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ABSTRACT: On the rooted $k$-ary tree we consider a 0-1 kinetically constrained spin
model in which the occupancy variable at each node is re-sampled with rate one
from the Bernoulli(p) measure iff all its children are empty. For this process
the following picture was conjectured to hold. As long as $p$ is below the
percolation threshold $p_c=1/k$ the process is ergodic with a finite relaxation
time while, for $p>p_c$, the process on the infinite tree is no longer ergodic
and the relaxation time on a finite regular sub-tree becomes exponentially
large in the depth of the tree. At the critical point $p=p_c$ the process on
the infinite tree is still ergodic but with an infinite relaxation time.
Moreover, on finite sub-trees, the relaxation time grows polynomially in the
depth of the tree.
The conjecture was recently proved by the second and forth author except at
criticality. Here we analyse the critical and quasi-critical case and prove for
the relevant time scales: (i) power law behaviour in the depth of the tree at
$p=p_c$ and (ii) power law scaling in $(p_c-p)^{-1}$ when $p$ approaches $p_c$
from below. Our results, which are very close to those obtained recently for
the Ising model at the spin glass critical point, represent the first rigorous
analysis of a kinetically constrained model at criticality.
11/2012;
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ABSTRACT: We present a comprehensive theoretical study of finite-size effects in the relaxation dynamics of glass-forming liquids. Our analysis is motivated by recent theoretical progress regarding the understanding of relevant correlation length scales in liquids approaching the glass transition. We obtain predictions both from general theoretical arguments and from a variety of specific perspectives: mode-coupling theory, kinetically constrained and defect models, and random first-order transition theory. In the last approach, we predict in particular a nonmonotonic evolution of finite-size effects across the mode-coupling crossover due to the competition between mode-coupling and activated relaxation. We study the role of competing relaxation mechanisms in giving rise to nonmonotonic finite-size effects by devising a kinetically constrained model where the proximity to the mode-coupling singularity can be continuously tuned by changing the lattice topology. We use our theoretical findings to interpret the results of extensive molecular dynamics studies of four model liquids with distinct structures and kinetic fragilities. While the less fragile model only displays modest finite-size effects, we find a more significant size dependence evolving with temperature for more fragile models, such as Lennard-Jones particles and soft spheres. Finally, for a binary mixture of harmonic spheres we observe the predicted nonmonotonic temperature evolution of finite-size effects near the fitted mode-coupling singularity, suggesting that the crossover from mode-coupling to activated dynamics is more pronounced for this model. Finally, we discuss the close connection between our results and the recent report of a nonmonotonic temperature evolution of a dynamic length scale near the mode-coupling crossover in harmonic spheres.
Physical Review E 09/2012; 86(3-1):031502. · 2.26 Impact Factor
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ABSTRACT: We consider the Fredrickson and Andersen one spin facilitated model (FA1f) on
an infinite connected graph with polynomial growth. Each site with rate one
refreshes its occupation variable to a filled or to an empty state with
probability $p\in[0,1]$ or $q=1-p$ respectively, provided that at least one of
its nearest neighbours is empty. We study the non-equilibrium dynamics started
from an initial distribution $\nu$ different from the stationary product
$p$-Bernoulli measure $\mu$. We assume that, under $\nu$, the mean distance
between two nearest empty sites is uniformly bounded. We then prove convergence
to equilibrium when the vacancy density $q$ is above a proper threshold $\bar
q<1$. The convergence is exponential or stretched exponential, depending on the
growth of the graph. In particular it is exponential on $\bbZ^d$ for $d=1$ and
stretched exponential for $d>1$. Our result can be generalized to other non
cooperative models.
05/2012;
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ABSTRACT: The East model is a particular one dimensional interacting particle system in
which certain transitions are forbidden according to some constraints depending
on the configuration of the system. As such it has received particular
attention in the physics literature as a special case of a more general class
of systems referred to as kinetically constrained models, which play a key role
in explaining some features of the dynamics of glasses. In this paper we give
an extensive overview of recent rigorous results concerning the equilibrium and
non-equilibrium dynamics of the East model together with some new improvements.
05/2012;
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ABSTRACT: We determine the finite size corrections to the large deviation function of
the activity in a kinetically constrained model (the Fredrickson-Andersen model
in one dimension), in the regime of dynamical phase coexistence. Numerical
results agree with an effective model where the boundary between active and
inactive regions is described by a Brownian interface.
11/2011;
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ABSTRACT: We consider two cases of kinetically constrained models, namely East and
FA-1f models. The object of interest of our work is the activity A(t) defined
as the total number of configuration changes in the interval [0,t] for the
dynamics on a finite domain. It has been shown in [GJLPDW1,GJLPDW2] that the
large deviations of the activity exhibit a non-equilibirum phase transition in
the thermodynamic limit and that reducing the activity is more likely than
increasing it due to a blocking mechanism induced by the constraints. In this
paper, we study the finite size effects around this first order phase
transition and analyze the phase coexistence between the active and inactive
dynamical phases in dimension 1. In higher dimensions, we show that the finite
size effects are also determined by the dimension and the choice of boundary
conditions.
01/2011;
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ABSTRACT: In this chapter we summarize recent developments in the study of kinetically constrained models (KCMs) as models for glass formers. After recalling the definition of the KCMs which we cover we study the possible occurrence of ergodicity breaking transitions and discuss in some detail how, before any such transition occurs, relaxation timescales depend on the relevant control parameter (density or temperature). Then we turn to the main issue: the prediction of KCMs for dynamical heterogeneities. We focus in particular on multipoint correlation functions and susceptibilities, and decoupling in the transport coefficients. Finally we discuss the recent view of KCMs as being at first order coexistence between an active and an inactive space-time phase. Comment: Chapter of "Dynamical heterogeneities in glasses, colloids, and granular media", Eds.: L. Berthier, G. Biroli, J-P Bouchaud, L. Cipelletti and W. van Saarloos (Oxford University Press, to appear), more info at http://w3.lcvn.univ-montp2.fr/~lucacip/DH_book.htm
09/2010;
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ABSTRACT: Kinetically constrained spin models (KCSM) are interacting particle systems which are intensively studied in physics literature
as models for systems undergoing glass or jamming transitions. KCSM leave on discrete lattices and evolve via a Glauber-like
dynamics which is reversible w.r.t. a simple product measure. The key feature is that the creation/destruction of a particle
at a given site can occur only if the current configuration satisfies proper local constraints. Due to the fact that creation/destruction
rates can be zero, the whole analysis of the long time behavior becomes quite delicate. From the mathematical point of view,
the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle
density remained open for most KCSM (with the exception of the East model in d=1 Aldous and P.Diaconis, J.Stat. Phys. 107(5–6):945–975 2002). Here we review a novel multi-scale approach which we have developed in Cancrini et al. (Probab. Theory Relat. Fields 140:459–504,
2008; Lecture Notes in Mathematics, vol.1970, pp.307–340, Springer, 2009) trough which we: (i) prove positivity of the spectral gap in the whole ergodic region for a wide class of KCSM on ℤ
d
, (ii) establish (sometimes optimal) bounds on the behavior of the spectral gap near the boundary of the ergodicity region
and (iii) prove pure exponential decay at equilibrium for the persistence function, i.e. the probability that the occupation
variable at the origin does not change before timet. Our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations.
In particular (i) above establishes exponential decay of auto-correlation functions disproving the stretched exponential decay
which had been conjecture for some KCSM and (ii) disproves some of the scalings which had been extrapolated from numerical
simulations for the relaxation times (inverse of the spectral gap).
08/2009: pages 741-752;
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ABSTRACT: We study bootstrap percolation (BP) on hyperbolic lattices obtained by regular tilings of the hyperbolic plane. Our work is motivated by the connection between the BP transition and the dynamical transition of kinetically constrained models, which are in turn relevant for the study of glass and jamming transitions. We show that for generic tilings there exists a BP transition at a nontrivial critical density, $0<\rho_c<1$. Thus, despite the presence of loops on all length scales in hyperbolic lattices, the behavior is very different from that on Euclidean lattices where the critical density is either zero or one. Furthermore, we show that the transition has a mixed character since it is discontinuous but characterized by a diverging correlation length, similarly to what happens on Bethe lattices and random graphs of constant connectivity.
07/2009;
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ABSTRACT: We analyze the relaxation to equilibrium for kinetically constrained spin
models (KCSM) when the initial distribution $\nu$ is different from the
reversible one, $\mu$. 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 $\bbZ$, 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 \cite{Aldous:2002p1074}, for which the constraint requires the two children
to be vacant. While the former model is ergodic at any $p\neq 1$, the latter
displays an ergodicity breaking transition at $p_c=1/2$. For the East we prove
exponential convergence to equilibrium with rate depending on the spectral gap
if $\nu$ is concentrated on any configuration which does not contain a forever
blocked site or if $\nu$ is a Bernoulli($p'$) product measure for any $p'\neq
1$. For the model on the binary tree we prove similar results in the regime
$p,p'<p_c$ and under the (plausible) assumption that the spectral gap is
positive for $p<p_c$. By constructing a proper test function we also prove that
if $p'>p_c$ and $p\leq p_c$ convergence to equilibrium cannot occur for all
local functions. Finally we present a very simple argument (different from the
one in \cite{Aldous:2002p1074}) based on a combination of combinatorial results
and ``energy barrier'' considerations, which yields the sharp upper bound for
the spectral gap of East when $p\uparrow 1$.
Journal of Statistical Physics 11/2008; · 1.40 Impact Factor
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ABSTRACT: Facilitated or kinetically constrained spin models (KCSM) are a class of
interacting particle systems reversible w.r.t. to a simple product measure.
Each dynamical variable (spin) is re-sampled from its equilibrium distribution
only if the surrounding configuration fulfills a simple local constraint which
\emph{does not involve} the chosen variable itself. Such simple models are
quite popular in the glass community since they display some of the peculiar
features of glassy dynamics, in particular they can undergo a dynamical arrest
reminiscent of the liquid/glass transitiom. Due to the fact that the jumps
rates of the Markov process can be zero, the whole analysis of the long time
behavior becomes quite delicate and, until recently, KCSM have escaped a
rigorous analysis with the notable exception of the East model. In these notes
we will mainly review several recent mathematical results which, besides being
applicable to a wide class of KCSM, have contributed to settle some debated
questions arising in numerical simulations made by physicists. We will also
provide some interesting new extensions. In particular we will show how to deal
with interacting models reversible w.r.t. to a high temperature Gibbs measure
and we will provide a detailed analysis of the so called one spin facilitated
model on a general connected graph.
01/2008;
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ABSTRACT: We introduce a new class of two-dimensional cellular automata with a bootstrap percolation-like dynamics. Each site can be
either empty or occupied by a single particle and the dynamics follows a deterministic updating rule at discrete times which
allows only emptying sites. We prove that the threshold density ρ
c
for convergence to a completely empty configuration is non trivial, 0<ρ
c
<1, contrary to standard bootstrap percolation. Furthermore we prove that in the subcritical regime, ρ<ρ
c
, emptying always occurs exponentially fast and that ρ
c
coincides with the critical density for two-dimensional oriented site percolation on ℤ2. This is known to occur also for some cellular automata with oriented rules for which the transition is continuous in the
value of the asymptotic density and the crossover length determining finite size effects diverges as a power law when the
critical density is approached from below. Instead for our model we prove that the transition is discontinuous and at the same time the crossover length diverges faster than any power law. The proofs of the discontinuity and the lower bound on the crossover length use a conjecture on the critical behaviour for
oriented percolation. The latter is supported by several numerical simulations and by analytical (though non rigorous) works
through renormalization techniques. Finally, we will discuss why, due to the peculiar mixed critical/first order character of this transition, the model is particularly relevant to study glassy and jamming transitions. Indeed, we will show that
it leads to a dynamical glass transition for a Kinetically Constrained Spin Model. Most of the results that we present are
the rigorous proofs of physical arguments developed in a joint work with D.S. Fisher.
Journal of Statistical Physics 12/2007; 130(1):83-112. · 1.40 Impact Factor
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ABSTRACT: We introduce a new class of two-dimensional cellular automata with a bootstrap percolation-like dynamics. Each site can be either empty or occupied by a single particle and the dynamics follows a deterministic updating rule at discrete times which allows only emptying sites. We prove that the threshold density $\rho_c$ for convergence to a completely empty configuration is non trivial, $0<\rho_c<1$, contrary to standard bootstrap percolation. Furthermore we prove that in the subcritical regime, $\rho<\rho_c$, emptying always occurs exponentially fast and that $\rho_c$ coincides with the critical density for two-dimensional oriented site percolation on $\bZ^2$. This is known to occur also for some cellular automata with oriented rules for which the transition is continuous in the value of the asymptotic density and the crossover length determining finite size effects diverges as a power law when the critical density is approached from below. Instead for our model we prove that the transition is {\it discontinuous} and at the same time the crossover length diverges {\it faster than any power law}. The proofs of the discontinuity and the lower bound on the crossover length use a conjecture on the critical behaviour for oriented percolation. The latter is supported by several numerical simulations and by analytical (though non rigorous) works through renormalization techniques. Finally, we will discuss why, due to the peculiar {\it mixed critical/first order character} of this transition, the model is particularly relevant to study glassy and jamming transitions. Indeed, we will show that it leads to a dynamical glass transition for a Kinetically Constrained Spin Model. Most of the results that we present are the rigorous proofs of physical arguments developed in a joint work with D.S.Fisher. Comment: 42 pages, 11 figures
09/2007;
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ABSTRACT: In Phys. Rev. Lett. 96, 035702 (2006) we introduced a class of kinetically constrained models which display a dynamical glass transition. We focused on a particular example: the "knights" model. As correctly pointed out by Jeng and Schwarz cond-mat/0612484, we overlooked some additional directed frozen structures of the knights model which are not simple directed percolation (DP) paths: these "thicker" directed structures lower the critical density. Here we argue that, nevertheless, the full directed processes are in the DP universality class and the T-junctions between perpendicular segments of these give rise to a jamming percolation transition with the universal properties discussed in our previous work. Moreover, we present other models for which all our previous results, included the value of the critical density, hold rigorously.
01/2007;
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ABSTRACT: We analyze the density and size dependence of the relaxation time for kinetically constrained spin models (KCSM) intensively studied in the physical literature as simple models sharing some of the features of a glass transition. KCSM are interacting particle systems on $\Z^d$ with Glauber-like dynamics, reversible w.r.t. a simple product i.i.d Bernoulli($p$) measure. The essential feature of a KCSM is that the creation/destruction of a particle at a given site can occur only if the current configuration of empty sites around it satisfies certain constraints which completely define each specific model. No other interaction is present in the model. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density $p$ remained open for most KCSM (with the notably exception of the East model in $d=1$ \cite{Aldous-Diaconis}). Here for the first time we: i) identify the ergodicity region by establishing a connection with an associated bootstrap percolation model; ii) develop a novel multi-scale approach which proves positivity of the spectral gap in the whole ergodic region; iii) establish, sometimes optimal, bounds on the behavior of the spectral gap near the boundary of the ergodicity region and iv) establish pure exponential decay for the persistence function. Our techniques are flexible enough to allow a variety of constraints and our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations.
Probability Theory and Related Fields 11/2006; · 1.53 Impact Factor
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ABSTRACT: We present a detailed physical analysis of the dynamical glass-jamming transition which occurs for the so called Knight models recently introduced and analyzed in a joint work with D.S.Fisher \cite{letterTBF}. Furthermore, we review some of our previous works on Kinetically Constrained Models. The Knights models correspond to a new class of kinetically constrained models which provide the first example of finite dimensional models with an ideal glass-jamming transition. This is due to the underlying percolation transition of particles which are mutually blocked by the constraints. This jamming percolation has unconventional features: it is discontinuous (i.e. the percolating cluster is compact at the transition) and the typical size of the clusters diverges faster than any power law when $\rho\nearrow\rho_c$. These properties give rise for Knight models to an ergodicity breaking transition at $\rho_c$: at and above $\rho_{c}$ a finite fraction of the system is frozen. In turn, this finite jump in the density of frozen sites leads to a two step relaxation for dynamic correlations in the unjammed phase, analogous to that of glass forming liquids. Also, due to the faster than power law divergence of the dynamical correlation length, relaxation times diverge in a way similar to the Vogel-Fulcher law.
05/2006;
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ABSTRACT: We analyze the density and size dependence of the relaxation time $\tau$ for kinetically constrained spin systems. These have been proposed as models for strong or fragile glasses and for systems undergoing jamming transitions. For the one (FA1f) or two (FA2f) spin facilitated Fredrickson-Andersen model at any density $\rho<1$ and for the Knight model below the critical density at which the glass transition occurs, we show that the persistence and the spin-spin time auto-correlation functions decay exponentially. This excludes the stretched exponential relaxation which was derived by numerical simulations. For FA2f in $d\geq 2$, we also prove a super-Arrhenius scaling of the form $\exp(1/(1-\rho))\leq \tau\leq\exp(1/(1-\rho)^2)$. For FA1f in $d$=$1,2$ we rigorously prove the power law scalings recently derived in \cite{JMS} while in $d\geq 3$ we obtain upper and lower bounds consistent with findings therein. Our results are based on a novel multi-scale approach which allows to analyze $\tau$ in presence of kinetic constraints and to connect time-scales and dynamical heterogeneities. The techniques are flexible enough to allow a variety of constraints and can also be applied to conservative stochastic lattice gases in presence of kinetic constraints.
04/2006;
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ABSTRACT: A new class of lattice gas models with trivial interactions but constrained dynamics is introduced. These models are proven to exhibit a dynamical glass transition: above a critical density rhoc ergodicity is broken due to the appearance of an infinite spanning cluster of jammed particles. The fraction of jammed particles is discontinuous at the transition, while in the unjammed phase dynamical correlation lengths and time scales diverge as exp[C(rhoc-rho)-mu]. Dynamic correlations display two-step relaxation similar to glass formers and jamming systems.
Physical Review Letters 02/2006; 96(3):035702. · 7.37 Impact Factor
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ABSTRACT: This paper has been withdrawn by the authors due to a mistake in the proof and a corresponding incorrect result. A correct rigorous analysis of a similar model is presented in ``Spiral Model: a cellular automaton with a discontinuous glass transition'', arXiv:0709.0378.
01/2006;
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ABSTRACT: We compute analytically and numerically the four-point correlation function that characterizes nontrivial cooperative dynamics in glassy systems within several models of glasses: elastoplastic deformations, mode-coupling theory (MCT), collectively rearranging regions (CRR's), diffusing defects, and kinetically constrained models (KCM's). Some features of the four-point susceptibility chi(4) (t) are expected to be universal: at short times we expect a power-law increase in time as t(4) due to ballistic motion (t(2) if the dynamics is Brownian) followed by an elastic regime (most relevant deep in the glass phase) characterized by a t or sqrt[t] growth, depending on whether phonons are propagative or diffusive. We find in both the beta and early alpha regime that chi(4) approximately t(mu), where mu is directly related to the mechanism responsible for relaxation. This regime ends when a maximum of chi(4) is reached at a time t= t(*) of the order of the relaxation time of the system. This maximum is followed by a fast decay to zero at large times. The height of the maximum also follows a power law chi(4) (t(*)) approximately t(*lambda). The value of the exponents mu and lambda allows one to distinguish between different mechanisms. For example, freely diffusing defects in d=3 lead to mu=2 and lambda=1 , whereas the CRR scenario rather predicts either mu=1 or a logarithmic behavior depending on the nature of the nucleation events and a logarithmic behavior of chi(4) (t(*)) . MCT leads to mu=b and lambda=1/gamma , where b and gamma are the standard MCT exponents. We compare our theoretical results with numerical simulations on a Lennard-Jones and a soft-sphere system. Within the limited time scales accessible to numerical simulations, we find that the exponent mu is rather small, mu<1 , with a value in reasonable agreement with the MCT predictions, but not with the prediction of simple diffusive defect models, KCM's with noncooperative defects, and CRR's. Experimental and numerical determination of chi(4) (t) for longer time scales and lower temperatures would yield highly valuable information on the glass formation mechanism.
Physical Review E 04/2005; 71(4 Pt 1):041505. · 2.26 Impact Factor
