Facilitated Spin Models: Recent and New Results

Cancrini, Nicoletta; Martinelli, Fabio; Roberto, Cyril; Toninelli, Cristina

doi:10.1007/978-3-540-92796-9_7

Cited by 21 publications

(23 citation statements)

References 30 publications

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“…Our result differs from the c = 1/ log 2 value incorrectly derived in [11]. As we clarify in [6], this is due to the fact that the relation between length and time scales extrapolated in [11] from coarsening dynamics does not lead to the correct equilibrium result unless relaxation on scales smaller than the typical distance of two vacancies is also taken into account.…”

contrasting

confidence: 90%

“…For all the models introduced in physics literature dynamics satisfies detailed balance w.r.t. to Bernoulli product measure (see instead [6] for the extension to models which are reversible w.r.t. high temperature Gibbs measures).…”

mentioning

confidence: 99%

“…Here we review our recent mathematical results [6][7][8] on KCM which have contributed to settle some debated questions arising in numerical simulations. In particular in [8] we have introduced a new technique trough which we obtain upper and lower bounds on the spectral gap of the Markov process and therefore on the relaxation time τ which, as we shall see, is directly related to the inverse of the spectral gap.…”

mentioning

confidence: 99%

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Kinetically constrained models

Garrahan¹,

Sollich²,

Toninelli³

2011

Dynamical Heterogeneities in Glasses, Colloids, and Granular Media

108

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Many fascinating questions still remain unsettled for condensed matter physicists who study the glass and jamming transitions. These phenomena, which lead to the formation of amorphous solids, occur in several microscopically different systems as supercooled liquids, colloidal suspensions and vibrated granular materials [9,27]. Basic glassy properties include a dramatic slowing down of dynamics when a proper external parameter is tuned (e.g. temperature is lowered for liquids) and the occurrence of a complicated relaxation: non exponential and spatially heterogeneous. When relaxation times become longer than experimental scales, equilibrium can no more be achieved: the system undergoes a dynamical arrest and freezes into an amorphous phase (the glass). The main issues in understanding these phenomena remain unsolved. In particular, it is not clear whether the dynamical arrest is due to the proximity of a phase transition and whether this is a static or purely dynamical one. The experiments on molecular liquids show that, if such an ideal glass transition occurs, it should have an unconventional behavior with mixed first and second order features. On the one hand, the divergence of relaxation times and the fact that both entropy and internal energy seem continuous is indicative of a second order transition. On the other hand a discontinuous order parameter is detected: the height of the plateau of the Fourier transform of the density-density correlation has a finite jump. This corresponds to the fact that the modulation of the microscopic density profile of the glass does not appear continuously from the flat liquid

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confidence: 90%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Kinetically constrained models

Garrahan¹,

Sollich²,

Toninelli³

2011

Dynamical Heterogeneities in Glasses, Colloids, and Granular Media

108

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“…There exists m > 0 and for any local function f there exists C f < ∞ such that: a) for any t > 0 [2].…”

Section: Non Equilibrium Results For the East Modelmentioning

confidence: 99%

Facilitated Oriented Spin Models: Some Non Equilibrium Results

Cancrini

Martinelli

Schonmann³

et al. 2010

J Stat Phys

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ABSTRACT. We begin the rigorous analysis of the relaxation to equilibrium for kinetically constrained spin models (KCSM) when the initial distribution ν is different from the reversible one, µ. This setting has been intensively studied in the physics literature to analyze the slow dynamics which follows a sudden quench from the liquid to the glass phase. We concentrate on two basic oriented KCSM: the East model on Z, for which the constraint requires that the East neighbor of the to-be-update vertex is vacant and the model on the binary tree introduced in [1], for which the constraint requires the two children to be vacant. It is important to observe that, while the former model is ergodic at any p = 1, the latter displays an ergodicity breaking transition at pc = 1/2. For the East we prove exponential convergence to equilibrium with rate depending on the spectral gap if ν is concentrated on any configuration which does not contain a forever blocked site or if ν is a Bernoulli(p ′ ) product measure for any p ′ = 1. For the model on the binary tree we prove similar results in the regime p, p ′ < pc and under the (plausible) assumption that the spectral gap is positive for p < pc. By constructing a proper test function, we also prove that if p ′ > pc and p pc convergence to equilibrium cannot occur for all local functions. Finally, in a short appendix, we present a very simple argument, different from the one given in [1], based on an elegant combination of some combinatorial results together with "energy barrier" considerations, which yields the sharp upper bound for the spectral gap of East when p ↑ 1.

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“…Finally, we recall that the Glauber version of KCLG, the so called Kinetically Constrained Spin Models (KCSM), have also been very much studied in physics literature [12,10,24,4,11] and that some features of their long time behaviors have been rigorously analyzed in [1,16,9,8]. In particular in [9] an important correction to the "exact solution" of the East model based on non rigorous methods was obtained.…”

Section: Definition 11 a Model Is Said To Be Non-cooperative If Itsmentioning

confidence: 99%

Kinetically Constrained Lattice Gases

Cancrini

Martinelli

Roberto³

et al. 2010

Commun. Math. Phys.

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ABSTRACT. Kinetically constrained lattice gases (KCLG) are interacting particle systems which show some of the key features of the liquid/glass transition and, more generally, of glassy dynamics. Their distintictive signature is the following: i) reversibility w.r.t. product i.i.d. Bernoulli measure at any particle density and ii) vanishing of the exchange rate across any edge unless the particle configuration around the edge satisfies a proper constraint besides hard core. Because of degeneracy of the exchange rates the models can show anomalous time decay in the relaxation process w.r.t. the usual high temperature lattice gas models particularily in the so called cooperative case, when the vacancies have to collectively cooperate in order for the particles to move through the systems. Here we focus on the Kob-Andersen (KA) model, a cooperative example widely analyzed in the physics literature, both in a finite box with particle reservoirs at the boundary and on the infinite lattice. In two dimensions (but our techniques extend to any dimension) we prove a diffusive scaling O(L 2 ) (apart from logarithmic corrections) of the relaxation time in a finite box of linear size L. We then use the above result to prove a diffusive decay 1/t (again apart from logarithmic corrections) of the spin-spin time autocorrelation function at any particle density, a result that has been sometimes questioned on the basis of numerical simulations. The techniques that we devise, based on a novel combination of renormalization and comparison with a long-range Glauber type constrained model, are robust enough to easily cover other choices of the kinetic constraints.

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Facilitated Spin Models: Recent and New Results

Cited by 21 publications

References 30 publications

Kinetically constrained models

Kinetically constrained models

Facilitated Oriented Spin Models: Some Non Equilibrium Results

Kinetically Constrained Lattice Gases

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