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Based on various results present in the literature, we elaborate a unifying cluster percolation approach, to interpret the dynamical arrest occurring in amorphous materials such as gels, glasses and granular materials. In the case of the sol-gel transition, this cluster approach predicts scaling laws relating the dynamical exponents to random percolation critical exponents. Interestingly, in mean-field such relations coincide with those predicted by the schematic continuous Mode Coupling Theory, known as model A. More appropriate to describe molecular glass transition is the schematic discontinuous Mode Coupling Theory known as model B. In this case a similar cluster approach and a diffusing defect mechanism predicts scaling laws, relating the dynamical exponents to static critical exponents of the bootstrap percolation. In finite dimensions a glass theory based on the Random First Order Transition suggests that the Mode Coupling Theory transition is only a crossover towards an ideal glass transition characterized by the divergence of cooperative rearranging regions. Interestingly, also this scenario can be mapped onto a mixed order percolation transition, where the order parameter jumps discontinuously at the transition, while the mean cluster size and the linear cluster dimension diverge. A similar mixed order percolation transition seems to apply also to the jamming transition.a#13

We study the segregation process of a granular mixture of discs on an horizontally oscillating tray by realistic molecular dynamics simulations. As found in the experiments, we observe the initially disordered mixture to segregate via the formation of stripes perpendicular to the driving direction. The segregation process turns out to be the result of a dynamical instability very similar to the Kelvin–Helmholtz instability observed in fluid mechanics, and it is not due to the depletion potential, i.e., to the effective potential existing between two big particles in a bath of smaller ones, but rather to a dynamical process whose nature is discussed.

We develop a novel approach that combines the ideas of inherent structures, free-volume theory and geometrical packing properties to derive a general theory to understanding the complex dynamics of glass-forming liquids, granular packings and amorphous solids. We show that the thermodynamical properties of these systems can be retrieved from the study of geometrical and topological properties of local configurations only. When applied to hard-sphere systems, the present theory describes the critical approach toward the random close-packing density with the configurational entropy that approaches zero, and the α-relaxation time that diverges according to the Vogel-Fulcher behavior following also the Adam-Gibbs relation. The equation of state derived for the stable fluid, the metastable fluid, and for the crystalline and amorphous solids, results in good quantitative agreement with the data available in the literature.

We present a novel theoretical approach to understanding the complex dynamics of glass-forming liquids, granular packings and amorphous solids. This theory, which is an elaboration of the free volume and inherent structure approaches, allows one to retrieve the thermodynamical properties of these systems from studies of geometrical and topological properties of local, static configurations alone. When applied to hard-sphere systems, the present theory reproduces with a good quantitative agreement the equation of state for the crystalline and the disordered glassy phases. Moreover, we find that, as the density approaches a critical value close to the random close-packing density, the configurational entropy approaches zero and the large relaxation time diverges according to the Vogel–Fulcher behaviour, following also the Adam–Gibbs relation.

The interplay between critical percolating clusters and critical points in physical systems and models is reviewed. The main emphasis is given to systems such as colloids and gels. Much information on the main mechanism leading to percolation in correlated systems can be obtained from the study of the Ising model. It is emphasized that different cluster definitions are necessary for studying different phenomena and that the possibility of experimentally detecting the percolation line is related to the cluster lifetime. Some results on the interplay between critical points and percolating clusters in fluids are also discussed.

We briefly review the basics ideas and results of a recently proposed statistical mechanical approach to granular materials. Using lattice models from standard Statistical Mechanics and results from a mean field replica approach and Monte Carlo simulations we find a jamming transition in granular media closely related to the glass transition in super-cooled liquids. These models reproduce the logarithmic relaxation in granular compaction and reversible-irreversible lines, in agreement with experimental data. The models also exhibit aging effects and breakdown of the usual fluctuation dissipation relation. It is shown that the glass transition may be responsible for the logarithmic relaxation and may be related to the cooperative effects underlying many phenomena of granular materials such as the Reynolds transition., 18 pages with 6 postscript figures. to appear in J.Phys: Cond. Mat

The sol-gel transition is studied introducing the bond fluctuation dynamics within the percolation model in order to investigate both static and dynamic properties. Computer simulations on a square lattice have shown that the static properties agree with the random percolation model and the self-diffusion coefficients vanish at the percolation threshold. From the self-diffusion coefficients the critical behaviour of the viscosity at the sol-gel transition is determined, giving an exponent .

We introduce a lattice model, in which frustration plays a crucial role, to describe relaxation properties of granular media. We show Monte Carlo results for compaction in the presence of vibrations and gravity, which compare well with experimental data.

A three-dimensional (3D) spin-glass model diluted with lattice gas variables is investigated and compared with glass-forming liquids. The spin variables which play the role of internal degrees of freedom exhibit strong fluctuations. While a critical length associated to these fluctuations diverges at the glass transition, where the diffusion coefficient is found to vanish, the density fluctuations do not show any critical behaviour.

The dynamical arrest of gels is the consequence of a well defined structural phase transition, leading to the formation of a spanning cluster of bonded particles. The dynamical glass transition, instead, is not accompanied by any clear structural signature. Nevertheless, both transitions are characterized by the emergence of dynamical heterogeneities. Reviewing recent results from numerical simulations, we discuss the behavior of dynamical heterogeneities in different systems and show that a clear connection with the structure exists in the case of gels. The emerging picture may be also relevant for the more elusive case of glasses. We show, as an example, that the relaxation process of a simple glass-forming model can be related to a reverse percolation transition and discuss further perspective in this direction., Published in the J. Stat. Mech. Special Issue of "The Role of Structure in Glassy and Jammed Systems "

The sluggish and heterogeneous dynamics of glass forming liquids is frequently associated to the transient coexistence of two phases of particles, respectively with an high and low mobility. In the absence of a dynamical order parameter that acquires a transient bimodal shape, these phases are commonly identified empirically, which makes difficult investigating their relation with the structural properties of the system. Here we show that the distribution of single particle diffusivities can be accessed within a Continuous Time Random Walk description of the intermittent motion, and that this distribution acquires a transient bimodal shape in the deeply supercooled regime, thus allowing for a clear identification of the two coexisting phase. In a simple two-dimensional glass forming model, the dynamic phase coexistence is accompanied by a striking structural counterpart: the distribution of the crystalline-like order parameter becomes also bimodal on cooling, with increasing overlap between ordered and immobile particles. This simple structural signature is absent in other models, such as the three-dimesional Kob-Andersen Lennard-Jones mixture, where more sophisticated order parameters might be relevant. In this perspective, the identification of the two dynamical coexisting phases opens the way to deeper investigations of structure-dynamics correlations., Comment: Published in the J. Stat. Mech. Special Issue "The Role of Structure in Glassy and Jammed Systems"

One of the challenges in soft and condensed matter over the last few years has been understanding the phenomena of glass and jamming transitions. A recent advance in the field is the idea that the dynamical heterogeneities play here the same role as the critical fluctuations in ordinary critical phenomena. This is due to the fact that the decay of density fluctuations in glasses and jammed systems takes place thanks to the dynamically correlated motions of groups of particles. In this paper, after a brief review of the properties of the dynamical heterogeneities in glasses we analyze the cases of chemical and colloidal gels, which are still intensely debated.

The question of transport in a random two-component mixture is addressed. To this end, two models are precisely formulated that effectively extend to two-component mixtures the de Gennes ant model of one-component systems. We develop and test a scaling theory, and we solve some of the problems associated with the limit where one of the components is superconducting. Our work provides the first practical realisation of the termite model of a random superconducting network, which performs a normal walk in the normal regions of the material but performs a new and unique form of random walk in the superconducting regions. We find that the divergence of the electrical conductivity at the percolation threshold can be described by this random walk, and that the critical exponent s is given by s = 1.3 f 0.1 in d = 2. If we perturb away from the pure superconduct- ing limit, we find that the electrical properties can still be described by the random walk model, with a crossover exponent I$ = 0.4 (d = 2). Moreover, we find that the diffusion constant in this region is described by a scaling form, so that data can be made to collapse upon a single curve whose form is governed by the exponents s and I$.

A theory of pair connectedness is developed for fluid as well as lattice systems when the presence of physical clusters of particles in the system is explicitly taken into account. Activity and density expansions, an Ornstein-Zernicke relation and the Percus-Yevick approximation are established in analogy with the theory of the pair-correlation function. A simple application to the percolation problem is given; for a lattice the results are compared with the known solution of the Bethe lattice. For a fluid system the theory is used to investigate the relation of percolation and condensation in a Van der Waals gas, the result shows that an infinite cluster of particles is formed in the gaseous phase, along the coexistence curve, before the critical point is reached.

An exact analytical solution of the time-dependent Ginzburg-Landau model in the large-N limit is presented for a quench to zero temperature. Standard dynamic scaling is obeyed when the order parameter is not conserved, while a novel form of scaling, due to two (marginally different) divergent lengths and characterized by infinitely many growth exponents, is found when the order parameter is conserved. The new scaling behaviour is expected to be a pattern common to other growth phenomena.

The Migdal-Kadanoff renormalisation group for two-dimensions is employed to obtain the global phase diagram for the site-bond correlated percolation problem. It is found that the Ising critical point (K=Kc,H=O) is a percolation point for a range of bond probability rho B such that 1>or= rho B>or=1-e-2Kc. In particular, as rho B approaches 1-e-2Kc, the percolation clusters become less compact and coincide with the Ising critical droplets.

Derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc. In particular, the author proves the relations p(dpij/dp)=( lambda ij) where p is the bond density, pij is the pair connectedness function and ( lambda ij) is the average number of cutting bonds between i and j. From this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi , diverges as mod p-pc mod -1. The remaining (multiply connected) bonds in the percolating backbone, which lump together in 'blobs', diverge with a dimensionality-dependent exponent. He also shows that in the cell renormalisation group of Reynolds et al. (1978, 1980) the 'thermal' eigenvalue is simply related to the average number of cutting bonds in the spanning cluster. He discusses a percolation model in which the 'blobs' can be controlled by varying a parameter, and study the influence on the critical exponents.

The free energies of directed lattice animals in good and theta -solvents, and the free energy of directed percolation are found by the use of a simple Flory-type approximation, which accounts for the inherent anisotropy of these systems. From these free energies, the authors obtain the upper critical dimension below which mean-field theory breaks down. They also calculate closed-form, dimension-dependent expressions for the parallel and perpendicular correlation length exponents, which characterise the asymptotic cluster shapes. These exponents are in excellent agreement with existing numerical data.

Rigorous inequalities are proved, which relate percolation probability, mean cluster size and pair connectedness respectively with magnetization, susceptibility and pair correlation function in ferromagnetic Ising models. In two dimensions the critical point is shown to be a percolation point, while in three dimensions this is not true.

The authors study the droplet size distribution of the correlated site-bond percolation model introduced by Coniglio and Klein (1980), and also the usual clusters of two-dimensional Ising models near the critical point. Equilibrium configurations of the Ising model with nearest-neighbour interaction and also one with nearest- and next-nearest-neighbour interactions are generated through a Monte Carlo simulation, and then a cluster analysis is performed. The exponents beta and gamma for the Coniglio and Klein droplet distribution are found to agree, for both the nearest-neighbour and the next-nearest-neighbour model, with the corresponding exponents of the Ising model. The usual Ising clusters diverge only at Tc in the Ising model with nearest-neighbour interaction but not for the model with next-nearest-neighbour interaction. The Potts model formulation is used to predict the behaviour of the droplet for general further-neighbour interactions.

A Potts model formulation of the statistics of branched polymers or lattice animals in a solvent is given. The Migdal-Kadanoff renormalisation group is employed to study the critical behaviour or fractal dimension of the branched polymer. Four different critical behaviours are found, corresponding to random animal, collapse or theta point, percolation and compact cluster. The theta point behaviour is described by a tricritical point while percolation corresponds to a higher-order critical point, where the effect of the solvent on the branched polymer is the same as the screening effect of the other clusters in percolation.

A general formalism is developed to obtain series expansions of the average number of physical clusters of particles in the framework of Mayer's theory. The special case of lattice systems is investigated in more detail and some preliminary results are given on the relation between percolation (namely the formation of an infinite cluster) and condensation in fluid systems in the lowest approximation (summation of chain diagrams).