Quantifying dimensionality: Bayesian cosmological model complexities

We demonstrate a measure for the effective number of parameters constrained by a posterior distribution in the context of cosmology. In the same way that the mean of the Shannon information (i.e. the Kullback-Leibler divergence) provides a measure of the strength of constraint between prior and posterior, we show that the variance of the Shannon information gives a measure of dimensionality of constraint. We examine this quantity in a cosmological context, applying it to likelihoods derived from Cosmic Microwave Background, large scale structure and supernovae data. We show that this measure of Bayesian model dimensionality compares favourably both analytically and numerically in a cosmological context with the existing measure of model complexity used in the literature.Comment: 14 pages, 9 figures. v2: updates post peer-review. v3: typographical correction to equation 3

Critical behaviour of thin films with quenched impurities

The critical behaviour of thin films containing quenched random impurities and inhomogeneities is investigated by the renormalization-group method. The finite-size crossover in impure films has been considered on the basis of the fundamental relationship between the effective spatial dimensionality and the characteristic lengths of the system. The difference between the critical properties of infinite systems and films is demonstrated and investigated. A new critical exponent, describing the scaling properties of the thickness of films with extended impurities has been deduced and calculated. A special attention is paid to the critical behaviour of real impure films.Comment: 27 pages LaTex; figures are available in the journal varian

Equation of state of a seven-dimensional hard-sphere fluid. Percus-Yevick theory and molecular dynamics simulations

Following the work of Leutheusser [Physica A 127, 667 (1984)], the solution to the Percus-Yevick equation for a seven-dimensional hard-sphere fluid is explicitly found. This allows the derivation of the equation of state for the fluid taking both the virial and the compressibility routes. An analysis of the virial coefficients and the determination of the radius of convergence of the virial series are carried out. Molecular dynamics simulations of the same system are also performed and a comparison between the simulation results for the compressibility factor and theoretical expressions for the same quantity is presented.Comment: 12 pages, 4 figures; v3: Equation (A.19) corrected (see http://dx.doi.org/10.1063/1.2390712