International Journal of Modern Physics C

This article presents new procedures in REDUCE language using the EXCALC package (adapted for IBM-PC machines) for algebraic programming in the Hamiltonian formulation of inflationary models based on a scalar field non-minimally coupled with gravity. The (3+1)-dimensional split of the action functional of the model and the constraint and dynamic equations are obtained. These equations are transposed in computer procedures. The results obtained after processing some examples of initial space-time models are presented here.

A class of numerical algorithms for solving the classical equations of motion in lattice gauge field theories which exactly fulfill the constraints imposed by the unitarity of the local group elements, by the local color charge conservation (Gauss law) and by the total energy conservation is constructed. The performance of these constrained algorithms is comparatively discussed.

Four different kinds of laminar flows between two parallel plates are investigated using the Lattice Boltzmann Method (LBM). The LBM accuracy is estimated in two cases using numerical fits of the parabolic velocity profiles and the kinetic energy decay curves, respectively. The error relative to the analytical kinematic viscosity values was found to be less than one percent in both cases. The LBM results for the unsteady development of the flow when one plate is brought suddenly at a constant velocity, are found in excellent agreement with the analytical solution. Because the classical Schlichting’s approximate solution for the entrance-region flow is not valid for small Reynolds numbers, a Finite Element Method solution was used in order to check the accuracy of the LBM results in this case.

We introduce a Monte Carlo method for the simulation of spin models with ferromagnetic long-range interactions in which the amount of time per spin-flip operation is independent of the system size, in spite of the fact that the interactions between each spin and all other spins are taken into account. We work out two algorithms for the q-state Potts model and discuss the generalization to systems with other interactions and to O(n) models. We illustrate the method with a simulation of the mean-field Ising model, for which we have also analytically calculated the leading finite-size correction to the dimensionless amplitude ratio <m²>²/<m⁴> at the critical temperature.

This paper describes a two-dimensional computer simulation of solid fracture that allows the body and the fragments to be followed well beyond the point of simple crack formation. The model is based on discrete particle computer simulations used for studying granular flows. Here, macroscopic polygonal solid are constructed by “gluing” together small elements. Depending on the stress conditions the glued bonds between the elements can respond elastically, undergo plastic failure or break, allowing cracks to propagate across the macroscopic particle along the boundaries between their microscopic constituents. In essence, this process creates a simulated material upon which breakage occurs. Several element shapes have been studied.

This paper uses the model described in Ref. 1 to simulate fracture in many simple systems with the goal of evaluating the advantages and deficiencies of the model. The examples include compressive failure of a rectangular sample, four-point shear failure of a beam and the impact of particles with a plate and binary impacts of particles. Where possible, the simulated results seem to be in good agreement with typical experimental results. Finally, a simulation of ball-milling, which involves the flow and fracture of many particles is shown to demonstrate the overall utility of the model.

The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.