International Journal of Modern Physics C

In this work we study the computing capabilities as well as some dynamical properties of an automaton called M4R. This automaton corresponds to the mixing of the energy profiles of two independent copies of the Q2R automaton with frustrations. We associate to each copy of a Q2R an equivalent automaton M2R, which, with the Margoluos neighborhood, exhibits the local changes of the Q2R energy.¹ By doing so we generalize the dynamics by upgrading M2R according to four partitions of the lattice. This new dynamics — called M4R — is based on a local rule which corresponds to the local energy change of two independent copies of Q2R.

The M4R model is reversible and conservative (magnetization is constant in time) and it has properties of a discrete billiard (as some of the hydrodynamics discrete versions of Navier-Stokes models).

Moreover, this automaton has powerful computing capabilities. In fact, by using some special configurations of M4R, we exhibit universal gates and register that allow us to code any algorithm.

An algorithm for calculating fractal dimensions by the box counting method is described. Parallel implementation of the algorithm is presented.

We have performed high precision finite size scaling measurements of the critical behavior of the Ising model on dynamically triangulated surfaces of genus 0.

The results are in perfect agreement with the matrix model predictions.^1,2 Values of the critical exponents equal to the regular lattice 2D Ising model are excluded by our error bars.

In order to accumulate sufficient statistics we used cluster methods which eliminated critical slowing down to a great extent.

Phase diagram of a binary system with quenched impurities has been studied with Monte Carlo computer simulation technique. Use of Swendsen-Wang algorithm makes it possi-ble to explore the vicinity of percolation transition which was difficult to explore with the traditional Metropolis method. For small vacancy concentration υ(υ=1−p, where p is the spin concentration) the critical temperature, T_c, decreases linearly with υ consistent with earlier results on this system. For larger values of v departure from linearity is observed with T_c decreasing to zero sharply near the percolation threshold. An estimate of the critical exponent describing this sharp drop of T_c is also given.

The derivatives of a Boolean function are defined up to any order. The Taylor and MacLaurin expansions of a Boolean function are thus obtained. The last corresponds to the ring sum expansion (RSE) of a Boolean function, and is a more compact form than the usual canonical disjunctive form. For totalistic functions the RSE allows the saving of a large number of Boolean operations. The algorithm has natural applications to the simulations of cellular automata using the multi-site coding technique. Several already published algorithms are analyzed, and expressions with fewer terms are generally found.

A program to compute the gluon self-energy diagram in class III nonrelativistic gauges has been constructed. It succeeded to show that axial gauges do not exhibit nonlocal counterterms. The high complexity of the calculations, the need to be able to control intermediate steps of calculations and the necessity to work inside a limited memory of a few Mbytes led us to give it special features which may serve outside the specific application for which it has been designed.

The structure of the program and its salient working features are described. It is divided into four modules, which are almost independent except that they are linked by their respective output. Files containing intermediate results of calculations are automatically generated. The amount of algebraic manipulations has been greatly reduced thanks to a careful choice of the representation of the expression adapted to each stage of the calculation. The adaptative character of this program is discussed as well as the generalization it needs to cover other types of Feynman diagram calculations.

In Path Integral Monte Carlo simulations the systems partition function is mapped to an equivalent classical one at the expense of a temperature-dependent Hamiltonian with an additional imaginary time dimension. As a consequence the standard relation linking the heat capacity C_v to the energy fluctuations, <E²>−<E>², which is useful in standard classical problems with temperature-independent Hamiltonian, becomes invalid. Instead, it gets replaced by the general relation

We investigate numerically totalistic cellular automata rules in four and five dimensions that have been recently proposed by Chaté and Manneville and that show periodic or quasiperiodic time sequences in the magnetization. We show how to implement such rules fast on a Connection Machine. We confirm the proposed behavior by analyzing the time and size dependence of the distance from the attractor. Also the dependence of the initial configuration, the behavior of very small systems and mean-field calculations are presented.

We report some selected recent developments in the finite-size scaling theory of critical phenomena occurring in systems with strong spatial anisotropies. Such systems are characterized by correlation lengths divergent with different exponents (ν_⊥, ν_||) along different directions. Attention is focused on the driven diffusive lattice gas that exhibits a second order nonequilibrium phase transition. We present in detail the phenomenology and its comparison with computer simulation. Novel features of finite-size effects in anisotropic nonequilibrium systems are emphasized.

The mathematical formalisms and computer enumerations of two polymeric models: trails and silhouettes are presented. Trails have been quite extensively studied, but their equivalence class, silhouettes are less familiar. By drawing parallelism from n→0 magnetic analog, the tricritical properties of silhouettes are studied both analytically and numerically. The tricriticality of trails, which is inaccessible by a renormalization group analysis in ε=4−D dimensions, can therefore only be investigated numerically. The tricritical exponents thus obtained indicate that trails belong to a distinct universality class from silhouettes.