38 research outputs found
Relaxation times of kinetically constrained spin models with glassy dynamics
We analyze the density and size dependence of the relaxation time 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 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 , we also prove a super-Arrhenius scaling of the form
. For FA1f in = we
rigorously prove the power law scalings recently derived in \cite{JMS} while in
we obtain upper and lower bounds consistent with findings therein.
Our results are based on a novel multi-scale approach which allows to analyze
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.Comment: 4 page
Phase diagram of the ABC model on an interval
The three species asymmetric ABC model was initially defined on a ring by
Evans, Kafri, Koduvely, and Mukamel, and the weakly asymmetric version was
later studied by Clincy, Derrida, and Evans. Here the latter model is studied
on a one-dimensional lattice of N sites with closed (zero flux) boundaries. In
this geometry the local particle conserving dynamics satisfies detailed balance
with respect to a canonical Gibbs measure with long range asymmetric pair
interactions. This generalizes results for the ring case, where detailed
balance holds, and in fact the steady state measure is known only for the case
of equal densities of the different species: in the latter case the stationary
states of the system on a ring and on an interval are the same. We prove that
in the N to infinity limit the scaled density profiles are given by (pieces of)
the periodic trajectory of a particle moving in a quartic confining potential.
We further prove uniqueness of the profiles, i.e., the existence of a single
phase, in all regions of the parameter space (of average densities and
temperature) except at low temperature with all densities equal; in this case a
continuum of phases, differing by translation, coexist. The results for the
equal density case apply also to the system on the ring, and there extend
results of Clincy et al.Comment: 52 pages, AMS-LaTeX, 8 figures from 10 eps figure files. Revision:
minor changes in response to referee reports; paper to appear in J. Stat.
Phy
Inhomogeneity-induced second-order phase transitions in Potts model on hierarchical lattices
The thermodynamics of the -state Potts model with arbitrary on a class
of hierarchical lattices is considered. Contrary to the case of the crystal
lattices, it has always the second-order phase transitions. The analytical
expressions fo the critical indexes are obtained, their dependencies on the
structural lattice pararmeters are studied and the scailing relations among
them are establised. The structural criterion of the inhomogeneity-induced
transformation of the transition order is suggested. The application of the
results to a description of critical phenomena in the dilute crystals and
substances confined in porous media is discussed.Comment: 9 pages, 2 figure
Jamming percolation and glassy dynamics
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 . These
properties give rise for Knight models to an ergodicity breaking transition at
: at and above 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.Comment: Submitted to the special issue of Journal of Statistical Physics on
Spin glasses and related topic
Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics
Kinetically constrained lattice models of glasses introduced by Kob and
Andersen (KA) are analyzed. It is proved that only two behaviors are possible
on hypercubic lattices: either ergodicity at all densities or trivial
non-ergodicity, depending on the constraint parameter and the dimensionality.
But in the ergodic cases, the dynamics is shown to be intrinsically cooperative
at high densities giving rise to glassy dynamics as observed in simulations.
The cooperativity is characterized by two length scales whose behavior controls
finite-size effects: these are essential for interpreting simulations. In
contrast to hypercubic lattices, on Bethe lattices KA models undergo a
dynamical (jamming) phase transition at a critical density: this is
characterized by diverging time and length scales and a discontinuous jump in
the long-time limit of the density autocorrelation function. By analyzing
generalized Bethe lattices (with loops) that interpolate between hypercubic
lattices and standard Bethe lattices, the crossover between the dynamical
transition that exists on these lattices and its absence in the hypercubic
lattice limit is explored. Contact with earlier results are made via analysis
of the related Fredrickson-Andersen models, followed by brief discussions of
universality, of other approaches to glass transitions, and of some issues
relevant for experiments.Comment: 59 page
Notes on the Third Law of Thermodynamics.I
We analyze some aspects of the third law of thermodynamics. We first review
both the entropic version (N) and the unattainability version (U) and the
relation occurring between them. Then, we heuristically interpret (N) as a
continuity boundary condition for thermodynamics at the boundary T=0 of the
thermodynamic domain. On a rigorous mathematical footing, we discuss the third
law both in Carath\'eodory's approach and in Gibbs' one. Carath\'eodory's
approach is fundamental in order to understand the nature of the surface T=0.
In fact, in this approach, under suitable mathematical conditions, T=0 appears
as a leaf of the foliation of the thermodynamic manifold associated with the
non-singular integrable Pfaffian form . Being a leaf, it cannot
intersect any other leaf const. of the foliation. We show that (N) is
equivalent to the requirement that T=0 is a leaf. In Gibbs' approach, the
peculiar nature of T=0 appears to be less evident because the existence of the
entropy is a postulate; nevertheless, it is still possible to conclude that the
lowest value of the entropy has to belong to the boundary of the convex set
where the function is defined.Comment: 29 pages, 2 figures; RevTex fil
Interpolating the Stage of Exponential Expansion in the Early Universe: a possible alternative with no reheating
In the standard picture, the inflationary universe is in a supercooled state
which ends with a short time, large scale reheating period, after which the
universe goes into a radiation dominated stage. An alternative is proposed here
in which the radiation energy density smoothly decreases all during an
inflation-like stage and with no discontinuity enters the subsequent radiation
dominated stage. The scale factor is calculated from standard Friedmann
cosmology in the presence of both radiation and vacuum energy density. A large
class of solutions confirm the above identified regime of non-reheating
inflation-like behavior for observationally consistent expansion factors and
not too large a drop in the radiation energy density. One dynamical realization
of such inflation without reheating is from warm inflation type scenarios.
However the solutions found here are properties of the Einstein equations with
generality beyond slow-roll inflation scenarios. The solutions also can be
continuously interpolated from the non-reheating type behavior to the standard
supercooled limit of exponential expansion, thus giving all intermediate
inflation-like behavior between these two extremes. The temperature of the
universe and the expansion factor are calculated for various cases.
Implications for baryongenesis are discussed. This non-reheating,
inflation-like regime also appears to have some natural features for a universe
that is between nearly flat and open.Comment: 26 pages, Latex, 2 figures, In press Physical Review
The three-dimensional random field Ising magnet: interfaces, scaling, and the nature of states
The nature of the zero temperature ordering transition in the 3D Gaussian
random field Ising magnet is studied numerically, aided by scaling analyses. In
the ferromagnetic phase the scaling of the roughness of the domain walls,
, is consistent with the theoretical prediction .
As the randomness is increased through the transition, the probability
distribution of the interfacial tension of domain walls scales as for a single
second order transition. At the critical point, the fractal dimensions of
domain walls and the fractal dimension of the outer surface of spin clusters
are investigated: there are at least two distinct physically important fractal
dimensions. These dimensions are argued to be related to combinations of the
energy scaling exponent, , which determines the violation of
hyperscaling, the correlation length exponent , and the magnetization
exponent . The value is derived from the
magnetization: this estimate is supported by the study of the spin cluster size
distribution at criticality. The variation of configurations in the interior of
a sample with boundary conditions is consistent with the hypothesis that there
is a single transition separating the disordered phase with one ground state
from the ordered phase with two ground states. The array of results are shown
to be consistent with a scaling picture and a geometric description of the
influence of boundary conditions on the spins. The details of the algorithm
used and its implementation are also described.Comment: 32 pp., 2 columns, 32 figure
What Broke First? Characterizing Sources of Structural Change Prior to the Great Recession
A Computational Mechanism for Unified Gain and Timing Control in the Cerebellum
Precise gain and timing control is the goal of cerebellar motor learning. Because the basic neural circuitry of the cerebellum is homogeneous throughout the cerebellar cortex, a single computational mechanism may be used for simultaneous gain and timing control. Although many computational models of the cerebellum have been proposed for either gain or timing control, few models have aimed to unify them. In this paper, we hypothesize that gain and timing control can be unified by learning of the complete waveform of the desired movement profile instructed by climbing fiber signals. To justify our hypothesis, we adopted a large-scale spiking network model of the cerebellum, which was originally developed for cerebellar timing mechanisms to explain the experimental data of Pavlovian delay eyeblink conditioning, to the gain adaptation of optokinetic response (OKR) eye movements. By conducting large-scale computer simulations, we could reproduce some features of OKR adaptation, such as the learning-related change of simple spike firing of model Purkinje cells and vestibular nuclear neurons, simulated gain increase, and frequency-dependent gain increase. These results suggest that the cerebellum may use a single computational mechanism to control gain and timing simultaneously