Tarcísio M. Rocha Filho

Lie Symmetries of Fokker-Planck Equations with Logarithmic diffusion and Drift Terms

In this work we address the problem of analyzing the symmetries of Fokker-Planck equations with logarithmic coefficients. Starting from a nonlinear reaction-diffusion equation with known symmetry algebra, we solve the inverse problem, namely, we find all equations in a given class that are invariant under this symmetry algebra. The class we consider is that of nonlinear Fokker-Planck equations for which the source term is a monomial in the distribution.

A novel approach for the stability problem in non-linear dynamical systems

We present a methodology for the determination of sufficient conditions for the existence of a Lyapunov function in a general class of non-linear dynamical systems. The algorithm can be applied in the cases where the system parameters are numerically specified or not. The numerical algorithm involves the resolution of a linear programming problem. The algebraic version is implemented using the MAPLE programming system in the package Lyapunov.

Solving the Vlasov equation for one-dimensional models with long range interactions on a GPU

Abstract We present a GPU parallel implementation of the numeric integration of the Vlasov equation in one spatial dimension based on a second order time-split algorithm with a local modified cubic-spline interpolation. We apply our approach to three different systems with long-range interactions: the Hamiltonian mean field, ring and self-gravitating sheet models. Speedups and accuracy for each model and different grid resolutions are presented.

Molecular dynamics for long-range interacting systems on graphic processing units

We present implementations of a fourth-order symplectic integrator on graphic processing units for three

Microcanonical Monte Carlo Study of One Dimensional Self-Gravitating Lattice Gas Models

N

Equivalence between nonlinear dynamical systems and urn processes

-body models with long-range interactions of general interest: the Hamiltonian Mean Field, Ring and two-dimensional self-gravitating models. We discuss the algorithms, speedups and errors using one and two GPU units. Speedups can be as high as 140 compared to a serial code, and the overall relative error in the total energy is of the same order of magnitude as for the CPU code. The number of particles used in the tests range from 10,000 to 50,000,000 depending on the model.

Performance Evaluation of two Parallel Programming Paradigms Applied to the Symplectic Integrator Running on COTS PC Cluster

Abstract In this study we present a microcanonical Monte Carlo investigation of one dimensional (1 − d) self-gravitating toy models. We study the effect of hard-core potentials and compare to the results obtained with softening parameters and also the effect of the topology on these systems. In order to study the effect of the topology in the system we introduce a model with the symmetry of motion in a line instead of a circle, which we denominate as 1 /r model. The hard-core particle potential introduces the effect of the size of particles and, consequently, the effect of the density of the system that is redefined in terms of the packing fraction of the system. The latter plays a role similar to the softening parameter ϵ in the softened particles’ case. In the case of low packing fractions both models with hard-core particles show a behavior that keeps the intrinsic properties of the three dimensional gravitational systems such as negative heat capacity. For higher values of the packing fraction the ring model behaves as the potential for the standard cosine Hamiltonian Mean Field model while for the 1 /r model it is similar to the one-dimensional systems. In the present paper we intend to show that a further simplification level is possible by introducing the lattice-gas counterpart of such models, where a drastic simplification of the microscopic state is obtained by considering a local average of the exact N-body dynamics.

The Lotka-Volterra canonical format

An equivalence is shown between a large class of deterministic dynamical systems and a class of stochastic processes, the balanced urn processes. These dynamical systems are governed by quasi-polynomial differential systems that are widely used in mathematical modeling while urn processes are actively studied in combinatorics and probability theory. The presented equivalence extends a theorem by Flajolet et al. (Flajolet, Dumas and Puyhaubert Discr. Math. Theor. Comp. Sc. AG - 2006, DMTCS Proceedings) already establishing an isomorphism between urn processes and a particular class of differential systems with monomial vector fields. The present result is based on the fact that such monomial differential systems are canonical forms for more general dynamical systems.

On the statistical interpretation of generalized entropies

There are two popular parallel programming paradigms available to high performance computing users such as engineering and physics professionals: message passing and distributed shared memory. It is interesting to have a comparative evaluation of these paradigms to choose the most adequate one. In this work, we present a performance comparison of these two programming paradigms using a computational physics problem as a case study. The self-gravitating ring model (Hamiltonian mean field model) for N particles is extensively studied in the literature as a simplified model for long range interacting systems in Physics. We parallelized and evaluated the performance of a simulation that uses the symplectic integrator to model an N particle system. From the obtained results it is possible to observe that message passing implementation of the symplectic integrator presents better results than distributed shared memory implementation.