633 research outputs found
Site Percolation and Phase Transitions in Two Dimensions
The properties of the pure-site clusters of spin models, i.e. the clusters
which are obtained by joining nearest-neighbour spins of the same sign, are
here investigated. In the Ising model in two dimensions it is known that such
clusters undergo a percolation transition exactly at the critical point. We
show that this result is valid for a wide class of bidimensional systems
undergoing a continuous magnetization transition. We provide numerical evidence
for discrete as well as for continuous spin models, including SU(N) lattice
gauge theories. The critical percolation exponents do not coincide with the
ones of the thermal transition, but they are the same for models belonging to
the same universality class.Comment: 8 pages, 6 figures, 2 tables. Numerical part developed; figures,
references and comments adde
Glass transition in granular media
In the framework of schematic hard spheres lattice models for granular media
we investigate the phenomenon of the ``jamming transition''. In particular,
using Edwards' approach, by analytical calculations at a mean field level, we
derive the system phase diagram and show that ``jamming'' corresponds to a
phase transition from a ``fluid'' to a ``glassy'' phase, observed 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.Comment: 7 pages, 4 figure
Relaxation properties in a lattice gas model with asymmetrical particles
We study the relaxation process in a two-dimensional lattice gas model, where
the interactions come from the excluded volume. In this model particles have
three arms with an asymmetrical shape, which results in geometrical frustration
that inhibits full packing. A dynamical crossover is found at the arm
percolation of the particles, from a dynamical behavior characterized by a
single step relaxation above the transition, to a two-step decay below it.
Relaxation functions of the self-part of density fluctuations are well fitted
by a stretched exponential form, with a exponent decreasing when the
temperature is lowered until the percolation transition is reached, and
constant below it. The structural arrest of the model seems to happen only at
the maximum density of the model, where both the inverse diffusivity and the
relaxation time of density fluctuations diverge with a power law. The dynamical
non linear susceptibility, defined as the fluctuations of the self-overlap
autocorrelation, exhibits a peak at some characteristic time, which seems to
diverge at the maximum density as well.Comment: 7 pages and 9 figure
Percolation and cluster Monte Carlo dynamics for spin models
A general scheme for devising efficient cluster dynamics proposed in a
previous letter [Phys.Rev.Lett. 72, 1541 (1994)] is extensively discussed. In
particular the strong connection among equilibrium properties of clusters and
dynamic properties as the correlation time for magnetization is emphasized. The
general scheme is applied to a number of frustrated spin model and the results
discussed.Comment: 17 pages LaTeX + 16 figures; will appear in Phys. Rev.
The jamming transition of Granular Media
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.Comment: 18 pages with 6 postscript figures. to appear in J.Phys: Cond. Ma
Heterogeneous slow dynamics in a two dimensional doped classical antiferromagnet
We introduce a lattice model for a classical doped two dimensional
antiferromagnet which has no quenched disorder, yet displays slow dynamics
similar to those observed in supercooled liquids. We calculate two-time spatial
and spin correlations via Monte Carlo simulations and find that for
sufficiently low temperatures, there is anomalous diffusion and
stretched-exponential relaxation of spin correlations. The relaxation times
associated with spin correlations and diffusion both diverge at low
temperatures in a sub-Arrhenius fashion if the fit is done over a large
temperature-window or an Arrhenius fashion if only low temperatures are
considered. We find evidence of spatially heterogeneous dynamics, in which
vacancies created by changes in occupation facilitate spin flips on
neighbouring sites. We find violations of the Stokes-Einstein relation and
Debye-Stokes-Einstein relation and show that the probability distributions of
local spatial correlations indicate fast and slow populations of sites, and
local spin correlations indicate a wide distribution of relaxation times,
similar to observ ations in other glassy systems with and without quenched
disorder.Comment: 12 pages, 17 figures, corrected erroneous figure, and improved
quality of manuscript, updated reference
Beyond Blobs in Percolation Cluster Structure: The Distribution of 3-Blocks at the Percolation Threshold
The incipient infinite cluster appearing at the bond percolation threshold
can be decomposed into singly-connected ``links'' and multiply-connected
``blobs.'' Here we decompose blobs into objects known in graph theory as
3-blocks. A 3-block is a graph that cannot be separated into disconnected
subgraphs by cutting the graph at 2 or fewer vertices. Clusters, blobs, and
3-blocks are special cases of -blocks with , 2, and 3, respectively. We
study bond percolation clusters at the percolation threshold on 2-dimensional
square lattices and 3-dimensional cubic lattices and, using Monte-Carlo
simulations, determine the distribution of the sizes of the 3-blocks into which
the blobs are decomposed. We find that the 3-blocks have fractal dimension
in 2D and in 3D. These fractal dimensions are
significantly smaller than the fractal dimensions of the blobs, making possible
more efficient calculation of percolation properties. Additionally, the
closeness of the estimated values for in 2D and 3D is consistent with the
possibility that is dimension independent. Generalizing the concept of
the backbone, we introduce the concept of a ``-bone'', which is the set of
all points in a percolation system connected to disjoint terminal points
(or sets of disjoint terminal points) by disjoint paths. We argue that the
fractal dimension of a -bone is equal to the fractal dimension of
-blocks, allowing us to discuss the relation between the fractal dimension
of -blocks and recent work on path crossing probabilities.Comment: All but first 2 figs. are low resolution and are best viewed when
printe
Non exponential relaxation in fully frustrated models
We study the dynamical properties of the fully frustrated Ising model. Due to
the absence of disorder the model, contrary to spin glass, does not exhibit any
Griffiths phase, which has been associated to non-exponential relaxation
dynamics. Nevertheless we find numerically that the model exhibits a stretched
exponential behavior below a temperature T_p corresponding to the percolation
transition of the Kasteleyn-Fortuin clusters. We have also found that the
critical behavior of this clusters for a fully frustrated q-state spin model at
the percolation threshold is strongly affected by frustration. In fact while in
absence of frustration the q=1 limit gives random percolation, in presence of
frustration the critical behavior is in the same universality class of the
ferromagnetic q=1/2-state Potts model.Comment: 7 pages, RevTeX, 11 figs, to appear on Physical Review
Thermodynamic versus Topological Phase Transitions: Cusp in the Kert\'esz Line
We present a study of phase transitions of the Curie--Weiss Potts model at
(inverse) temperature , in presence of an external field . Both
thermodynamic and topological aspects of these transitions are considered. For
the first aspect we complement previous results and give an explicit equation
of the thermodynamic transition line in the -- plane as well as the
magnitude of the jump of the magnetization (for . The signature
of the latter aspect is characterized here by the presence or not of a giant
component in the clusters of a Fortuin--Kasteleyn type representation of the
model. We give the equation of the Kert\'esz line separating (in the
-- plane) the two behaviours. As a result, we get that this line
exhibits, as soon as , a very interesting cusp where it
separates from the thermodynamic transition line
Scaling properties in off equilibrium dynamical processes
In the present paper, we analyze the consequences of scaling hypotheses on
dynamic functions, as two times correlations . We show, under general
conditions, that must obey the following scaling behavior , where the scaling variable is
and , two
undetermined functions. The presence of a non constant exponent
signals the appearance of multiscaling properties in the dynamics.Comment: 6 pages, no figure
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