Formally Verified Approximations of Definite Integrals

International audienceFinding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis. This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system. Our approach is not based on traditional quadrature methods such as Newton-Cotes formulas. Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting. This work has been integrated to the CoqInterval library

A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities

It is shown that tunnelling splittings in ergodic double wells and resonant widths in ergodic metastable wells can be approximated as easily-calculated matrix elements involving the wavefunction in the neighbourhood of a certain real orbit. This orbit is a continuation of the complex orbit which crosses the barrier with minimum imaginary action. The matrix element is computed by integrating across the orbit in a surface of section representation, and uses only the wavefunction in the allowed region and the stability properties of the orbit. When the real orbit is periodic, the matrix element is a natural measure of the degree of scarring of the wavefunction. This scarring measure is canonically invariant and independent of the choice of surface of section, within semiclassical error. The result can alternatively be interpretated as the autocorrelation function of the state with respect to a transfer operator which quantises a certain complex surface of section mapping. The formula provides an efficient numerical method to compute tunnelling rates while avoiding the need for the exceedingly precise diagonalisation endemic to numerical tunnelling calculations.Comment: Submitted to Annals of Physics. This work has been submitted to Academic Press for possible publicatio

Two-time Green's functions and spectral density method in nonextensive quantum statistical mechanics

We extend the formalism of the thermodynamic two-time Green's functions to nonextensive quantum statistical mechanics. Working in the optimal Lagrangian multipliers representation, the $q$-spectral properties and the methods for a direct calculation of the two-time $q$% -Green's functions and the related $q$-spectral density ($q$ measures the nonextensivity degree) for two generic operators are presented in strict analogy with the extensive ($q=1$) counterpart. Some emphasis is devoted to the nonextensive version of the less known spectral density method whose effectiveness in exploring equilibrium and transport properties of a wide variety of systems has been well established in conventional classical and quantum many-body physics. To check how both the equations of motion and the spectral density methods work to study the $q$-induced nonextensivity effects in nontrivial many-body problems, we focus on the equilibrium properties of a second-quantized model for a high-density Bose gas with strong attraction between particles for which exact results exist in extensive conditions. Remarkably, the contributions to several thermodynamic quantities of the $q$-induced nonextensivity close to the extensive regime are explicitly calculated in the low-temperature regime by overcoming the calculation of the $q$ grand-partition function.Comment: 48 pages, no figure

Transport coefficients in high temperature gauge theories: (II) Beyond leading log

Results are presented of a full leading-order evaluation of the shear viscosity, flavor diffusion constants, and electrical conductivity in high temperature QCD and QED. The presence of Coulomb logarithms associated with gauge interactions imply that the leading-order results for transport coefficients may themselves be expanded in an infinite series in powers of 1/log(1/g); the utility of this expansion is also examined. A next-to-leading-log approximation is found to approximate the full leading-order result quite well as long as the Debye mass is less than the temperature.Comment: 38 pages, 6 figure

Multi-dimensional Gaussian fluctuations on the Poisson space

We study multi-dimensional normal approximations on the Poisson space by means of Malliavin calculus, Stein's method and probabilistic interpolations. Our results yield new multi-dimensional central limit theorems for multiple integrals with respect to Poisson measures -- thus significantly extending previous works by Peccati, Sol\'e, Taqqu and Utzet. Several explicit examples (including in particular vectors of linear and non-linear functionals of Ornstein-Uhlenbeck L\'evy processes) are discussed in detail.Comment: 40 page

Equilibrium solutions of the shallow water equations

A statistical method for calculating equilibrium solutions of the shallow water equations, a model of essentially 2-d fluid flow with a free surface, is described. The model contains a competing acoustic turbulent {\it direct} energy cascade, and a 2-d turbulent {\it inverse} energy cascade. It is shown, nonetheless that, just as in the corresponding theory of the inviscid Euler equation, the infinite number of conserved quantities constrain the flow sufficiently to produce nontrivial large-scale vortex structures which are solutions to a set of explicitly derived coupled nonlinear partial differential equations.Comment: 4 pages, no figures. Submitted to Physical Review Letter

Gaussian Approximations of Multiple Integrals

Fix an integer k, and let I(l), l=1,2,..., be a sequence of k-dimensional vectors of multiple Wiener-It\^o integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as l diverges, the law of I(l) is asymptotically close (for example, in the sense of Prokhorov's distance) to the law of a k-dimensional Gaussian vector having the same covariance matrix as I(l). The main feature of our results is that they require minimal assumptions (basically, boundedness of variances) on the asymptotic behaviour of the variances and covariances of the elements of I(l). In particular, we will not assume that the covariance matrix of I(l) is convergent. This generalizes the results proved in Nualart and Peccati (2005), Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As shown in Marinucci and Peccati (2007b), the criteria established in this paper are crucial in the study of the high-frequency behaviour of stationary fields defined on homogeneous spaces.Comment: 15 page