A note on quantum chaology and gamma approximations to eigenvalue spacings for infinite random matrices

Quantum counterparts of certain simple classical systems can exhibit chaotic behaviour through the statistics of their energy levels and the irregular spectra of chaotic systems are modelled by eigenvalues of infinite random matrices. We use known bounds on the distribution function for eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real symmetric matrices and show that gamma distributions, which have an important uniqueness property, can yield an approximation to the GOE distribution. That has the advantage that then both chaotic and non chaotic cases fit in the information geometric framework of the manifold of gamma distributions, which has been the subject of recent work on neighbourhoods of randomness for general stochastic systems. Additionally, gamma distributions give approximations, to eigenvalue spacings for the Gaussian unitary ensemble (GUE) of infinite random hermitian matrices and for the Gaussian symplectic ensemble (GSE) of infinite random hermitian matrices with real quaternionic elements, except near the origin. Gamma distributions do not precisely model the various analytic systems discussed here, but some features may be useful in studies of qualitative generic properties in applications to data from real systems which manifestly seem to exhibit behaviour reminiscent of near-random processes.Comment: 9 pages, 5 figures, 2 tables, 27 references. Updates version 1 with data and references from feedback receive

Low error measurement-free phase gates for qubus computation

We discuss the desired criteria for a two-qubit phase gate and present a method for realising such a gate for quantum computation that is measurement-free and low error. The gate is implemented between qubits via an intermediate bus mode. We take a coherent state as the bus and use cross-Kerr type interactions between the bus and the qubits. This new method is robust against parameter variations and is thus low error. It fundamentally improves on previous methods due its deterministic nature and the lack of approximations used in the geometry of the phase rotations. This interaction is applicable both to solid state and photonic qubit systems.Comment: 6 pages, 4 figures. Published versio

Some recent work in Frechet geometry

Some recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Frechet structure needed for applications. A summary is given of recent results on hypercyclicity of operators on Frechet spaces.Comment: 14 pages 48 reference

Flow Induced by the Impulsive Motion of an Infinite Flat Plate in a Dusty Gas

Flow Induced by the Impulsive Motion of an Immite Flat Plate in a Dusty Gas. The problem of flow induced by an infinite flat plate suddenly set into motion parallel to its own plane in an incompressible dusty gas is of considerable physical interest in its own right as well as because of its close relation to the non-linear, steady (constant-pressure) laminar boundary layer. Its solution provides complete and exact information about modifications of the boundary layer growth and skin friction due to particle-fluid interaction. Moreover, it provides a basis for judging the accuracy of approximations which have been employed in more complex problems of viscous fluid-particle motion. The uncoupled thermal Rayleigh problem for small relative temperature differences is directly inferred and this answers questions about the modifications of the surface heat transfer rate and about the possibility of similarity with the velocity boundary layer. Similarity is possible when, in addition to a Prandtl number of unity, the streamwise relaxation processes are also similar

On the entropy flows to disorder

Gamma distributions, which contain the exponential as a special case, have a distinguished place in the representation of near-Poisson randomness for statistical processes; typically, they represent distributions of spacings between events or voids among objects. Here we look at the properties of the Shannon entropy function and calculate its corresponding flow curves. We consider univariate and bivariate gamma, as well as Weibull distributions which also include exponential distributions.Comment: Enlarged version of original. 11 pages, 6 figures, 15 reference