High precision single-cluster Monte Carlo measurement of the critical exponents of the classical 3D Heisenberg model

We report measurements of the critical exponents of the classical three-dimensional Heisenberg model on simple cubic lattices of size $L^3$ with $L$ = 12, 16, 20, 24, 32, 40, and 48. The data was obtained from a few long single-cluster Monte Carlo simulations near the phase transition. We compute high precision estimates of the critical coupling $K_c$, Binder's parameter $U^* and the critical exponents$\nu,\beta / \nu, \eta$, and$\alpha / \nu$, using extensively histogram reweighting and optimization techniques that allow us to keep control over the statistical errors. Measurements of the autocorrelation time show the expected reduction of critical slowing down at the phase transition as compared to local update algorithms. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors in finite-size scaling analyses.Comment: 4 pages, (contribution to the Lattice92 proceedings) 1 postscript file as uufile included. Preprints FUB-HEP 21/92 and HLRZ 89/92. (note: first version arrived incomplete due to mailer problems

Finite-Size Scaling Study of the Three-Dimensional Classical Heisenberg Model

We use the single-cluster Monte Carlo update algorithm to simulate the three-dimensional classical Heisenberg model in the critical region on simple cubic lattices of size $L^3$ with $L=12, 16, 20, 24, 32, 40$, and $48$. By means of finite-size scaling analyses we compute high-precision estimates of the critical temperature and the critical exponents, using extensively histogram reweighting and optimization techniques. Measurements of the autocorrelation time show the expected reduction of critical slowing down at the phase transition. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors in finite-size scaling analyses.Comment: 9 pages, FUB-HEP 9/92, HLRZ Preprint 56/92, August 199

High-Temperature Series Analyses of the Classical Heisenberg and XY Model

Although there is now a good measure of agreement between Monte Carlo and high-temperature series expansion estimates for Ising ($n=1$) models, published results for the critical temperature from series expansions up to 12{\em th} order for the three-dimensional classical Heisenberg ($n=3$) and XY ($n=2$) model do not agree very well with recent high-precision Monte Carlo estimates. In order to clarify this discrepancy we have analyzed extended high-temperature series expansions of the susceptibility, the second correlation moment, and the second field derivative of the susceptibility, which have been derived a few years ago by L\"uscher and Weisz for general $O(n)$ vector spin models on $D$-dimensional hypercubic lattices up to 14{\em th} order in $K \equiv J/k_B T$. By analyzing these series expansions in three dimensions with two different methods that allow for confluent correction terms, we obtain good agreement with the standard field theory exponent estimates and with the critical temperature estimates from the new high-precision MC simulations. Furthermore, for the Heisenberg model we reanalyze existing series for the susceptibility on the BCC lattice up to 11{\em th} order and on the FCC lattice up to 12{\em th} order.Comment: 15 pages, Latex, 2 PS figures not included. FUB-HEP 18/92 and HLRZ 76/9

A stochastic Lagrangian representation of the 3-dimensional incompressible Navier-Stokes equations

In this paper we derive a representation of the deterministic 3-dimensional Navier-Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for the Euler equations of ideal fluids is used to recover the velocity field. This method admits a self-contained proof of local existence for the nonlinear stochastic system, and can be extended to formulate stochastic representations of related hydrodynamic-type equations, including viscous Burgers equations and LANS-alpha models.Comment: v4: Minor corrections to bibliography, and final version that will apear in CPAM. v3: Minor corrections to the algebra in the last section. v2: Minor changes to introduction and refferences. 14 pages, 0 figure

Maintenance Models for Systems subject to Measurable Deterioration

Complex engineering systems such as bridges, roads, flood defence structures, and power pylons play an important role in our society. Unfortunately such systems are subject to deterioration, meaning that in course of time their condition falls from higher to lower, and possibly even to unacceptable, levels. Maintenance actions such as inspection, local repair and replacement should be done to retain such systems in or restore them to acceptable operating conditions. After all, the economic consequences of malfunctioning infrastructure systems can be huge. In the life-cycle management of engineering systems, the decisions regarding the timing and the type of maintenance depend on the temporal uncertainty associated with the deterioration. Hence it is of importance to model this uncertainty. In the literature, deterioration models based on Brownian motion and gamma process have had much attention, but a thorough comparison of these models lacks. In this thesis both models are compared on several aspects, both in a theoretical as well as in an empirical setting. Moreover, they are compared with physical process models, which can capture structural insights into the underlying process. For the latter a new framework is developed to draw inference. Next, models for imperfect maintenance are investigated. Finally, a review is given for systems consisting of multiple components

Finite size effects on measures of critical exponents in d=3 O(N) models

We study the critical properties of three-dimensional O(N) models, for N=2,3,4. Parameterizing the leading corrections-to-scaling for the $\eta$ exponent, we obtain a reliable infinite volume extrapolation, incompatible with previous Monte Carlo values, but in agreement with $\epsilon$-expansions. We also measure the critical exponent related with the tensorial magnetization as well as the $\nu$ exponents and critical couplings.Comment: 12 pages, 2 postscript figure

Parallelized Hybrid Monte Carlo Simulation of Stress-Induced Texture Evolution

A parallelized hybrid Monte Carlo (HMC) methodology is devised to quantify the microstructural evolution of polycrystalline material under elastic loading. The approach combines a time explicit material point method (MPM) for the mechanical stresses with a calibrated Monte Carlo (cMC) model for grain boundary kinetics. The computed elastic stress generates an additional driving force for grain boundary migration. The paradigm is developed, tested, and subsequently used to quantify the effect of elastic stress on the evolution of texture in nickel polycrystals. As expected, elastic loading favors grains which appear softer with respect to the loading direction. The rate of texture evolution is also quantified, and an internal variable rate equation is constructed which predicts the time evolution of the distribution of orientations.Comment: 20 pages, 8 figure

Three-State Anti-ferromagnetic Potts Model in Three Dimensions: Universality and Critical Amplitudes

We present the results of a Monte Carlo study of the three-dimensional anti-ferromagnetic 3-state Potts model. We compute various cumulants in the neighbourhood of the critical coupling. The comparison of the results with a recent high statistics study of the 3D XY model strongly supports the hypothesis that both models belong to the same universality class. From our numerical data for the anti-ferromagnetic 3-state Potts model we obtain for the critical coupling \coup_c=0.81563(3), and for the static critical exponents $\gamma /\nu=1.973(9)$ and $\nu=0.664(4)$.Comment: 18pages + 3figures, KL-TH-94/5 , CERN-TH.7183/9

Critical Behaviour of the 3D XY-Model: A Monte Carlo Study

We present the results of a study of the three-dimensional $XY$-model on a simple cubic lattice using the single cluster updating algorithm combined with improved estimators. We have measured the susceptibility and the correlation length for various couplings in the high temperature phase on lattices of size up to $L=112$. At the transition temperature we studied the fourth-order cumulant and other cumulant-like quantities on lattices of size up to $L=64$. From our numerical data we obtain for the critical coupling \coup_c=0.45420(2), and for the static critical exponents $\gamma /\nu=1.976(6)$ and $\nu=0.662(7)$.Comment: 24 pages (4 PS-Figures Not included, Revtex 3.O file), report No.: CERN-TH.6885/93, KL-TH-93/1