Hamiltonian mechanics on discrete manifolds

The mathematical/geometric structure of discrete models of systems, whether these models are obtained after discretization of a smooth system or as a direct result of modeling at the discrete level, have not been studied much. Mostly one is concerned regarding the nature of the solutions, but not much has been done regarding the structure of these discrete models. In this paper we provide a framework for the study of discrete models, speci?cally we present a Hamiltonian point of view. To this end we introduce the concept of a discrete calculus

Classical Tensors and Quantum Entanglement I: Pure States

The geometrical description of a Hilbert space asociated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.Comment: 16 pages, 1 figure, to appear in Int. J. Geom. Meth. Mod. Phy

Statistics and Nos\'e formalism for Ehrenfest dynamics

Quantum dynamics (i.e., the Schr\"odinger equation) and classical dynamics (i.e., Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well defined Poisson bracket allows to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of a Ehrenfest system, it is straightforward to extend the formalism of Nos\'e (invented to do constant temperature Molecular Dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.Comment: 28 pages, 1 figure. Published version. arXiv admin note: substantial text overlap with arXiv:1010.149

Investigar e inovar na educação em ciências para um futuro sustentável

No pico de uma real situação de emergência planetária, a educação torna-se a melhor aliada de uma luta global com vista a um desenvolvimento sustentável. Para concretizar a Década da Educação para um Futuro Sustentável, a investigação em educação em ciências e a correspondente inovação na formação de professores e no ensino, apresentam-se entre os contributos mais fortes, amplos e eficazes. Parte do nosso contributo, que se apresenta neste artigo, tem passado pelo desenvolvimento de alguns estudos situados no quadro teórico que sustenta a educação CTS e assentes em temáticas centrais para a educação para a sustentabilidade ambiental: os transportes e a mobilidade, o uso da água, a fome no mundo, a preservação da biodiversidade. A aposta tem-se dirigido para o ensino nos primeiros anos através do desenho de propostas didácticas validadas por especialistas e em sala de aula e utilizadas quer no ensino quer como ferramentas de formação inicial e contínua de professores