87,813 research outputs found

### Geometry, thermodynamics, and finite-size corrections in the critical Potts model

We establish an intriguing connection between geometry and thermodynamics in
the critical q-state Potts model on two-dimensional lattices, using the q-state
bond-correlated percolation model (QBCPM) representation. We find that the
number of clusters of the QBCPM has an energy-like singularity for q different
from 1, which is reached and supported by exact results, numerical simulation,
and scaling arguments. We also establish that the finite-size correction to the
number of bonds, has no constant term and explains the divergence of related
quantities as q --> 4, the multicritical point. Similar analyses are applicable
to a variety of other systems.Comment: 12 pages, 6 figure

### Mapping functions and critical behavior of percolation on rectangular domains

The existence probability $E_p$ and the percolation probability $P$ of the
bond percolation on rectangular domains with different aspect ratios $R$ are
studied via the mapping functions between systems with different aspect ratios.
The superscaling behavior of $E_p$ and $P$ for such systems with exponents $a$
and $b$, respectively, found by Watanabe, Yukawa, Ito, and Hu in [Phys. Rev.
Lett. \textbf{93}, 190601 (2004)] can be understood from the lower order
approximation of the mapping functions $f_R$ and $g_R$ for $E_p$ and $P$,
respectively; the exponents $a$ and $b$ can be obtained from numerically
determined mapping functions $f_R$ and $g_R$, respectively.Comment: 17 pages with 6 figure

### Universal scaling functions for bond percolation on planar random and square lattices with multiple percolating clusters

Percolation models with multiple percolating clusters have attracted much
attention in recent years. Here we use Monte Carlo simulations to study bond
percolation on $L_{1}\times L_{2}$ planar random lattices, duals of random
lattices, and square lattices with free and periodic boundary conditions, in
vertical and horizontal directions, respectively, and with various aspect ratio
$L_{1}/L_{2}$. We calculate the probability for the appearance of $n$
percolating clusters, $W_{n},$ the percolating probabilities, $P$, the average
fraction of lattice bonds (sites) in the percolating clusters, $_{n}$
($_{n}$), and the probability distribution function for the fraction $c$
of lattice bonds (sites), in percolating clusters of subgraphs with $n$
percolating clusters, $f_{n}(c^{b})$ ($f_{n}(c^{s})$). Using a small number of
nonuniversal metric factors, we find that $W_{n}$, $P$, $_{n}$
($_{n}$), and $f_{n}(c^{b})$ ($f_{n}(c^{s})$) for random lattices, duals
of random lattices, and square lattices have the same universal finite-size
scaling functions. We also find that nonuniversal metric factors are
independent of boundary conditions and aspect ratios.Comment: 15 pages, 11 figure

### Renormalization group approach to an Abelian sandpile model on planar lattices

One important step in the renormalization group (RG) approach to a lattice
sandpile model is the exact enumeration of all possible toppling processes of
sandpile dynamics inside a cell for RG transformations. Here we propose a
computer algorithm to carry out such exact enumeration for cells of planar
lattices in RG approach to Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett.
{\bf 59}, 381 (1987)] and consider both the reduced-high RG equations proposed
by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. {\bf 72}, 1690
(1994)] and the real-height RG equations proposed by Ivashkevich [Phys. Rev.
Lett. {\bf 76}, 3368 (1996)]. Using this algorithm we are able to carry out RG
transformations more quickly with large cell size, e.g. $3 \times 3$ cell for
the square (sq) lattice in PVZ RG equations, which is the largest cell size at
the present, and find some mistakes in a previous paper [Phys. Rev. E {\bf 51},
1711 (1995)]. For sq and plane triangular (pt) lattices, we obtain the only
attractive fixed point for each lattice and calculate the avalanche exponent
$\tau$ and the dynamical exponent $z$. Our results suggest that the increase of
the cell size in the PVZ RG transformation does not lead to more accurate
results. The implication of such result is discussed.Comment: 29 pages, 6 figure

### A proposed generalized constitutive equation for nonlinear para-isotropic materials

Finite element models of varying complexities were used to solve problems in solid mechanics. Particular emphasis was given to concrete which is nonisotropic at any level of deformation and is also nonlinear in terms of stress-strain relationships

### Random-cluster multi-histogram sampling for the q-state Potts model

Using the random-cluster representation of the $q$-state Potts models we
consider the pooling of data from cluster-update Monte Carlo simulations for
different thermal couplings $K$ and number of states per spin $q$. Proper
combination of histograms allows for the evaluation of thermal averages in a
broad range of $K$ and $q$ values, including non-integer values of $q$. Due to
restrictions in the sampling process proper normalization of the combined
histogram data is non-trivial. We discuss the different possibilities and
analyze their respective ranges of applicability.Comment: 12 pages, 9 figures, RevTeX

### Probability-Changing Cluster Algorithm for Potts Models

We propose a new effective cluster algorithm of tuning the critical point
automatically, which is an extended version of Swendsen-Wang algorithm. We
change the probability of connecting spins of the same type, $p = 1 - e^{- J/
k_BT}$, in the process of the Monte Carlo spin update. Since we approach the
canonical ensemble asymptotically, we can use the finite-size scaling analysis
for physical quantities near the critical point. Simulating the two-dimensional
Potts models to demonstrate the validity of the algorithm, we have obtained the
critical temperatures and critical exponents which are consistent with the
exact values; the comparison has been made with the invaded cluster algorithm.Comment: 4 pages including 5 eps figures, RevTeX, to appear in Phys. Rev. Let

### Exact Ampitude Ratio and Finite-Size Corrections for the M x N Square Lattice Ising Model The :

Let f, U and C represent, respectively, the free energy, the internal energy
and the specific heat of the critical Ising model on the square M x N lattice
with periodic boundary conditions. We find that N f and U are well-defined odd
function of 1/N. We also find that ratios of subdominant (N^(-2 i - 1))
finite-size corrections amplitudes for the internal energy and the specific
heat are constant. The free energy and the internal energy at the critical
point are calculated asymtotically up to N^(-5) order, and the specific heat up
to N^(-3) order.Comment: 18 pages, 4 figures, to be published in Phys. Rev. E 65, 1 February
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