Probabilities of spanning clusters in critical site percolation are computed. Square and cubic lattices with the free and periodic boundaries and with different aspect ratios are investigated. Finite size scaling of probabilities were discussed. Finally, the detailed comparison with the existing analytical results are discussed.
[LC42.02] Aspects of Universality in Percolation
Robert Ziff, Christian Lorenz (University of Michigan), Peter Kleban (University of Maine)
Besides the familiar universality of critical exponents and scaling functions near the critical point, the phenomenon of universality is evident in many additional aspects of percolation. For finite systems precisely at the (infinite-system) critical threshold, there exist numerous shape-dependent universal properties, such as the excess cluster number, cross-configuration probability, and certain amplitude ratios. These quantities also depend upon boudary conditions, including the twist in the boundary when periodic b.c.'s are used. (They are still considered universal because they do not depend upon the underlying model or lattice that is used.) Explicit expressions for a torus with a twist are derived. The infinite-system size distribution n_s \sim s^-tau can be reformulated into a universal form by stating that the number of clusters whose x- or y-dimension is greater than a length L, per unit area L^2, is a constant (0.116). The universality of the corrections-to-scaling exponent is also discussed.
[LC42.03] Static and dynamic critical behaviors of fully coordinated percolation
Eduardo Cuansing, Jae Hwa Kim, Hisao Nakanishi (Purdue University)
We study the static and dynamic critical behaviors of the fully coordinated percolation clusters on the square lattice. Fully coordinated percolation refers to the geometrical phase transition of the connectivity among the sites all of whose neighbors are occupied. Our preliminary results from Monte Carlo simulations (static exponents) and normal mode analyses (dynamic exponents) suggest that this problem is in the same universality class with ordinary percolation statically but not so dynamically. This may be attributable to the local correlations introduced by the full coordination requirements, somewhat similar to loopless or loop-enhanced percolation problems.(S. Muherjee, D. J. Jacobs and H. Nakanishi, Phys. Rev E 53), 1470 (1996). We also present a breadth first Monte Carlo algorithm to implement the full coordination statistics simultaneously as the percolation cluster is grown in simulation.
[LC42.04] Ordering in a High-q Antiferromagnetic Potts Model on a Simple Cubic Lattice
Shafiqur Rahman (Allegheny College, Meadville, PA 16335)
We have investigated the ordering in four-, five- and six-state antiferromagnetic Potts models on a simple cubic lattice by using an efficient cluster flipping Monte-Carlo algorithm(1). Unlike the q=3 case(2), which moves from a broken-sublattice-symmetry state to a rotationally-symmetric state before reaching the paramagnetic transition as temperature is increased, the higher q-states show much simpler behavior. For the q=4 case, we find that at small temperatures, two of the states are distributed randomly on one sublattice while the other two are distributed randomly on the other sublattice. The system undergoes a transition to the paramagnetic state at a finite temperature. For the q=5 case, the transition to the paramagnetic state appears to be at T=0. The q=6 case is currently under investigation. Preliminary results show no transition at finite temperature.
1 R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86
(1987). 2 S. Rahman, E. Rush and R.H.Swendsen, Phys. Rev. B
58, 9125 (1998).
[LC42.05] Phase transitions in the q=5 state Potts model in an uniform external magnetic field.
Nilay Roy, David Landau (Center for Simulational Physics, University of Georgia, Athens, Georgia.)
The q=5 state ferromagnetic Potts Model in d=2 dimensions
has been the focus of investigation for many years. There
has been no attempt yet to fully understand the phase
behaviour of this model in the entire space of positive and
negative uniform external magnetic field which couples to
only one of the states. In the absence of a magnetic field
there is known to be a weak first order transition. However
the nature of transitions in a magnetic field is still
unclear. We use a hybrid Metropolis-Cluster algorithm with
techniques of reweighting and finite size scaling including
the Lee-Kosterlitz method to determine the location of the
phase boundary for both positive and negative fields. These
results are compared to that obtained using a third order
variational cumulant expansion technique.
[LC42.06] Effects of quenched disorder in the two-dimensional Potts model
Ricardo Paredes V. (IVIC, Venezuela), Johnny Valbuena (PDVSA, Intevep)
Motivated by recent experiments on phase behavior of systems
confined in porous media, we have studied the effect of
randomness on the nature of the phase transition in the two
dimensional Potts model. To model the effects of the porous
matrix we introduce a random distribution of couplings,
\cal P (J_ij)=p\delta (J_ij-J_1)+(1-p)\delta
(J_ij-J_2), in the q state Potts Hamiltonian. An
extensive Monte Carlo study is made on this system for
q=5. We studied two different cases of disorder (a)
J_1/J_2\to \infty and p=0.8 and (b) J_1/J_2=10 and
p=0.5. We observed, in both cases, that the weak first
order transition that appears in the pure case, changes two
a second order transition. A finite size scaling analysis
shows that, the correlation length exponent, \nu is close
to 1 and the best fit to the dependence of the specific
heat on system size is logarithmic. This suggests that both
cases belong to the universality class of the Ising model.
In contrast, the magnetic exponents point to a different
universality class.
[LC42.07] Critical dynamics of Heisenberg antiferromagnets: simulation versus theory and experiment
Shan-Ho Tsai, Alex Bunker, D. P. Landau (Center for Simulational Physics, The University of Georgia, Athens, GA 30602 USA)
We use spin-dynamics techniques to perform large-scale simulations of the dynamic behavior of the classical Heisenberg antiferromagnet in three dimensions. Time-evolutions of spin configurations are determined numerically from coupled equations of motion for individual spins using a new algorithm(M. Krech, Alex Bunker, D. P. Landau, Computer Phys. Comm. 111), 1 (1998) based on Suzuki-Trotter decompositions of exponential operators. The dynamic structure factor is calculated from the space- and time-displaced spin-spin correlation function. We study the width of spin-wave peaks as a function of the wave-vector for temperature below but close to the critical value. Our results are compared with some theoretical predictions which in turn disagree with very recent experimental results. Our preliminary data indicate that hydrodynamic theory only holds for wave-vectors which are far from the Brillouin zone boundary.
[LC42.08] Nonequilibrium relaxation of the 3dim \pm J EA model
Fugao Wang (Center for Simulational Physics, University of Georgia, Athens, GA30602)
By studying correlation functions of two replicas with the damage spreading method, we estimate the spin glass transition temperature of the 3D \pm J EA model using Campbell's criterion D(T_\mbox \scriptsize g,t \rightarrow \infty)=1/2 or q_\mbox \scriptsize d^\mbox \scriptsize AB(T_\mbox \scriptsize g, t \rightarrow \infty)=0, where D and q_\mbox \scriptsize d^\mbox \scriptsize AB are the Hamming distance and the overlap of the two replicas, respectively. The two replicas evolve with the same sequence of the random numbers and with initial spin-configurations related to each other by S_i^\mbox \scriptsize A(t=0) =-S_i^\mbox\scriptsize B(t=0) (for all i). From our simulational data for system sizes up to L=64, the spin glass transition temperature is estimated as T_\mbox \scriptsize g=1.18\pm 0.06. The replica simulational method and the damage spreading method are compared with each other. The relaxation exponents of the magnetization and the correlation functions are also estimated in the vicinity of T_\mbox\scriptsize g on a large lattice and the dynamical critical exponent z is estimated as z\simeq 6.0. [0.2cm]
[LC42.09] Monte Carlo study of phase separation in Si_1-xGe_x with the Stillinger-Weber potential
John Lees, D. P. Landau (Center for Simulational Physics, University of Georgia)
We investigate the structural properties of silicon-germanium binary alloys using Monte Carlo simulations. We employ the Stillinger-Weber model(F.H. Stillinger and T.A. Weber, Phys. Rev. B 32), 5262 (1985). at constant pressure allowing lattice vibrations in the canonical ensemble. In the model, we use a LxLxL box with a distortable lattice where the box volume fluctuates under constant pressure. Atoms are allowed to attempt small moves off-lattice and exchange position with a nearest neighbor atom of unlike type using Metropolis rejection criteria. We study the phase separation of the binary mixture into Si-rich and Ge-rich domains in the two-phase coexistence region. The structure factor is obtained under different temperatures and concentrations of the constituents. We then compare those results to those obtained for the rigid lattice model.
[LC42.10] High-precision Monte Carlo study of the 3D XY model
Peter Olsson (UmeåUniversity, Sweden)
The 3D XY model with spin interaction of the Villain type is studied by means of extensive Monte Carlo simulations. The critical behavior is extracted by finite size scaling analysis with corrections to scaling of the fourth-order cumulant for system sizes 8^3 through 64^3. The benefit from the study is twofold. First, a systematic way to perform this kind of analyses is suggested. This includes ideas on how to decide which data should be used in the attempted data collapse. Second, the critical temperature as well as the universal values for the exponents and the fourth-order cumulant at criticality are obtained with high precision.
[LC42.11] Dynamical Simulation of a Phase Transition in a Model Gravitating System
Bruce Miller, Paige Youngkins (Texas Christian University)
We present recent developments in the study of an interacting gravitational system of concentric, spherical, mass shells. The existence of two distinct phases is predicted from mean field theory. We report the results of dynamical simulations carried out under the conditions of constant energy (microcanonical ensmble), constant temperature (canonical ensemble with energy exchange with a reservoir) and constant chemical potential (grand canonical ensemble with both particle and energy exchange). Both simulations and theory predict that, in contrast to chemical systems, the physics of each environment is different. We describe the simulation results and compare them with the theoretical predictions.