1. **Fractal Growth in Quenched disordered media.**

In this field we have studied, both theoretically and numerically, different fractal growth models characterized by an extremal dynamics in a medium with quenched disorder. In particular we focused on models connected to percolation theory.

The first part of the work has been devoted to the formulation of a new mathematical transformation called **Run Time Statistics** (RTS). This transformation allows to represent a general deterministic extremal growth dynamics in a quenched disordered medium as a stochastic growth dynamics without quenched disorder but with memory [1]. Through this transformation it is possible to understand the mechanisms generating strong spatio-temporal correlations in many invasion processes of fluids in porous media.

The RTS transformation has been also combined with a standard method for the calculation of the fractal dimension in stochastic growth models called **Fixed Scale Transformation** (FST). This has permitted to study quantitatively the fractal properties of the geometrical structures generated by the growth process. The main application of this method has concerned the **Invasion Percolation** (IP) model and other related models [2][3][4]. IP is a model introduced in ’80’s for the problem of one fluid displacing another from a porous medium under the action of only capillary forces. It generates, with no-fine tuning of external parameters, by invasion the same fractal percolating cluster fuond at the critical point of the ordinary percolation. The theoretical results found by using the RTS+FST technic show a perfect agreement with the numerical ones obtained by Monte Carlo simulations of the growth process [15].

Other works on this subject focused on mathematical questions about the RTS as the study of the importance of the spatial correlations in its formulation [5]. Moreover it has been possible to generalize the RTS to include in its field of applicability growth models characterized by both stochastic dynamics and quenched disorder. In this way it has been possible to take into account the presence of a thermal noise in modified IP-like models [23]. Finally the RTS+FST method has been appropriately modified in such a way to study the finite size properties of the IP model [9][12].

2. **Bak-Sneppen evolution model.**

The Bak-Sneppen model (BS) is a simple model of co-evolution between interacting species. It was developed to show how self-organized criticality may explain key features of the fossil record, such as the distribution of sizes of extinction events and the phenomenon of punctuated equilibria. The BS cellular automaton for biological evolution has been studied through the use of the RTS transformation. This model, characterized by an extremal dynamics in a disordered medium, in the last years has become one of the prototypes of the concept of SOC as its dynamics self-stabilizes spontaneously in a stationary state characterized by power law distributed avalanches of evolution activity. Through an appropriate version of the RTS we have given an analytical estimate of both the critical point and the exponent characterizing the size distribution of the activity avalanches [26][34].

3. **Contact processes in disordered media.**

**Contact processes** (CP) are lattice cellular automata introduced mainly to study auto-catalytic chemical reactions and diffusion of epidemics in populations. Their physics is very intersting and presents in general a very particular dynamical second order phase transition between an **active** and an **absorbing** phase. In homogeneous media they are strictly related to the **Directed Percolation** (DP) theory and to the **Reggeon Field Theory** (RFT) with multiplicative noise. We have studied a CP in a disordered medium [13]. The application of an appropriate modification of the RTS has permitted to represent a CP in a disordered medium as a stochastic CP in a homogeneous medium but with a strong memory, and therefore to introduce a relative field theory to study the critical properties of the model.

— AndreaGabrielli – 10 Nov 2005

4. **Self-organized criticality and Absorbing phase transitions.**

Self-organized criticality (SOC) has been proposed by Bak, Tang and Wiesenfeld as a general framework to understand the occurence of power laws in nature. Slowly driven systems would develop long-range correlations spontaneously without the fine tuning of external parameters like in second-order phase transitions. It has been recognized later that this definition is somewhat ambigous since in SOC models the driving rate act as a control parameter.

The prototypical SOC model is the so-called sandpile model, because it was inspired by the dynamics of sand. In the sandpile model grains are dropped on a lattice, they can pile up until a specified height is reached, after which they fall on the neighboring sites. In this way avalanches propagate trough the system until they fall out of the boundaries.

We have studied the sandpile model by use of a mean-field approximation, which maps the model to a branching process. It is then possible to study critical exponents and to study how the critical stationary state is reached. We have analyzed the effect of dissipation and showed that the driving rate acts as a control parameter in analogy with other non-equilibrium critical phenomena. Beyond mean-field theory, critical exponents for sandpile and forest-fire model have been obtained using the dynamically-driven renormalization group.

The picture that emerges from those studies is that SOC is a particular non equilibrium critical phenomenon with a steady-state. SOC peculiaraty is that the critical values of the control parameters (driving rate, dissipation) is set to zero and therefore these systems are somewhat less sensitive to fine tuning than ordinary phase transitions..

We explored the connection between self-organized criticality and phase transitions in models with absorbing states. Sandpile models are found to exhibit criticality only when a pair of relevant parameters – dissipation e and driving field h – are set to their critical values. The critical values of epsilon and h are both equal to zero. The first is due to the absence of saturation (no bound on energy) in the sandpile model, while the second result is common to other absorbing-state transitions. The original definition of the sandpile model places it at the point ( e=0, h=0): it is critical by definition. We argue power-law avalanche distributions are a general feature of models with infinitely many absorbing configurations, when they are subject to slow driving at the critical point. Our assertions are supported by simulations of the sandpile at e=h=0 and fixed energy density (no drive, periodic boundaries), and of the slowly-driven pair contact process. We formulate a field theory for the sandpile model, in which the order parameter is coupled to a conserved energy density, which plays the role of an effective creation rate.

We study sandpile models as closed systems, with conserved energy density playing the role of an external parameter. The critical energy density marks a nonequilibrium phase transition between active and absorbing states. Several fixed-energy sandpiles are studied in extensive simulations of stationary and transient properties, as well as the dynamics of roughening in an interface-height representation.

For more informations read our pedagogical review paper:

R. Dickman, M. A. Mu�oz, A. Vespignani and S. Zapperi, “Paths to Self-organized criticality” Braz. J. Phys. 30, 27 (2000).

— StefanoZapperi – 21 Nov 2005