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We study the elasticity behaviour at the sol-gel transition with a percolation dynamic model via computer simulations on a square lattice. From the fluctuations in the center-of-mass motion of the percolating cluster we determine the critical behaviour of the macroscopic elastic constant of the gel phase, growing above the gel point with a critical exponent mu similar or equal to 2.7 +/- 0.1. We also study time autocorrelation functions and find that the relaxation times diverge, as the percolation threshold is approached from below, with a critical exponent s similar or equal to 1.9 +/- 0.1.

A q-state frustrated Potts model is introduced which generalizes the Kasteleyn-Fortuin formalism to frustrated systems. For q = 2 the Ising spin is recovered. For q = 1 it gives the frustrated percolation model, which combines frustration and connectivity features and might be relevant to systems like gels of glasses. The solution on a decorated lattice shows that a line of critical temperatures T(p)(q) appears when frustration is introduced. T(p)(q) is the percolation temperature where the clusters used in the Swendsen and Wang dynamics diverge. The critical behaviour at T(p)(q) is found to be the same as the ferromagnetic q/2 state Potts model, implying the universality class of the ferromagnetic 1/2 state Potts model for frustrated percolation.

We study the BTW-height model of self-organized criticality on a square lattice with some long range connections giving to the lattice the character of small world network. We find that as function of the fraction $p$ of long ranged bonds the power law of the avalanche size and lifetime distribution changes following a crossover scaling law with crossover exponents 2/3 and 1 for size and lifetime respectively., 7 figures

We use the method of damage spreading to study the time, t , taken for a spin at a distance r to be damaged for the first time in Ising models with heat-bath dynamics. We report time-dependent scaling exponents, d t ≃ 2.0 (1D), 2.52 (2D), 2.26 (3D) and (2.0) (4D), if we assume that t ∾ r dt . We also analyze our data assuming various forms of logarithmic corrections and we find that d t ∾ 2.25 or less (2D) and ∾2.0 (3D). We argue that it is extremely plausible that d t is equal to z , the dynamic critical exponent, and thus our estimate of z , in two dimensions is higher than the “consensus” value of 2.14, but we find good agreement in one, three and four dimensions with the exact and expected values.

Clustering in time and space is a widely accepted feature of seismicity. Much more questionable is the existence of magnitude correlations. The standard approach to time-dependent seismic hazard generally assumes that magnitudes are independent and therefore, in principle, the next earthquake magnitude is unpredictable. We will show that an earthquake magnitude depends on previous ones: earthquakes occur with higher probability close, not only in time and space, but also in magnitude to previous events. Moreover, the magnitude difference fixes the characteristic temporal and spatial scale controlling correlations between events. As a consequence, the next earthquake tends to have a magnitude similar but smaller than the previous one. We will discuss a dynamical scaling relation between energy, time and space distance that reproduces the main statistical properties of experimental catalogs.

Earthquakes are phenomena of great complexity, however some simple general laws govern the statistics of their occurrence. Some of these most important laws exhibit scale invariance, as the Gutenberg- Richter law and the Omori law. The origin of these scaling behaviours is not yet fully understood and a natural fondamental question concerns the existence of these features also in other complex phenomena. A direct inspection of experimental catalogues has shown that the stochastic processes underlying solar flare and earthquake occurrence have universal properties. Another intensively debated question is the ex- istence of correlations between magnitudes of subsequent earthquakes. Our recent analysis of the Southern California Catalogue has shown that non-zero magnitude correlations exist. A branching model based on a dynamical scaling hypothesis, relating magnitude to time, reproduces the hierarchical organization in time and magnitude of events and the observed magnitude correlations.

Through "damage spreading" numerical simulations, we study the Ising spin glass in three dimensions with ±J near-neighbour interactions. The results are compared to other simulation data; we conclude that the damage spreading method can provide good estimates of the transition temperatures in this type of system. A discussion is given of phase transitions in spin glasses in terms of the geometry of phase space.

The 3D ± J Ising spin glass has been shown in massive simulations by Ogielski to have strongly non-exponential relaxation between T g and the ferromagnetic ordering temperature T G . “Damage spreading” numerical data by Derrida and collaborators have indicated anomalous behaviour in a similar temperature range. The results from the two different types of simulation are compared and discussed in terms of the geometry of the phase space of the system; conclusions are drawn on the phase transitions in spin glasses.

We discuss how the anomalous increase of the viscosity in colloidal systems with short-range attraction can be related to the formation of long-living clusters. Based on molecular dynamics and Monte Carlo numerical simulations of different models, we propose a similar picture for colloidal gelation at low and intermediate volume fractions. On this basis, we analyze the distinct role played by the formation of long-living bonds and the crowding of the particles in the slow dynamics of attractive colloidal systems.

We review some results on the dynamics of gelation phenomena, obtained via a lattice model and via molecular dynamics using a DLVO potential. This study allowed us to make a connection between classical gelation and the phenomenology of colloidal systems, suggesting that gelation phenomena in attractive colloids at low temperature and low volume fraction can be described in terms of a two-line scenario.

We have studied the time and temperature dependence of the autocorrelation function and the ‘damage’ distance for ± J Ising spin glasses in dimensions 2, 3, 4 and 6 and mean field using simulations with heat bath dynamics. From the damage data we observe an upper critical temperature T D , which we suggest can be identified with a change from a simple to a ‘labyrinthine’ phase space structure, and we also estimate the glass temperature T g . In higher dimensions our T g values are below series expansion values.

The spreading of the Hamming distance or damage has been investigated for ± J Ising spin glasses under heat bath dynamics. Dimensions d = 2, 3, 4, 6 and mean field were studied. For finite dimensions, the damage goes to zero at long times above a temperature T D ( d ). Accurate values of this critical temperature were obtained, together with certain critical exponents. The spin glass ordering temperatures T g were also estimated from the damage spreading data. The results are compared with other work and discussed from a phase space approach.

The percolation properties of geometrical clusters related to spin fluctuations have been investigated for the 3d ± J Ising spin glass. The percolation transition is found at a temperature Tp ~= 3.92, far from the spin glass critical temperature. The critical exponents are consistent with the random percolation exponents.

We investigate the breakdown properties of a random resistor-fuse network in which each network element behaves as a linear resistor if the voltage drop is less than a threshold value, but then ``burns out'' and changes irreversibly to an insulator for larger voltages. We consider a fully occupied network in which each resistor has the same resistance (in the linear regime), and with the threshold voltage drop uniformly distributed over the range ${v}_{\mathrm{\ensuremath{-}}}$=1-w/2 to ${v}_{+}$=1+w/2 (0\ensuremath{\le}2). The breakdown properties of this model depend crucially on w, and also on L, the linear dimension of the network. For sufficiently small w, ``brittle'' fracture occurs, in which catastrophic breaking is triggered by the failure of a vanishingly small fraction of bonds in the network. In this regime, the average voltage drop per unit length required to break the network, 〈${v}_{b}$〉, varies as ${v}_{\mathrm{\ensuremath{-}}}$+O(1/${L}^{2}$), and L\ensuremath{\rightarrow}\ensuremath{\infty}, and the distribution of breakdown voltages decays exponentially in ${v}_{b}$. By probabilistic arguments, we also establish the existence of a transition between this brittle regime and a ``ductile'' regime at a critical value of w=${w}_{c}$(L), which approaches 2, as L\ensuremath{\rightarrow}\ensuremath{\infty}. This suggests that the fuse network fails by brittle fracture in the thermodynamic limit, except in the extreme case where the distribution of bond strengths includes the value zero. The ductile regime, wg${w}_{c}$(L), is characterized by crack growth which is driven by increases in the external potential, before the network reaches the breaking point. For this case, numerical simulations indicate that the average breaking potential decreases as 1/(lnL${)}^{y}$, with y\ensuremath{\le}0.8, and that the distribution of breakdown voltages has a double experimental form. Numerical simulations are also performed to provide a geometrical description of the details of the breaking process as a function of w.

We investigate the damage or Hamming distance between two configurations of Ising spins. We find an exact relation between the difference of the two possible types of damage and the spin-spin correlation function, which is generally valid. For the specific case of ferromagnetic interactions, heat bath dynamics and same sequence of random numbers, this relation involves only one type of damage. The susceptibility and the magnetization can also be expressed in terms of the damage. Numerical determination of the damage for the 2d Ising model is not only an efficient way to calculate correlation functions but also gives access to spin fluctuations visualized as clusters of damaged sites which have a fractal dimension d−β/ν at Tc and whose size distribution is also related to static exponents.

Critical fluctuations, growth or disorder can produce fractal structures which have unusual physical properties due to their scale invariance. Particularly interesting are the so called multifractals measures that can exist on these fractals. We will introduce the concept of multifractals and the most common way to formalize them via the function f (α). We will present three situations in which multifractality appears: percolation, growth phenomena and fracture.

By studying the time evolution of two configurations submitted to the same thermal noise, we investigate phase space properties of the three-dimensional ±J Ising spin glass. Our results for the distance between states are consistent with the picture of phase space having a multivalley structure below Tc. By fixing the damage at one site, we also study the Hamming distance between the two configurations at its critical point, where the distribution of probability of getting damaged n times is found to be multifractal.

We discuss the scaling properties of a scalar model for fracture: The random fuse network with randomly distributed breaking thresholds. We present the current-voltage breaking characteristics for the system and we discuss its universal scaling behaviour. We consider then the distribution of local currents in the network and its moments: This distribution has multifractal properties just before the system breaks fully apart, whereas it exhibits constant gap scaling at the maximum of the breaking characteristics, that is at the end of the disorder controlled regime when the catastrophic regime sets in. All the presented results appear to be universal with respect to different distributions of quenched disorder in the thresholds and other models for fracture.

The authors study the time evolution of the damage and the macroscopic distance between two configurations of three-dimensional +or-J Ising spin glasses in an external magnetic field. Similarly to the zero-field case, a phase transition and three temperature regimes are found for the damage. The multifractal properties of the distribution of probability for a site to be damaged n times are also investigated at the onset of the chaotic phase.

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 order to study the viscoelastic properties at the gelation transition we have introduced a percolation dynamic model. Using percolation and bond-fluctuation dynamics it is suited to study the critical behaviour of the viscoelastic properties and shows complex relaxation functions. The results obtained via numerical simulation are in good agreement with some theoretical predictions and experimental results. (C) 2002 Elsevier Science B.V. All rights reserved.

We propose a simple statistical mechanics model for the study of the dynamics of gelling systems. It is based on percolation and bond-fluctuation dynamics for the bond vectors: we study the critical viscoelastic properties and the relaxation patterns in the case of irreversible gelation and the results obtained are discussed by means of scaling arguments. By introducing the idea of a finite lifetime for the bonds and by simply tuning this timescale, the model can be made to present very different dynamics and relaxation patterns corresponding to different gelling phenomenology.

The elastic response in polymeric gels is studied by means of a percolation dynamic model. By numerical simulations the fluctuations in the gyration radius and in the center-of-mass motion of the percolating cluster are determined. Their scaling behavior at the gelation threshold gives a critical exponent for the elastic modulus f approximately 2.5+/-0.1 in agreement with the prediction f=dnu.