Convergent sequences of perturbative approximations for the anharmonic oscillator I. Harmonic approach

We present numerical evidence that a simple variational improvement of the ordinary perturbation theory of the quantum anharmonic oscillator can give a convergent sequence of approximations even in the extreme strong coupling limit, the purely anharmonic case. Some of the new techniques of this paper can be extended to renormalizable field theories.Comment: 29 pages, 12 Postscript figures available through anonymous ftp at ftp://algol.lpm.univ-montp2.fr ; replaces earlier version which could not be postscripted presumably due to lack of figures.uu fil

Truncation method for Green's functions in time-dependent fields

We investigate the influence of a time dependent, homogeneous electric field on scattering properties of non-interacting electrons in an arbitrary static potential. We develop a method to calculate the (Keldysh) Green's function in two complementary approaches. Starting from a plane wave basis, a formally exact solution is given in terms of the inverse of a matrix containing infinitely many 'photoblocks' which can be evaluated approximately by truncation. In the exact eigenstate basis of the scattering potential, we obtain a version of the Floquet state theory in the Green's functions language. The formalism is checked for cases such as a simple model of a double barrier in a strong electric field. Furthermore, an exact relation between the inelastic scattering rate due to the microwave and the AC conductivity of the system is derived which in particular holds near or at a metal-insulator transition in disordered systems.Comment: to appear in Phys. Rev. B., 21 pages, 3 figures (ps-files

Inference for reaction networks using the Linear Noise Approximation

We consider inference for the reaction rates in discretely observed networks such as those found in models for systems biology, population ecology and epidemics. Most such networks are neither slow enough nor small enough for inference via the true state-dependent Markov jump process to be feasible. Typically, inference is conducted by approximating the dynamics through an ordinary differential equation (ODE), or a stochastic differential equation (SDE). The former ignores the stochasticity in the true model, and can lead to inaccurate inferences. The latter is more accurate but is harder to implement as the transition density of the SDE model is generally unknown. The Linear Noise Approximation (LNA) is a first order Taylor expansion of the approximating SDE about a deterministic solution and can be viewed as a compromise between the ODE and SDE models. It is a stochastic model, but discrete time transition probabilities for the LNA are available through the solution of a series of ordinary differential equations. We describe how a restarting LNA can be efficiently used to perform inference for a general class of reaction networks; evaluate the accuracy of such an approach; and show how and when this approach is either statistically or computationally more efficient than ODE or SDE methods. We apply the LNA to analyse Google Flu Trends data from the North and South Islands of New Zealand, and are able to obtain more accurate short-term forecasts of new flu cases than another recently proposed method, although at a greater computational cost

Computation of biochemical pathway fluctuations beyond the linear noise approximation using iNA

The linear noise approximation is commonly used to obtain intrinsic noise statistics for biochemical networks. These estimates are accurate for networks with large numbers of molecules. However it is well known that many biochemical networks are characterized by at least one species with a small number of molecules. We here describe version 0.3 of the software intrinsic Noise Analyzer (iNA) which allows for accurate computation of noise statistics over wide ranges of molecule numbers. This is achieved by calculating the next order corrections to the linear noise approximation's estimates of variance and covariance of concentration fluctuations. The efficiency of the methods is significantly improved by automated just-in-time compilation using the LLVM framework leading to a fluctuation analysis which typically outperforms that obtained by means of exact stochastic simulations. iNA is hence particularly well suited for the needs of the computational biology community.Comment: 5 pages, 2 figures, conference proceeding IEEE International Conference on Bioinformatics and Biomedicine (BIBM) 201

Computing A Glimpse of Randomness

A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly non-computable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 64 bits of a Chaitin Omega: 0000001000000100000110001000011010001111110010111011101000010000. Full description of programs and proofs will be given elsewhere.Comment: 16 pages; Experimental Mathematics (accepted

QmeQ 1.0: An open-source Python package for calculations of transport through quantum dot devices

QmeQ is an open-source Python package for numerical modeling of transport through quantum dot devices with strong electron-electron interactions using various approximate master equation approaches. The package provides a framework for calculating stationary particle or energy currents driven by differences in chemical potentials or temperatures between the leads which are tunnel coupled to the quantum dots. The electronic structures of the quantum dots are described by their single-particle states and the Coulomb matrix elements between the states. When transport is treated perturbatively to lowest order in the tunneling couplings, the possible approaches are Pauli (classical), first-order Redfield, and first-order von Neumann master equations, and a particular form of the Lindblad equation. When all processes involving two-particle excitations in the leads are of interest, the second-order von Neumann approach can be applied. All these approaches are implemented in QmeQ. We here give an overview of the basic structure of the package, give examples of transport calculations, and outline the range of applicability of the different approximate approaches.Comment: 34 pages, 10 figure