366 research outputs found
Potts-Percolation-Gauss Model of a Solid
We study a statistical mechanics model of a solid. Neighboring atoms are
connected by Hookian springs. If the energy is larger than a threshold the
"spring" is more likely to fail, while if the energy is lower than the
threshold the spring is more likely to be alive. The phase diagram and
thermodynamic quantities, such as free energy, numbers of bonds and clusters,
and their fluctuations, are determined using renormalization-group and
Monte-Carlo techniques.Comment: 10 pages, 12 figure
Percolation on the average and spontaneous magnetization for q-states Potts model on graph
We prove that the q-states Potts model on graph is spontaneously magnetized
at finite temperature if and only if the graph presents percolation on the
average. Percolation on the average is a combinatorial problem defined by
averaging over all the sites of the graph the probability of belonging to a
cluster of a given size. In the paper we obtain an inequality between this
average probability and the average magnetization, which is a typical extensive
function describing the thermodynamic behaviour of the model
Poisson approximations for the Ising model
A -dimensional Ising model on a lattice torus is considered. As the size
of the lattice tends to infinity, a Poisson approximation is given for the
distribution of the number of copies in the lattice of any given local
configuration, provided the magnetic field tends to and the
pair potential remains fixed. Using the Stein-Chen method, a bound is given
for the total variation error in the ferromagnetic case.Comment: 25 pages, 1 figur
Rejection-free Geometric Cluster Algorithm for Complex Fluids
We present a novel, generally applicable Monte Carlo algorithm for the
simulation of fluid systems. Geometric transformations are used to identify
clusters of particles in such a manner that every cluster move is accepted,
irrespective of the nature of the pair interactions. The rejection-free and
non-local nature of the algorithm make it particularly suitable for the
efficient simulation of complex fluids with components of widely varying size,
such as colloidal mixtures. Compared to conventional simulation algorithms,
typical efficiency improvements amount to several orders of magnitude
Invaded cluster algorithm for equilibrium critical points
A new cluster algorithm based on invasion percolation is described. The
algorithm samples the critical point of a spin system without a priori
knowledge of the critical temperature and provides an efficient way to
determine the critical temperature and other observables in the critical
region. The method is illustrated for the two- and three-dimensional Ising
models. The algorithm equilibrates spin configurations much faster than the
closely related Swendsen-Wang algorithm.Comment: 13 pages RevTex and 4 Postscript figures. Submitted to Phys. Rev.
Lett. Replacement corrects problem in printing figure
First-Order Phase Transition in Potts Models with finite-range interactions
We consider the -state Potts model on , , ,
with Kac ferromagnetic interactions and scaling parameter \ga. We prove the
existence of a first order phase transition for large but finite potential
ranges. More precisely we prove that for \ga small enough there is a value of
the temperature at which coexist Gibbs states. The proof is obtained by a
perturbation around mean-field using Pirogov-Sinai theory. The result is valid
in particular for , Q=3, in contrast with the case of nearest-neighbor
interactions for which available results indicate a second order phase
transition. Putting both results together provides an example of a system which
undergoes a transition from second to first order phase transition by changing
only the finite range of the interaction.Comment: Soumis pour publication a Journal of statistical physics - version
r\'{e}vis\'{e}
Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0
Denoting as the chromatic polynomial for coloring an -vertex
graph with colors, and considering the limiting function , a fundamental question in graph theory is the
following: is analytic or not at the origin
of the plane? (where the complex generalization of is assumed). This
question is also relevant in statistical mechanics because
, where is the ground state entropy of the
-state Potts antiferromagnet on the lattice graph , and the
analyticity of at is necessary for the large- series
expansions of . Although is analytic at for many
, there are some for which it is not; for these, has no
large- series expansion. It is important to understand the reason for this
nonanalyticity. Here we give a general condition that determines whether or not
a particular is analytic at and explains the
nonanalyticity where it occurs. We also construct infinite families of graphs
with functions that are non-analytic at and investigate the
properties of these functions. Our results are consistent with the conjecture
that a sufficient condition for to be analytic at is
that is a regular lattice graph . (This is known not to be a
necessary condition).Comment: 22 pages, Revtex, 4 encapsulated postscript figures, to appear in
Phys. Rev.
Absence of Phase Transition for Antiferromagnetic Potts Models via the Dobrushin Uniqueness Theorem
We prove that the -state Potts antiferromagnet on a lattice of maximum
coordination number exhibits exponential decay of correlations uniformly at
all temperatures (including zero temperature) whenever . We also prove
slightly better bounds for several two-dimensional lattices: square lattice
(exponential decay for ), triangular lattice (), hexagonal
lattice (), and Kagom\'e lattice (). The proofs are based on
the Dobrushin uniqueness theorem.Comment: 32 pages including 3 figures. Self-unpacking file containing the tex
file, the needed macros (epsf.sty, indent.sty, subeqnarray.sty, and
eqsection.sty) and the 3 ps file
Theory of continuum percolation I. General formalism
The theoretical basis of continuum percolation has changed greatly since its
beginning as little more than an analogy with lattice systems. Nevertheless,
there is yet no comprehensive theory of this field. A basis for such a theory
is provided here with the introduction of the Potts fluid, a system of
interacting -state spins which are free to move in the continuum. In the limit, the Potts magnetization, susceptibility and correlation functions
are directly related to the percolation probability, the mean cluster size and
the pair-connectedness, respectively. Through the Hamiltonian formulation of
the Potts fluid, the standard methods of statistical mechanics can therefore be
used in the continuum percolation problem.Comment: 26 pages, Late
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