1,721,041 research outputs found
COALESCING AND BRANCHING SIMPLE SYMMETRIC EXCLUSION PROCESS
No full textMotivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph G = (V, E) dual to the biased voter model on G. Our main goal is tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as vertical bar V vertical bar -> infinity. Our results allow us to recover very directly and improve to l(p)-mixing, p >= 2, and to more general graphs, the mixing time results of Pillai and Smith for the Fredrickson-Andersen one spin facilitated (FA-1f) KCM on the discrete d-dimensional torus. In view of applications to the more complex FA-jf KCM, j > 1, we also extend part of the analysis to an analogous process with a more general product state space
Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality
No full textOn 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 vacant. For this process the following picture was conjectured to hold. As long as p is below the percolation threshold pc = 1/k the process is ergodic with a finite relaxation time while, for p > pc, 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 = pc 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 behavior in the depth of the tree at 13 p = pc and (ii) power law scaling in (pc − p)−1 when p approaches pc 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
Classical versus quantum structures: The case of pyramidal molecules
No full textIn a previous paper we proposed a model to describe a gas of pyramidal molecules interacting via dipole-dipole interactions. The interaction modifies the tunneling properties between the classical equilibrium configurations of the sin- gle molecule and, for sufficiently high pressure, the molecules become localized in these classical configurations. The model explains quantitatively the shift to zero- frequency of the inversion line observed upon increase of the pressure in a gas of ammonia or deuterated ammonia. Here we analyze further the model especially with respect to stability questions
Interaction Induced Localization in a Gas of Pyramidal Molecules
No full textWe propose a model to describe a gas of pyramidal molecules interacting via dipole-dipole interactions. The interaction modifies the tunneling properties between the classical equilibrium configurations of the single molecule and, for sufficiently high pressure, the molecules become localized in these classical configurations. We explain quantitatively, without free parameters, the shift to zero frequency of the inversion line observed upon increase of the pressure in a gas of ammonia or deuterated ammonia. For sufficiently high pressures, our model suggests the existence of a superselection rule for states of different chirality in substituted derivatives
Environment induced localization and superselection rules in a gas of pyramidal molecules
No full textVle propose a model to describe a gas of pyramidal molecules interacting via dipole-dipole interactions. A cooperative effect induced by the interaction modifies the tunneling properties between the classical equilibrium configurations of the single molecule. The model suggests that, for sufficiently high gas density, the molecules become localized in these classical configurations. On this basis it is possible to explain the shift and the disappearance of the inversion line observed upon increase of the pressure in a gas of ammonia or deuterated ammonia. The same mechanism also accounts for the presence of stable optical activity of certain pyramidal molecules. We discuss the concept of environment induced superselection rule which has been invoked in connection with this problem
Kinetically Constrained Lattice Gases
No full textKinetically 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 particularly 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(L2) (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 density-density 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
Facilitated Oriented Spin Models: Some Non Equilibrium Results
No full textWe perform the rigorous analysis of the relaxation to equilibrium for some facilitated
or kinetically constrained spin models (KCSM) when the initial distribution ! 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 AD
model on the binary tree introduced in Aldous and Diaconis (J. Stat. Phys. 107(5–6):945–
975, 2002), 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 ! is concentrated on any configuration which
does not contain a forever blocked site or if ! 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 Aldous and Diaconis (J. Stat. Phys.
107(5–6):945–975, 2002), based on a 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
Fredrickson-Andersen one spin facilitated model out of equilibrium.
No full textWe 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 ∈ [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 ν different
from the stationary product p-Bernoulli measure μ. We assume that, under ν, 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 q < ̄ 1. The convergence is exponential or stretched exponential, depending on the growth of the graph. In
particular it is exponential on Z
d
for d = 1 and stretched exponential for d > 1. Our result can be generalized to other non cooperative models
Aging through hierarchical coalescence processes in the east model
No full textWe rigorously analyze the low temperature non-equilibrium dynamics of the
East model, a special example of a one dimensional oriented kinetically constrained
particle model, when the initial distribution is different from the reversible one and for
times much smaller than the global relaxation time. 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. In the limit of zero temperature (i.e. a vanishing
density of vacancies) and for initial distributions such that the vacancies form a
renewal process, we prove that the density of vacancies, the persistence function and the
two-time autocorrelation function behave as staircase functions with several plateaux.
Furthermore the two-time autocorrelation function displays an aging behavior. We also
provide a sharp description of the statistics of the domain length as a function of time, a
domain being the interval between two consecutive vacancies.When the initial renewal
process has finite mean, our results confirm (and generalize) previous findings of the
physicists for the restricted case of a product Bernoulli measure. However we show
that a different behavior appears when the initial domain distribution is in the attraction
domain of a α-stable law. All the above results actually follow from a more general result
which says that the low temperature dynamics of the East model is very well described
by that of a certain hierarchical coalescence process, a probabilistic object which can be
viewed as a hierarchical sequence of suitably linked coalescence processes and whose
asymptotic behavior has been recently studied in Faggionato et al. (Universality in one
dimensional hierarchical 1059 coalescence processes. Preprint, 2011)
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