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We determine the dimensional dependence of the percolative exponents of the jamming transition via numerical simulations in four and five spatial dimensions. These novel results complement literature ones, and establish jamming as a mixed first-order percolation transition, with critical exponents $\beta =0$, $\gamma = 2$, $\alpha = 0$ and the finite size scaling exponent $\nu^* = 2/d$ for values of the spatial dimension $d \geq 2$. We argue that the upper critical dimension is $d_u=2$ and the connectedness length exponent is $\nu =1$., Comment: 11 pages, 3 figures, VSI of Physica A in memory of Dietrich Stauffer

The standard phase-ordering process is obtained by quenching a system, like the Ising model, to below the critical point. This is usually done with periodic boundary conditions to insure ergodicity breaking in the low temperature phase. With this arrangement the infinite system is known to remain permanently out of equilibrium, i.e. there exists a well defined asymptotic state which is time-invariant but different from the ordered ferromagnetic state. In this paper we establish the critical nature of this invariant state, by demonstrating numerically that the quench dynamics with periodic and antiperiodic boundary conditions are indistinguishable one from the other. However while the asymptotic state does not coincide with the equilibrium state for the periodic case, it coincides instead with the equilibrium state of the antiperiodic case, which in fact is critical. The specific example of the Ising model is shown to be one instance of a more general phenomenon, since an analogous picture emerges in the spherical model, where boundary conditions are kept fixed to periodic, while the breaking or preserving of ergodicity is managed by imposing the spherical constraint either sharply or smoothly., Comment: 18 pages, 9 figures

By changing the control parameters, many physical systems reach a slow dynamics regime followed by an arrested or a quasi-arrested state. Examples, among others, are gels and glasses. In this paper, we discuss some experimental and theoretical results in polymer and colloidal systems, where gel and glass transitions interfere, and use models from Mode Coupling Theory (MCT) to illustrate the rich phenomenology observed. The continuous and the discontinuous transition lines, found in the MCT models, are considered suitable to describe respectively the gel and the glass transitions, so we suggest that the interplay between gel and glass may be interpreted in terms of the $F_{13}$ MCT model, clarifying also the origin of logarithmic decays often observed in such systems. In particular, the theoretical predictions of the MCT in the $F_{13}$ model are compared with Molecular Dynamics simulations in model systems for chemical gels and charged colloids.

The ferromagnetic transition in the Ising model is the paradigmatic example of ergodicity breaking accompanied by symmetry breaking. It is routinely assumed that the thermodynamic limit is taken with free or periodic boundary conditions. More exotic symmetry-preserving boundary conditions, like cylindrical antiperiodic, are less frequently used for special tasks, such as the study of phase coexistence or the roughening of an interface. Here we show, instead, that when the thermodynamic limit is taken with these boundary conditions, a novel type of transition takes place below $T_c$ (the usual Ising transition temperature) without breaking neither ergodicity nor symmetry. Then, the low temperature phase is characterized by a regime (condensation) of strong magnetization's fluctuations which replaces the usual ferromagnetic ordering. This is due to critical correlations perduring for all T below Tc. The argument is developed exactly in the $d=1$ case and numerically in the d=2 case., 13 pages, 10 figures

The dynamical facilitation scenario, by which localized relaxation events promote nearby relaxation events in an avalanching process, has been suggested as the key mechanism connecting the microscopic and the macroscopic dynamics of structural glasses. Here we investigate the statistical features of this process via the numerical simulation of a model structural glass. First we show that the relaxation dynamics of the system occurs through particle jumps that are irreversible, and that cannot be decomposed in smaller irreversible events. Then we show that each jump does actually trigger an avalanche. The characteristic of this avalanche change on cooling, suggesting that the relaxation dynamics crossovers from a noise dominated regime where jumps do not trigger other relaxation events, to a regime dominated by the facilitation process, where a jump trigger more relaxation events., Comment: 8 pages, 6 figures

We investigate the glass and the jamming transitions of hard spheres in finite dimensions $d$, through a revised cell theory, that combines the free volume and the Random First Order Theory (RFOT). Recent results show that in infinite dimension the ideal glass transition and jamming transitions are distinct, while based on our theory we argue that they indeed coincide for finite $d$. As a consequence, jamming results into a percolation transition described by RFOT, with a static length diverging with exponent $\nu=2/d$, which we verify through finite size scaling, and standard critical exponents $\alpha = 0$, $\beta = 0$ and $\gamma = 2$ independent on $d$.

The mechanical and transport properties of jammed materials originate from an underlying per- colating network of contact forces between the grains. Using extensive simulations we investigate the force-percolation transition of this network, where two particles are considered as linked if their interparticle force overcomes a threshold. We show that this transition belongs to the random percolation universality class, thus ruling out the existence of long-range correlations between the forces. Through a combined size and pressure scaling for the percolative quantities, we show that the continuous force percolation transition evolves into the discontinuous jamming transition in the zero pressure limit, as the size of the critical region scales with the pressure., Comment: 5 pages, 5 figures

A long standing problem in glassy dynamics is the geometrical interpretation of clusters and the role they play in the observed scaling laws. In this context, the mode-coupling theory (MCT) of type-A transition and the sol-gel transition are both characterized by a structural arrest to a disordered state in which the long-time limit of the correlator continuously approaches zero at the transition point. In this paper, we describe a cluster approach to the sol-gel transition and explore its predictions, including universal scaling laws and a new stretched relaxation regime close to criticality. We show that while MCT consistently describes gelation at mean-field level, the percolation approach elucidates the geometrical character underlying MCT scaling laws.

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

The evaluation of the long term stability of a material requires the estimation of its long-time dynamics. For amorphous materials such as structural glasses, it has proven difficult to predict the long-time dynamics starting from static measurements. Here we consider how long one needs to monitor the dynamics of a structural glass to predict its long--time features. We present a detailed characterization of the statistical features of the single-particle intermittent motion of structural glasses, and show that single--particle jumps are the irreversible events leading to the relaxation of the system. This allows to evaluate the diffusion constant on the time--scale of the jump duration, which is small and temperature independent, well before the system enters the diffusive regime. The prediction is obtained by analyzing the particle trajectories via a parameter-free algorithm., Comment: 6 pages, 5 figures, accepted for publication in Soft Matter (21 May 2014)

We have performed a detailed analysis of the local dynamics and local energies of the equilibrium, paramagnetic phase of the d = 2 and d = 3 ±J Ising spin glass model. Here we discuss our recently reported observations l) that while the average flip rate and average energy decrease monotonically with decreasing temperature, both the flip rate and energy of an increasing fraction of spins increase as the glass transition is approached on cooling. These findings are consistent with recent experimental results for the frequency-dependent magnetic susceptibility of an insulating spin glass, which showed that the approach to the glass transition could be detected from the high frequency behavior. In an effort to better understand the effects of frustration as the glass transition is approached, and to elucidate recent claims of the presence and ramifications of dynamic heterogeneities significantly above the glass transition temperature Tg in glass-forming materials with self-induced frustration, 2 )- 7 ) we are currently exploring the relationship between microstructure and local dynamics in a variety of systems, including supercooled liquids with self-induced disorder 8 ) and paramagnets with random, quenched disorder. 1 ) In particular, we have recently performed a detailed Monte Carlo simulation study of the paramagnetic phase of the Ising spin glass to characterize the hetero­ geneities induced in this system by the quenched disorder, and to test the model for the emergence of fast processes recently reported in an experimental study of an Ising-like spin glass. 4 ) In the Ising spin glass model, exchange interactions Jij = ±J are fixed randomly to the edges of a lattice, and Ising spins with values

We investigate the heterogeneous dynamics in a model, where chemical gelation and glass transition interplay, focusing on the dynamical susceptibility. Two independent mechanisms give raise to the correlations, which are manifested in the dynamical susceptibility: one is related to the presence of permanent clusters, while the other is due to the increase of particle crowding as the glass transition is approached. The superposition of these two mechanisms originates a variety of different behaviours. We show that these two mechanisms can be unentangled considering the wave vector dependence of the dynamical susceptibility., Comment: The final publication is available at Springer via http://dx.doi.org/10.1140/epjst/e2016-60175-x

The structural arrest of a polymeric suspension might be driven by an increase of the cross-linker concentration, which drives the gel transition, as well as by an increase of the polymer density, which induces a glass transition. These dynamical continuous (gel) and discontinuous (glass) transitions might interfere, since the glass transition might occur within the gel phase, and the gel transition might be induced in a polymer suspension with glassy features. Here we study the interplay of these transitions by investigating via event-driven molecular dynamics simulation the relaxation dynamics of a polymeric suspension as a function of the cross-linker concentration and the monomer volume fraction. We show that the slow dynamics within the gel phase is characterized by a long sub-diffusive regime, which is due both to the crowding as well as to the presence of a percolating cluster. In this regime, the transition of structural arrest is found to occur either along the gel or along the glass line, depending on the length scale at which the dynamics is probed. Where the two lines meet there is no apparent sign of higher order dynamical singularity. Logarithmic behavior typical of A3 singularity appears inside the gel phase along the glass transition line. These findings seem to be related to the results of the mode coupling theory for the F13 schematic model.

Kinetic facilitated models and the Mode Coupling Theory (MCT) model B are within those systems known to exhibit a discontinuous dynamical transition with a two step relaxation. We consider a general scaling approach, within mean field theory, for such systems by considering the behavior of the density correlator and the dynamical susceptibility -^2. Focusing on the Fredrickson and Andersen (FA) facilitated spin model on the Bethe lattice, we extend a cluster approach that was previously developed for continuous glass transitions by Arenzon et al (Phys. Rev. E 90, 020301(R) (2014)) to describe the decay to the plateau, and consider a damage spreading mechanism to describe the departure from the plateau. We predict scaling laws, which relate dynamical exponents to the static exponents of mean field bootstrap percolation. The dynamical behavior and the scaling laws for both density correlator and dynamical susceptibility coincide with those predicted by MCT. These results explain the origin of scaling laws and the universal behavior associated with the glass transition in mean field, which is characterized by the divergence of the static length of the bootstrap percolation model with an upper critical dimension d_c=8., Comment: 16 pages, 9 figures

We perform molecular dynamic (MD) simulations of frictional non-thermal particles driven by an externally applied shear stress. After the system jams following a transient flow, we probe its mechanical response in order to clarify whether the resulting solid is 'fragile'. We find the system to respond elastically and isotropically to small perturbations of the shear stress, suggesting absence of fragility. These results are interpreted in terms of the energy landscape of dissipative systems. For the same values of the control parameters, we check the behaviour of the system during a stress cycle. Increasing the maximum stress value, a crossover from a visco-elastic to a plastic regime is observed., 6 pages, 9 figures, accepted in Granular Matter on 01-02-2012

The dynamical susceptibility has been introduced to characterize the dynamical heterogeneities in glass forming liquids. We have used it as a tool to investigate the slow dynamics of other disordered systems such as gels, granular media and spin glasses. We review here the results obtained via numerical simulations of different model systems. The comparative study of the behaviour of the dynamical susceptibility sheds some light on the significant differences in the complex slow dynamics of glasses, spin glasses, granular media, irreversible gels, and colloidal gels.

Particles in structural glasses rattle around temporary equilibriumpositions, that seldom change through a process which is much faster than the relaxation time, known as particle jump. Since the relaxation of the system is due to the accumulation of many such jumps, it could be possible to connect the single particle short time motion to the macroscopic relaxation by understanding the features of the jump dynamics. Here we review recent results in this research direction, clarifying the features of particles jumps that have been understood and those that are still under investigation, and examining the role of particle jumps in different theories of the glass transition., Comment: 10 pages, 4 figures, Review article

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.

A system under gravity, made up of hard spheres with an internal degree of freedom, is studied on a 3d lattice under a Monte Carlo “tap” dynamics. A “jamming” transition similar to that found in real experiment is found. To understand the nature of such transition a simplified version of the model is also studied in a mean field approximation, using the statistical mechanics approach a la Edwards. A phase transition from a “fluid” to a “glassy” phase is found when crystallization is avoided. Interestingly, the nature of such a “glassy” phase turns out to be the same found in mean field models for glass formers.

We briefly review a Statistical Mechanics approach for the description of granular materials following the original ideas developed by Edwards. We show that Edwards’ assumptions works very well for a class of lattice models and we discuss some recent analytic results obtained in such a framework on their phase diagram, jamming transition and mixing/segregation properties.

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.

A cell theory for glassforming liquids, amorphous solids and granular packings, is developed by combining geometrical properties of local packings with the ideas of inherent structures and free volume theory. We show that for hard-spheres the present theory reproduces well the equation of state, the approach to zero of the configurational entropy and the divergence of large relaxation times following both the Vogel–Tammann–Fulcher behavior and the Adam–Gibbs relation.

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.

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 Tg(r) below which the system falls out of equilibrium. Tg(r) approaches logarithmically in r the ideal glass temperature T0 = lim r→0Tg(r), where the relaxation time diverges á la Vogel-Fulcher, similarly to some experimental observations. Well above Tg(r) a simple fluid behavior is observed. As Tg is approached from above a characteristic wave vector kd divides an high momenta equilibrated region, where a fluid-like behavior is obeyed, from the non thermalized modes with k < kd, for which time translational invariance lacks. Below Tg 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.

Clusters in frustrated systems are studied in the context of the frustrated percolation model. This model, which contains frustration and connectivity as an essential ingredient, exhibits a large degree of complexity in both static and dynamics. The bond version of the model maps on the spin glass model, while the site version can be applied to a large variety of frustrated systems such as glasses and granular materials.

We show that the relaxation dynamics near a glass transition with continuous ergodicity breaking can be endowed with a geometric interpretation based on percolation theory. At the mean-field level this approach is consistent with the mode-coupling theory (MCT) of type-A liquid-glass transitions and allows one to disentangle the universal and nonuniversal contributions to MCT relaxation exponents. Scaling predictions for the time correlation function are successfully tested in the ${\mathsf{F}}_{12}$ schematic model and facilitated spin systems on a Bethe lattice. Our approach immediately suggests the extension of MCT scaling laws to finite spatial dimensions and yields predictions for dynamic relaxation exponents below an upper critical dimension of 6.

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 present a study of molecular dynamics at the sol-gel transition. Application of percolation and bond-fluctuation dynamics provides critical exponents for viscoelastic properties in a strong covalent gel and shows a complex non-exponential relaxation behaviour. The results are in good agreement with experimental findings and offer interesting pointers for discussions.

In this paper we study the 3d frustrated lattice gas model in the annealed version, where the disorder is allowed to evolve in time with a suitable kinetic constraint. Although the model does not exhibit any thermodynamic transition it shows a diverging peak at some characteristic time in the dynamical non-linear susceptibility, similar to the results on the p-spin model in mean field and Lennard-Jones mixture recently found by Donati et al. [cond-mat/9905433]. Comparing these results to those obtained in the model with quenched interactions, we conclude that the critical behavior of the dynamical susceptibility is reminiscent of the thermodynamic transition present in the quenched model, and signaled by the divergence of the static non-linear susceptibility, suggesting therefore a similar mechanism also in supercooled glass-forming liquids., 8 pages, 14 figures

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

We review the basic ideas which have been developed to characterize geometrically the equilibrium phase transitions. This approach elucidates many concepts like propagation of correlations, upper critical dimensionality, breakdown of hyperscaling, strong universality, and other basic ideas in phase transitions. More recently the extension of the geometrical approach to frustrated systems like spin glasses has lead to the definition of a new geometrical model called frustrated percolation. Application of this model to complex systems such as glasses and granular materials is also briefly reviewed.

In the four-dimensional Edwards–Anderson ±J spin glasses, we observe that the quenched energies distribution from T=2.5 are similar to those observed for the equilibrium states at T=1.0. We show that although these states appear close there are subtle differences which are only seen when we try to quench the system from T=2.5 for a longer number of Monte Carlo steps. The T=2.5 states do not evolve to those seen for the states quenched from T=1.0. We try to identify regions responsible for the subtle differences.

To understand the origin of the dynamical transition, between high temperature exponential relaxation and low temperature nonexponential relaxation, that occurs well above the static transition in glassy systems, a frustrated spin model, with and without disorder, is considered. The model has two phase transitions, the lower being a standard spin glass transition (in presence of disorder) or fully frustrated Ising (in absence of disorder), and the higher being a Potts transition. Monte Carlo results clarify that in the model with (or without) disorder the precursor phenomena are related to the Griffiths (or Potts) transition. The Griffiths transition is a vanishing transition which occurs above the Potts transition and is present only when disorder is present, while the Potts transition which signals the effect due to frustration is always present. These results suggest that precursor phenomena in frustrated systems are due either to disorder and/or to frustration, giving a consistent interpretation also for the limiting cases of Ising spin glass and of Ising fully frustrated model, where also the Potts transition is vanishing. This interpretation could play a relevant role in glassy systems beyond the spin systems case., Comment: Completely rewritten. New data. New results

In the framework of recently introduced frustrated lattice gas models, we study the out-of-equilibrium dynamical processes during the compaction in granular media. We find irreversible-reversible cycles in agreement with recent experimental observations, along which we can individuate an equivalent “glass transition,” Gg. In analogy with the phenomenology of glassy systems we find aging effects during the compaction process. In particular, we find that the two-time density correlation function Cst, t 0 d asymptotically scales as a function of the single variable ln st 0 dy lnstd. [S0031-9007(98)08224-6]

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.

Density relaxation in a frustrated lattice gas subject to gravity and vibrations is explored via Monte Carlo simulations. A comparison to some recent experimental results on compaction of granular media gives good agreement, and interesting predictions are possible.

We develop a Hamiltonian formalism for the frustrated percolation model. This is a cluster model which has recently received attention for its relevance to glassy systems and granular materials. The exact solution of the model on the frustrated hierarchical lattice of Mc Kay et al. exhibits a percolation transition with critical exponents of the ferromagnetic 1 2 - state Potts model and a spin glass transition characterized by chaotic renormalization group trajectories. Finally, we discuss the model in light of recent experiments.

We describe a Monte Carlo renormalization group approach to the calculation of critical behavior for percolation models. This approach can be utilized to determine the renormalized bond probabilities and the values of the critical exponents. We illustrate the method for two-dimensional bond percolation, but the method is also applicable to other percolation models and other dimensions.

The concepts of connectivity, localization, and frustration are explored in relation to glass formation in amorphous materials. First, the concept of eigenclusters to geometrically characterize correlations in amorphous materials is introduced, and discussed in detail for both the Ising ferromagnet and Ising spin glass models. Second, a new, glass-forming percolation model that contains frustration as the essential ingredient, and exhibits two percolation transitions, is discussed. This new model gives new insights into frustrated systems, and applications to the Ising spin glass model and other glass-forming systems are discussed. In particular, we propose the possibility that the occurrence of a percolation-type transition at temperatures above the glass transition temperature may be a general feature of glass-forming systems. The important role of computer simulations in probing the mechanism of glass formation is emphasized.