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Monte Carlo simulations can be used to explore many unsolved systems in Statistical Mechanics. The behavior of two Monte Carlo Algorithms, the Metropolis Algorithm and Wolff Algorithm, was analyzed through simulations of the ferromagnetic Ising Model, and then these simulations were applied to studies of the antiferromagnetic Ising Model and Hard Squares Model. Further study would include looking at the Hard Hexagon system, the Hard-Constraint model of Hard Hexagons, and the Ising Model on a three-dimensional antiferromagnetic face-centered-cubic lattice. Study of the latter model might elucidate behavior around "super-degenerate points" that is believed to exist in this model. The studies done so far have encountered useful results. The Metropolis Algorithm was found to provide a good qualitative description of each system studied, although it often returned inaccurate individual measurements. The Wolff algorithm was found to yield excellent results with shorter run-times, but could not give good values for the order parameter at zero field. A Metropolis algorithm was used to explore the Hard Squares system, which is not analytically solvable and has only been studied through series expansions. The results that we obtained seem to duplicate the results obtained through series expansion. Further study along these lines promises to yield useful results.
This book is based on research carried out by the author in close collabora tion with a number of colleagues. In particular, I wish to thank Per Bak, A. John Berlinsky, Hans C. Fogedby, Barry Frank, S. 1. Knak Jensen, David Mukamel, David Pink, and Martin Zuckermann for fruitful and extremely stimulating cooperation. It is a pleasure for me to note that active interaction with most of these colleagues is still continuing. The work has been performed at several different institutions, notably the Department of Chemistry, Aarhus University, Denmark, and the Depart ment of Physics, University of British Columb~a, Canada. I wish to thank the Department of Chemistry at Aarhus University for providing me with splen did research facilities over the years. From May 1980 to August 1981, I visited the Department of Physics at the University of British Columbia and I would like to express my sincere gratitude to members ofthe department for provi ding me with excellent working conditions. My special thanks are due to Professor Myer Bloom who introduced me to the field of phase transitions in biological membranes and in whose biomembrane group I found an extre mely stimulating scientific atmosphere happily married with a most agreeable social climate. During the last two years when a major part ofthis work was carried out, I was supported by AlS De Danske Spritfabrikker through their Jubilreumsle gat of 1981. Their support is gratefully acknowledged.
This unique volume provides a comprehensive overview of exactly solved models in statistical mechanics by looking at the scientific achievements of F Y Wu in this and related fields, which span four decades of his career. The book is organized into topics ranging from lattice models in condensed matter physics to graph theory in mathematics, and includes the author's pioneering contributions. Through insightful commentaries, the author presents an overview of each of the topics and an insider's look at how crucial developments emerged. With the inclusion of important pedagogical review articles by the author, Exactly Solved Models is an indispensable learning tool for graduate students, and an essential reference and source book for researchers in physics and mathematics as well as historians of science.
We have studied the finite-size behavior at magnetic phase transitions by using extensive Monte Carlo simulations. For the second-order transition in the simple cubic Ising model, we have investigated the critical behavior by implementing the Wolff cluster flipping algorithm and data analysis with histogram reweighting in quadruple precision arithmetic. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, we have obtained the critical quantities with precision that exceeds all previous Monte Carlo estimates. For the first-order ``spin-flop" transition in the 3D anisotropic Heisenberg antiferromagnet in an external field, we have explored the finite-size behavior of the transition between the Ising-like antiferromagnetic state and the canted, $XY$-like state. Finite-size scaling for a first-order phase transition where a continuous symmetry is broken is developed using an approximation of Gaussian probability distributions with a phenomenological ``degeneracy" factor, $q$, included. Our theory yields $q = pi$, and it predicts that for large linear dimension $L$ the field dependence of all moments of the order parameters as well as the fourth-order cumulants exhibit universal intersections, where the values of these intersections can be expressed in terms of the factor $q$. The agreement between our theory and high-resolution multicanonical simulation data implies a heretofore unknown universality can be invoked for first-order phase transitions.
"This thesis focuses on the effects of both correlated and non-correlated disorder on non-equilibrium phase transitions, specifically those found in the d-dimensional contact process. These effects are studied by means of extensive Monte-Carlo simulations. The scaling behavior of various parameters is evaluated for both cases, and the results are compared with theory. For the correlated disorder case, the stationary density in the vicinity of the transition is also examined, and found to be smeared. The behavior in both cases can be understood as the results of rare regions where the system is locally free of disorder. For point-like defects, i.e., uncorrelated disorder, the rare regions are of finite size and cannot undergo a true phase transition. Instead, they fluctuate slowly which gives rise to Griffiths effects. In contrast, if the rare regions are infinite in at least one dimension, a stronger effect occurs: each rare region can independently undergo the phase transition and develop a nonzero steady state density. This leads to a smearing of the global transition"--Abstract, leaf iv.