Colored activity in self-organized critical interface dynamics

Pinning-depinning transition in a stochastic growth model for the evolution of cell colony fronts in a disordered medium

We study a stochastic lattice model for cell colony growth, which takes into account proliferation, diffusion, and rotation of cells, in a culture medium with quenched disorder. The medium is composed of sites that inhibit any possible change in the internal state of the cells, representing the disorder, as well as by active medium sites that do not interfere with the cell dynamics. By means of Monte Carlo simulations we find that the velocity of the growing interface, which is taken as the order parameter of the model, strongly depends on the density of active medium sites (ρ_{A}). In fact, the model presents a (continuous) second-order pinning-depinning transition at a certain critical value of ρ_{A}^{crit}, such as, for ρ_{A}>ρ_{A}^{crit}, the interface moves freely across the disordered medium, but for ρ_{A}<ρ_{A}^{crit} the interface becomes irreversible pinned by the disorder. By determining the relevant critical exponents, our study reveals that within the depinned phase the interface can be rationalized in terms of the Kardar-Parisi-Zhang universality class, but when approaching the critical threshold, the nonlinear term of the Kardar-Parisi-Zhang equation tends to vanish and then the pinned interface belongs to the quenched Edwards-Wilkinson universality class.

Activity statistics, avalanche kinetics, and velocity correlations in surface growth

We investigate the complex spatiotemporal dynamics in avalanche driven surface growth by means of scaling theory. We study local activity statistics, avalanche kinetics, and temporal correlations in the global interface velocity, obtaining different scaling relationships among the involved critical exponents depending on how far from or close to a critical point the system is. Our scaling arguments are very general and connect local and global magnitudes through several scaling relationships. We expect our results to be applicable in a wide range of systems exhibiting interface kinetic roughening driven by avalanches of local activity, either critical or not. As an example we apply the scaling theory to analyze avalanches and roughening of forced-flow imbibition fronts in excellent agreement with phase-field numerical integrations.

Adaptive coevolutionary networks: a review

Adaptive networks appear in many biological applications. They combine topological evolution of the network with dynamics in the network nodes. Recently, the dynamics of adaptive networks has been investigated in a number of parallel studies from different fields, ranging from genomics to game theory. Here we review these recent developments and show that they can be viewed from a unique angle. We demonstrate that all these studies are characterized by common themes, most prominently: complex dynamics and robust topological self-organization based on simple local rules.

Logarithmic corrections in directed percolation

We study directed percolation at the upper critical transverse dimension d=4, where critical fluctuations induce logarithmic corrections to the leading (mean-field) behavior. Viewing directed percolation as a kinetic process, we address the following properties of directed percolation clusters: the mass (the number of active sites or particles), the radius of gyration, and the survival probability. Using renormalized dynamical field theory, we determine the leading and the next to leading logarithmic corrections for these quantities. In addition, we calculate the logarithmic corrections to the equation of state that describes the stationary homogeneous particle density in the presence of a homogeneous particle source.

Self-organized interface growth with the negative nonlinearity in a random medium

We introduce two self-organized growth models that describe the motion of the driven interfaces in random media including the Kardar-Parisi-Zhang (KPZ) nonlinearity. One model follows the quenched KPZ equation with a positive nonlinear term, while the other model follows the quenched KPZ equation with a negative nonlinear term. By obtaining the critical exponents for two models, we confirm that the sign of the KPZ nonlinear term does not affect the universality class.

Interface dynamics and kinetic roughening in fractals

We consider the dynamics and kinetic roughening of single-valued interfaces in two-dimensional fractal media. Assuming that the local height difference distribution function of the fronts obeys Levý statistics with a well-defined power-law decay exponent, we derive analytic expressions for the local scaling exponents. We also show that the kinetic roughening of the interfaces displays anomalous scaling and multiscaling in the relevant correlation functions. For invasion percolation models, the exponents can be obtained from the fractal geometry of percolation clusters. Our predictions are in excellent agreement with numerical simulations.

Multifractal properties of resistor diode percolation

Focusing on multifractal properties we investigate electric transport on random resistor diode networks at the phase transition between the nonpercolating and the directed percolating phase. Building on first principles such as symmetries and relevance we derive a field theoretic Hamiltonian. Based on this Hamiltonian we determine the multifractal moments of the current distribution that are governed by a family of critical exponents [psi(l)]. We calculate the family [psi(l)] to two-loop order in a diagrammatic perturbation calculation augmented by renormalization group methods.

Nonlinear random resistor diode networks and fractal dimensions of directed percolation clusters

We study nonlinear random resistor diode networks at the transition from the nonpercolating to the directed percolating phase. The resistor-like bonds and the diode-like bonds under forward bias voltage obey a generalized Ohm's law V approximately I(r). Based on general grounds such as symmetries and relevance we develop a field theoretic model. We focus on the average two-port resistance, which is governed at the transition by the resistance exponent straight phi(r). By employing renormalization group methods we calculate straight phi(r) for arbitrary r to one-loop order. Then we address the fractal dimensions characterizing directed percolation clusters. Via considering distinct values of the nonlinearity r, we determine the dimension of the red bonds, the chemical path, and the backbone to two-loop order.

Transport on directed percolation clusters

We study random lattice networks consisting of resistorlike and diodelike bonds. For investigating the transport properties of these random resistor diode networks we introduce a field-theoretic Hamiltonian amenable to renormalization group analysis. We focus on the average two-port resistance at the transition from the nonpercolating to the directed percolating phase and calculate the corresponding resistance exponent straight phi to two-loop order. Moreover, we determine the backbone dimension D(B) of directed percolation clusters to two-loop order. We obtain a scaling relation for D(B) that is in agreement with well known scaling arguments.

Anomalous scaling in depinning transitions

It is demonstrated that the renormalization group (RG) flows of depinning transitions do not depend on whether the driving force or the system velocity is kept constant. This allows for a comparison between RG results and corresponding self-organized critical models. However, close to the critical point, scaling functions cross over to forms that can have singular behavior not seen in equilibrium thermal phase transitions. These can be different for the constant force and constant velocity driving modes, leading to different apparent critical exponents. This is illustrated by comparing extremal dynamics for interface depinning with RG results, deriving the change in apparent exponents. Thus, care has to be exercised in such comparisons.

Spatial-temporal correlations in the process to self-organized criticality

A different type of spatial-temporal correlation in the process approaching the self-organized criticality is investigated for the two simple models for biological evolution. The changed behaviors of the position with minimum barrier are shown to be quantitatively different in the two models. Different results of the correlation are given for the two models. We argue that the correlation can be used, together with the power-law distributions, as criteria for self-organized criticality.

Barkhausen avalanches in anisotropic ferromagnets with 180 degrees domain walls

We show that Barkhausen noise in two-dimensional disordered ferromagnets with extended domain walls is characterized by the avalanche size exponent tau(s)=1.54 at low disorder. With increasing disorder the characteristic domain size is reduced relative to the system size due to nucleation of new domains and a dynamic phase transition occurs to the scaling behavior with tau(s)=1.30. The exponents decrease at finite driving rate. The results agree with recently observed behavior in amorphous Metglas and Fe-Co-B ribbons when the applied anisotropic stress is varied.

Self-organized growth model for a driven interface in random media

We introduce a simple self-organized growth model in the quenched Edwards-Wilkinson universality class. The roughness and growth exponents are obtained for the model as alpha approximately 1.15 and beta approximately 0.89, respectively. These values of exponents are in good agreement with previous results obtained from numerical integrations or discretized model simulations of the quenched Edwards-Wilkinson equation. The velocity of a driven interface is found to be independent of the slope of the tilted substrate.