The Statistics of -Statistics.

Almost two decades ago, Ernesto P. Borges and Bruce M. Boghosian embarked on the intricate task of composing a manuscript to honor the profound contributions of Constantino Tsallis to the realm of statistical physics, coupled with a concise exploration of -Statistics.

A generalization of the standard map and its statistical characterization.

From the statistical mechanical point of view, area-preserving maps have great potential and importance. These maps exhibit chaotic and regular behavior separately or together in the available phase space as the control parameter changes. Several works on these maps, e.

An economically realistic asset exchange model.

Despite their highly idealized nature, certain agent-based models of asset exchange, proposed for the most part by physicists and mathematicians, have been shown to exhibit remarkable agreement with empirical wealth distribution data. While this agre- ement is comforting, there is widespread sentiment that further progress will require a detailed under- standing of the connection between these idealized models and the more realistic microeconomic models of exchange used by economists. In this paper, we examine that connection for a three-parameter asset exchange model, the Affine Wealth Model (AWM), that has demonstrated fraction-of-a-per cent agreement with empirical wealth data.

The lattice Fokker-Planck equation for models of wealth distribution.

Recent work on agent-based models of wealth distribution has yielded nonlinear, non-local Fokker-Planck equations whose steady-state solutions describe empirical wealth distributions with remarkable accuracy using only a few free parameters. Because these equations are often used to solve the 'inverse problem' of determining the free parameters given empirical wealth data, there is much impetus to find fast and accurate methods of solving the 'forward problem' of finding the steady state corresponding to given parameters. In this work, we derive and calibrate a lattice Boltzmann equation for this purpose.

A New Pathology in the Simulation of Chaotic Dynamical Systems on Digital Computers.

Systematic distortions are uncovered in the statistical properties of chaotic dynamical systems when represented and simulated on digital computers using standard IEEE floating-point numbers. This is done by studying a model chaotic dynamical system with a single free parameter β, known as the generalized Bernoulli map, many of whose exact properties are known. Much of the structure of the dynamical system is lost in the floating-point representation.

Kinetics of wealth and the Pareto law.

An important class of economic models involve agents whose wealth changes due to transactions with other agents. Several authors have pointed out an analogy with kinetic theory, which describes molecules whose momentum and energy change due to interactions with other molecules. We pursue this analogy and derive a Boltzmann equation for the time evolution of the wealth distribution of a population of agents for the so-called Yard-Sale Model of wealth exchange.

Robust numerical method for integration of point-vortex trajectories in two dimensions.

The venerable two-dimensional (2D) point-vortex model plays an important role as a simplified version of many disparate physical systems, including superfluids, Bose-Einstein condensates, certain plasma configurations, and inviscid turbulence. This system is also a veritable mathematical playground, touching upon many different disciplines from topology to dynamic systems theory. Point-vortex dynamics are described by a relatively simple system of nonlinear ordinary differential equations which can easily be integrated numerically using an appropriate adaptive time stepping method.

Unstable periodic orbits in the Lorenz attractor.

We apply a new method for the determination of periodic orbits of general dynamical systems to the Lorenz equations. The accuracy of the expectation values obtained using this approach is shown to be much larger and have better convergence properties than the more traditional approach of time averaging over a generic orbit. Finally, we discuss the relevance of the present work to the computation of unstable periodic orbits of the driven Navier-Stokes equations, which can be simulated using the lattice Boltzmann method.

New variational principles for locating periodic orbits of differential equations.

We present new methods for the determination of periodic orbits of general dynamical systems. Iterative algorithms for finding solutions by these methods, for both the exact continuum case, and for approximate discrete representations suitable for numerical implementation, are discussed. Finally, we describe our approach to the computation of unstable periodic orbits of the driven Navier-Stokes equations, simulated using the lattice Boltzmann equation.

Real science at the petascale.

We describe computational science research that uses petascale resources to achieve scientific results at unprecedented scales and resolution. The applications span a wide range of domains, from investigation of fundamental problems in turbulence through computational materials science research to biomedical applications at the forefront of HIV/AIDS research and cerebrovascular haemodynamics. This work was mainly performed on the US TeraGrid 'petascale' resource, Ranger, at Texas Advanced Computing Center, in the first half of 2008 when it was the largest computing system in the world available for open scientific research.

Towards the simplest hydrodynamic lattice-gas model.

It has been known since 1986 that it is possible to construct simple lattice-gas cellular automata whose hydrodynamics are governed by the Navier-Stokes equations in two dimensions. The simplest such model heretofore known has six bits of state per site on a triangular lattice. In this work, we demonstrate that it is possible to construct a model with only five bits of state per site on a Kagome lattice.

Vortex core identification in viscous hydrodynamics.

We describe a software package designed for the investigation of topological fluid dynamics with a novel algorithm for locating and tracking vortex cores. The package is equipped with modules for generating desired vortex knots and links and evolving them according to the Navier-Stokes equations, while tracking and visualizing them. The package is parallelized using a message passing interface for a multiprocessor environment and makes use of a computational steering library for dynamic user intervention.

Entropic lattice Boltzmann model for Burgers's equation.

Entropic lattice Boltzmann models are discrete-velocity models of hydrodynamics that possess a Lyapunov function. This feature makes them useful as nonlinearly stable numerical methods for integrating hydrodynamic equations. Over the last few years, such models have been successfully developed for the Navier-Stokes equations in two and three dimensions, and have been proposed as a new category of subgrid model of turbulence.

Lattice gas simulations of dynamical geometry in one dimension.

We present numerical results obtained using a lattice gas model with dynamical geometry. The (irreversible) macroscopic behaviour of the geometry (size) of the lattice is discussed in terms of a simple scaling theory and obtained numerically. The emergence of irreversible behaviour from the reversible microscopic lattice gas rules is discussed in terms of the constraint that the macroscopic evolution be reproducible.

Galilean-invariant lattice-Boltzmann models with H theorem.

We demonstrate that the requirement of Galilean invariance determines the choice of H function for a wide class of entropic lattice-Boltzmann models for the incompressible Navier-Stokes equations. The required H function has the form of the Burg entropy for D=2, and of a Tsallis entropy with q=1-(2/D) for D>2, where D is the number of spatial dimensions. We use this observation to construct a fully explicit, unconditionally stable, Galilean-invariant, lattice-Boltzmann model for the incompressible Navier-Stokes equations, for which attainable Reynolds number is limited only by grid resolution.