Scaling and Disordered Systems

Preface.

ANTONIO CONIGLIO: Curriculum Vitae.

Program of the Workshop.

Contents.

Renormalization group and Coulomb gas mappings are used to derive theoretical predictions for the corrections to the exactly known asymptotic fractal masses of the hull, external perimeter, singly connected bonds and total mass of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality. For q = 4 these include exact logarithmic (as well as log-log) corrections.

We present a percolation dynamic model for the study of dynamics at the sol-gel transition. Percolation and bond-fluctuation dynamics result to be suited to study the critical behaviour of the viscoelastic properties and show a complex relaxation behaviour. The results obtained via numerical simulation on the cubic lattice are in good agreement with some theoretical predictions and experimental results.

In a first part, we study the backbone connecting two given sites of a two-dimensional lattice separated by an arbitrary distance r in a system of size L. We find a scaling form for the average backbone mass and we also propose a scaling form for the probability distribution P(M_B) of backbone mass for a given r. For r ≈ L, P(M_B) is peaked around L^d_B, whereas for r ≪ L, P(M_B) decreases as a power law, , with τ_B ≃ 1.20 ± 0.03. The exponents ψ and τ_B satisfy the relation ψ = d_B(τ_B - 1), and ψ is the codimension of the backbone, ψ = d - d_B. In a second part, we study the multifractal spectrum of the current in the two-dimensional random resistor network at the percolation threshold. Our numerical results suggest that in the infinite system limit, the probability distribution behaves for small i as P(i) ~ 1/i where i is the current. As a consequence, the moments of i of order q ≤ q_c = 0 diverge with system size, and all sets of bonds with current values below the most probable one have the fractal dimension of the backbone. Hence we hypothesize that the backbone can be described in terms of only (i) blobs of fractal dimension d_B and (ii) high current carrying bonds of fractal dimension going from d_red to d_B, where d_red is the fractal dimension of the red bonds carrying the maximal current.

A growth model, introduced to model the development of branched polymers in an heterogeneous environment, gives rise to clusters whose boundary is either faceted or rough. We study the transition between these two morphologies as a function of the parameters of the model, impurity concentration and branching ratio. The phase diagram is first obtained by direct numerical simulations, using an original algorithm, based on a self-regulated search of a critical point. Then an analytic computation of the phase boundary is proposed based on a simple approximation. The obtained phase boundary is in good agreement with the numerical results. The nature of the transition is discussed.

In this work we study an attractive micellar system for which the percolation curve terminates near the critical point. We have studied such an intriguing situation by means of scattering (elastic and dynamical) and viscoelasticity experiments. Obtained data are accounted by considering in a proper way the fractal clustering processes typical of percolating systems and the related scaling concepts. We observe that the main role in the system structure and dynamics it is played by the cluster's partial screening of hydrodynamic interaction. This behaves on approaching the percolation threshold dramatic effects on the system rheological properties and on the density decay relaxations. The measured correlation functions assume a stretched exponential form and the system becomes strongly viscoelastic. The overall behavior of the measured dynamical and structural parameters indicates, that in the present micellar system, the clustering process originates dilute, poly-disperse and swelling structures. Finally, this originates an interesting situation observed in the present experiment. As it has been previously, proposed by A. Coniglio et al., percolation clusters can be considered to be "Ising clusters" with the same properties as the Fisher's critical droplets. Therefore at the critical point the percolation connectedness length (ξ_p) can be assumed as the diverging correlation length (ξ_p ≡ ξ) and the mean cluster size diverges as the susceptibility.

In this paper finite size scaling techniques are used to study the universality class of thermally diluted Ising systems, in which the realization of the disposition of magnetic atoms and vacancies is taken from the local distribution of spins in the pure original Ising model at criticality. The critical temperature, the critical exponents and therefore the universality class of these thermally diluted Ising systems depart markedly from the ones of short range correlated disordered systems. This result is in agreement with theoretical predictions previously made by Weinrib and Halperin for systems with long range correlated disorder.

In this paper some critical aspects of the behaviour of breaking lattices subject to slow driving forces are briefly reviewed. In particular, fluctuations in the response to external solicitations are discussed.

We give a review of the properties of low-temperature magnetic fluctuations in a finite-size 2D XY system. The behavior of such a simple model closely resembles that of several complex systems, notably the spectrum of power fluctuations in enclosed turbulent flow and interface width in surface growth problems. It suggests moreover new ideas in the study of aging phenomena in disordered systems.

We present a short review of experimental and theoretical aspects of granular compaction, and we discuss in more details the behaviour of the so-called Random Tetris Model, a model of particles diffusing on a lattice, subject to gravity and geometrical constraints. We show how this model reproduces the experimental phenomenology, e.g. slow relaxation, irreversible/reversible cycles, memory effects. The study of the density profiles allows to interpret these results.

The relaxation time scale in glassy materials is derived within a model of anomalous defect diffusion. The effect of the defects on ion-doped polymeric glasses is to produce a stretched exponential waiting time distribution for ion jumps. The characteristic time scale for ion jumps is connected to the temperature and pressure dependent concentration of mobile defects. The resultant expression for ionic conductivity is compared with experimental results for the polymer electrolyte poly (propylene glycol) (PPG) containing LiCF₃SO₃.

In this paper we study the 3D frustrated lattice gas model in the quenched and annealed versions. In the first case, the dynamical non-linear susceptibility grows monotonically as a function of time, until reaching a plateau that corresponds to the static value. The static non-linear susceptibility diverges at some density, signaling the presence of a thermodynamical transition. In the annealed version, where the disorder is allowed to evolve in time with a suitable kinetic constraint, the thermodynamics of the model is trivial, and the static non-linear susceptibility does not show any singularity. Nevertheless, the model shows a maximum in the dynamical non-linear susceptibility at a characteristic value of the time. Approaching the density corresponding to the singularity of the quenched model, both the maximum and the characteristic time diverge. We conclude that the critical behavior of the dynamical susceptibility in the annealed model is related to the divergence of the static susceptibility in the quenched case. This suggests a similar mechanism also in supercooled glass-forming liquids, where an analogous behavior in the dynamical non-linear susceptibility is observed.

We consider the dynamics of supercooled fluids subject to a continuous quenching procedure, with cooling rate r = dT(t)/dt. The analysis is carried out analytically in the framework of a mean field schematic model recently introduced.¹ We show the existence of a glass temperature T_g(r) below which the system falls out of equilibrium. T_g(r) approaches logarithmically in r the ideal glass temperature T₀ = lim_r→0 T_g(r), where the relaxation time diverges á la Vogel-Fulcher, similarly to some experimental observations. Well above T_g(r) a simple fluid behavior is observed. As T_g is approached from above a characteristic wave vector k_d divides an high momenta equilibrated region, where a fluid-like behavior is obeyed, from the non thermalized modes with k < k_d, for which time translational invariance lacks. Below T_g the system is found in a globally off-equilibrium glassy state characterized by a logarithmic decay of the density fluctuations and aging. The two time correlator decays as an enhanced power law.

We study the phase separation of a binary mixture in uniform shear flow in the framework of the continuum convection-diffusion equation based on a Ginzburg-Landau free energy. This equation is solved both numerically and in the context of large-N approximation. Our results show the existence of domains with two typical sizes, whose relative abundance changes in time. As a consequence log-time periodic oscillations are observed in the behavior of most thermodynamic observables.

We consider a frustrated spin model with a glassy dynamics characterized by a slow component and a fast component in the relaxation process. The slow process involves variables with critical behavior at finite temperature T_p and has a global character like the (structural) α-relaxation of glasses. The fast process has a more local character and can be associated to the β-relaxation of glasses. At temperature T > T_p the fast relaxation follows the non-Arrhenius behavior of the slow variables. At T ≲ T_p the fast variables have an Arrhenius behavior, resembling the α - β bifurcation of fragile glasses. The model allows us to analyze the relation between the dynamics and the thermodynamics.

The transport and the relaxation properties of a biatomic supercooled liquid are studied by molecular-dynamics methods. Both translational and rotational jumps are evidenced. At lower temperatures their waiting-time distributions decay as a power law at short times. The Stokes-Einstein relation (SE) breaks down at a temperature which is close to the onset of the intermittency. A precursor effect of the SE breakdown is observed as an apparent stick-slip transition. The breakdown of Debye-Stokes-Einstein law for rotational motion is also observed. On cooling, the changes of the rotational correlation time τ₁ and the translational diffusion coefficient at low temperatures are fitted by power laws over more than three and four orders of magnitude, respectively. A less impressive agreement is found for τ_l with l = 2 - 4 and the rotational diffusion coefficient.

The present paper provides a short review of recent progresses in understanding out of equilibrium vortex behaviours in type II superconductors by use of a schematic coarse grained vortex model. In particular, it is possible to depict a unifying scenario for magnetic and transport properties of off-equilibrium vortex matter, ranging from the reentrant phase diagram, to magnetisation loops, "anomalous" 2-nd peak, logarithmic creep, "anomalous" finite creep rate for T → 0, "memory" and "irreversibility" in I-V characteristics, "rejuvenation" and "stiffening" of the system response.

Here we study the zero temperature dynamics of the Sherrington Kirkpatrick model and we investigate the statistical properties of the configurations that are obtained in the large time limit. We find that the replica symmetry is broken (in a weak sense). We also present some general considerations on the synchronic approach to the off-equilibrium dynamics, that has motivated the present study.

A memory function equation and scaling relationships were used for the physical interpretation of the Cole-Cole exponent. The correspondence between the relaxation time, the geometrical properties, the self-diffusion coefficient and the Cole-Cole exponent was established. Using this approach the dielectric relaxation spectra of the polymer-water mixtures and the glass transition process in the nylon 6,6 quenched, crystalline and micro-composite samples were analyzed.

We review the properties of the Parking Lot Model and their connection with the phenomenology of vibrated granular materials. New simulation results concerning the out-of-equilibrium, aging behavior of the model are presented. We investigate in particular the relation between two-time response and correlation functions and the so-called violation of the fluctuation-dissipation theorem.

The phase-ordering dynamics with conserved scalar order parameter is studied simulating the Cahn-Hilliard equation and the kinetic Ising model. In both cases a preasymptotic multiscaling regime is found revealing that the late stage of phase-ordering is always approached through a crossover from multiscaling to standard scaling, independently from the nature of the microscopic dynamics.

Characterizing the complex atmospheric variability at all pertinent temporal and spatial scales remains one of the most important challenges to scientific research today.^1–5 The main issues are to quantify, within reasonably narrow limits, the potential extent of global warming, and to downscale the global results in order to describe and quantify the regional implications of global change.

We analyze the nonlinear relaxation of a complex ecosystem composed of many interacting species. The ecological system is described by generalized Lotka-Volterra equations with a multiplicative noise. The transient dynamics is studied in the framework of the mean field theory and with random interaction between the species. We focus on the statistical properties of the asymptotic behaviour of the time integral of the ith population and on the distribution of the population and of the local field.

We consider a one-parameter kinetic model for a fluctuating interface which can be thought of as an infinite string decorated with infinitely many closed strings. Numerical simulations show that a number of scaling exponents describing this string system may be related to the Kardar-Parisi-Zhang exponents. However, as the average velocity of the infinite string is taken to zero, and the string system becomes an isotropic fractal set, we also find new exponents which cannot be reduced to previously known ones.

Stochastic multiline evolution in square lattice is studied. It turns out that the emerging patterns evolve subdiffusively, which is characterized by the exponent. Possible origin of such a slow behavior is discussed, and some elucidation, supporting the small fractional value is given. A notion of (dynamic) phase transition concept may sometimes help in understanding the presented random kinetic behavior.

The dynamics of majority rule voting in hierarchical structures is studied using concepts from collective phenomena in physics. In the case of a two-party competition a very simple model to a democratic dictatorship is presented. For each running group, a critical threshold (in the overall support) is found to ensure full and total power at the hierarchy top. However, the respective value of this threshold may vary a lot from one party to the other. It is this difference which creates the dictatorian nature of the democratic voting system. While climbing up the hierarchy, the initial majority-minority ratio can be reversed at the profit of actual running party. Such a reversal is shown to be driven by the natural inertia of being in power. The model could shed light on last century Eastern European Communist collapse.

Generalized fractional relaxation equations based on generalized Riemann-Liouville derivatives are combined with a simple short time regularization and solved exactly. The solution involves generalized Mittag-Leffler functions. The associated frequency dependent susceptibilities are related to symmetrically broadened Cole-Cole susceptibilities occurring as Johari Goldstein β-relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.

We report pattern formation of complex stripes with binary and ternary granular mixtures. Ternary mixtures lead to a particular ordering of the strates which was not accounted for in former explanations. Bouncing grains are found to have an important effect on strate formation. A complementary mechanism for self-stratification of binary and ternary granular mixtures is proposed. This mechanism leads to a simple model. Eventually we report the observation of self-stratification for a binary mixture of grains of the same size but with different shapes.

The study of the properties of cosmic structures in the universe is one of the most fascinating subject of the modern cosmology research. Far from being predicted, the large scale structure of the matter distribution is a very recent discovery, which continuosly exhibits new features and issues. We have faced such topic along two directions; from one side we have studied the correlation properties of the cosmic structures, that we have found substantially different from the commonly accepted ones. From the other side, we have studied the statistical properties of the very simplified system, in the attempt to capture the essential ingredients of the formation of the observed strucures.

The fractional diffusion equation is derived from the master equation of continuous time random walks (CTRWs) via a straightforward application of the Gnedenko-Kolmogorov limit theorem. The Cauchy problem for the fractional diffusion equation is solved in various important and general cases. The meaning of the proper diffusion limit for CTRWs is discussed.

We study a model of generalized-Hebbian learning in asymmetric oscillatory neural networks modeling cortical areas such as hippocampus and olfactory cortex. The learning rule is based on the synaptic plasticity observed experimentally, in particular long-term potentiation and long-term depression of the synaptic efficacies depending on the relative timing of the pre- and postsynaptic activities during learning. The learned memory or representational states can be encoded by both the amplitude and the phase patterns of the oscillating neural populations, enabling more efficient and robust information coding than in conventional models of associative memory or input representation. Depending on the class of nonlinearity of the activation function, the model can function as an associative memory for oscillatory patterns (nonlinearity of class II) or can generalize from or interpolate between the learned states, appropriate for the function of input representation (nonlinearity of class I). In the former case, simulations of the model exhibits a first order transition between the "disordered state" and the "ordered" memory state.

This talk briefly reviews the subject of fluid flow through disordered media. First, we use two-dimensional percolation networks as a simple model for porous media to investigate the dynamics of viscous penetration when the ratio between the viscosities of displaced and injected fluids is very large. The results indicate the possibility that viscous displacement through critical percolation networks constitutes a single universality class, independent of the viscosity ratio. We also focus on the sorts of considerations that may be necessary to move statistical physics from the description of idealized flows in the limit of zero Reynolds number to more realistic flows of real fluids moving at a nonzero velocity, when inertia effects may become relevant. We discuss several intriguing features, such as the surprisingly change in behavior from a "localized" to a "delocalized" flow structure (distribution of flow velocities) that seems to occur at a critical value of Re which is significantly smaller than the critical value of Re where turbulence sets in.

Computational models for social phenomena are reviewed: Bonabeau et al. for the formation of social hierarchies, Donangelo and Sneppen for the replacement of barter by money, Solomon and Weisbuch for marketing percolation, and Sznajd for political persuasion. Finally we review how to destroy the internet.

How general are Boltzmann-Gibbs statistical mechanics and standard Thermodynamics? What classes of systems have thermostatistical properties, in particular equilibrium properties, that are correctly described by these formalisms? The answer is far from trivial. It is, however, clear today that these relevant and popular formalisms are not universal, this is to say that they have a domain of validity that it would be important to define precisely. A few remarks on these questions are herein presented, in particular in relation to nonextensive statistical mechanics, introduced a decade ago with the aim to cover at least some (but most probably not all) of the natural systems that are out of this domain of validity.

We investigate the effect on the roughness of microcrack nucleation ahead of a propagating planar crack and study the structure of the damage zone. To this end we consider a quasi-two dimensional random fuse model, confining the crack between two horizontal plates. The two-dimensional geometry introduces a characteristic length in the problem, limiting the crack roughness. The damage ahead of the crack does not appear to change the scaling properties of the model, which are well described by gradient percolation.

Author Index.