two-component generalizations of binary spreading processes in one dimension. In these models pair annihilation: AA->0, BB->0, explicit particle diffusion and binary pair production processes compete with each other. Several versions with spatially different productions have been explored and shown that for the cases: 2A->3A, 2B->3B and 2A->2AB, 2B->2BA a phase transition occurs at zero production rate ($\sigma=0$), that belongs to the class of N-component, asymmetric branching and annihilating random walks, characterized by the order parameter exponent $\beta=2$. In the model with particle production: AB->ABA, BA-> BAB a phase transition point can be located at $\sigma_c=0.3253$ that belongs to the class of the one-component binary spreading processes.Comment: 5 pages, 5 figure

Test of Local Scale Invariance from the direct measurement of the response function in the Ising model quenched to and to below TCT_C

In order to check on a recent suggestion that local scale invariance [M.Henkel et al. Phys.Rev.Lett. {\bf 87}, 265701 (2001)] might hold when the dynamics is of Gaussian nature, we have carried out the measurement of the response function in the kinetic Ising model with Glauber dynamics quenched to $T_C$ in $d=4$, where Gaussian behavior is expected to apply, and in the two other cases of the $d=2$ model quenched to $T_C$ and to below $T_C$, where instead deviations from Gaussian behavior are expected to appear. We find that in the $d=4$ case there is an excellent agreement between the numerical data, the local scale invariance prediction and the analytical Gaussian approximation. No logarithmic corrections are numerically detected. Conversely, in the $d=2$ cases, both in the quench to $T_C$ and to below $T_C$, sizable deviations of the local scale invariance behavior from the numerical data are observed. These results do support the idea that local scale invariance might miss to capture the non Gaussian features of the dynamics. The considerable precision needed for the comparison has been achieved through the use of a fast new algorithm for the measurement of the response function without applying the external field. From these high quality data we obtain $a=0.27 \pm 0.002$ for the scaling exponent of the response function in the $d=2$ Ising model quenched to below $T_C$, in agreement with previous results.Comment: 24 pages, 6 figures. Resubmitted version with improved discussions and figure

The universal behavior of one-dimensional, multi-species branching and annihilating random walks with exclusion

A directed percolation process with two symmetric particle species exhibiting exclusion in one dimension is investigated numerically. It is shown that if the species are coupled by branching ($A\to AB$, $B\to BA$) a continuous phase transition will appear at zero branching rate limit belonging to the same universality class as that of the dynamical two-offspring (2-BARW2) model. This class persists even if the branching is biased towards one of the species. If the two systems are not coupled by branching but hard-core interaction is allowed only the transition will occur at finite branching rate belonging to the usual 1+1 dimensional directed percolation class.Comment: 3 pages, 3 figures include

Numerical schemes for continuum models of reaction-diffusion systems subject to internal noise

We present new numerical schemes to integrate stochastic partial differential equations which describe the spatio-temporal dynamics of reaction-diffusion (RD) problems under the effect of internal fluctuations. The schemes conserve the nonnegativity of the solutions and incorporate the Poissonian nature of internal fluctuations at small densities, their performance being limited by the level of approximation of density fluctuations at small scales. We apply the new schemes to two different aspects of the Reggeon model namely, the study of its non-equilibrium phase transition and the dynamics of fluctuating pulled fronts. In the latter case, our approach allows to reproduce quantitatively for the first time microscopic properties within the continuum model.Comment: 5 pages, 3 figures, Accepted for publication in Physical Review E as a Rapid Communicatio

Contact process with long-range interactions: a study in the ensemble of constant particle number

We analyze the properties of the contact process with long-range interactions by the use of a kinetic ensemble in which the total number of particles is strictly conserved. In this ensemble, both annihilation and creation processes are replaced by an unique process in which a particle of the system chosen at random leaves its place and jumps to an active site. The present approach is particularly useful for determining the transition point and the nature of the transition, whether continuous or discontinuous, by evaluating the fractal dimension of the cluster at the emergence of the phase transition. We also present another criterion appropriate to identify the phase transition that consists of studying the system in the supercritical regime, where the presence of a "loop" characterizes the first-order transition. All results obtained by the present approach are in full agreement with those obtained by using the constant rate ensemble, supporting that, in the thermodynamic limit the results from distinct ensembles are equivalent

Non-equilibrium Phase Transitions with Long-Range Interactions

This review article gives an overview of recent progress in the field of non-equilibrium phase transitions into absorbing states with long-range interactions. It focuses on two possible types of long-range interactions. The first one is to replace nearest-neighbor couplings by unrestricted Levy flights with a power-law distribution P(r) ~ r^(-d-sigma) controlled by an exponent sigma. Similarly, the temporal evolution can be modified by introducing waiting times Dt between subsequent moves which are distributed algebraically as P(Dt)~ (Dt)^(-1-kappa). It turns out that such systems with Levy-distributed long-range interactions still exhibit a continuous phase transition with critical exponents varying continuously with sigma and/or kappa in certain ranges of the parameter space. In a field-theoretical framework such algebraically distributed long-range interactions can be accounted for by replacing the differential operators nabla^2 and d/dt with fractional derivatives nabla^sigma and (d/dt)^kappa. As another possibility, one may introduce algebraically decaying long-range interactions which cannot exceed the actual distance to the nearest particle. Such interactions are motivated by studies of non-equilibrium growth processes and may be interpreted as Levy flights cut off at the actual distance to the nearest particle. In the continuum limit such truncated Levy flights can be described to leading order by terms involving fractional powers of the density field while the differential operators remain short-ranged.Comment: LaTeX, 39 pages, 13 figures, minor revision