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Kåre Olaussen

Publications and source records attributed to Kåre Olaussen.

At least 19 recordsLinked to original sources
arXiv2016

Statistical mechanics approach to lattice field theory

The mean spherical approximation (MSA) is a closure relation for pair correlation functions (two-point functions) in statistical physics. It can be applied to a wide range of systems, is computationally fairly inexpensive, and when properly applied and interpreted lead to rather good results. In this paper we promote its applicability to euclidean quantum field theories formulated on a lattice, by demonstrating how it can be used to locate the critical lines of a class of multi-component bosonic models. The MSA has the potential to handle models lacking a positive definite integration measure, which therefore are difficult to investigate by Monte-Carlo simulations.

hep-th↗
arXiv2015

Python Classes for Numerical Solution of PDE's

We announce some Python classes for numerical solution of partial differential equations, or boundary value problems of ordinary differential equations. These classes are built on routines in \texttt{numpy} and \texttt{scipy.sparse.linalg} (or \texttt{scipy.linalg} for smaller problems).

physics.comp-ph↗
arXiv2015

A Python Class for Higher-Dimensional Schrödinger Equations

We announce a Python class for numerical solution of Schr{ö}dinger equations in one or more space dimensions, employing some recently developed general classes for numerical solution of partial differential equations, and routines from \texttt{numpy} and \texttt{scipy.sparse.linalg} (or \texttt{scipy.linalg} for smaller problems).

physics.comp-ph↗
arXiv2015

Properties of the Virial Expansion and Equation of State of Ideal Quantum Gases in Arbitrary Dimensions

The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality $d$. In particular, the convergence radius $ρ_c(d)$ of the expansion is particulary large at {\em exactly\/} $d=3$ dimensions, $ρ_c(3) = 7.1068\ldots \times \lim_{d\to3} ρ_c(d)$. The same phenomenon occurs in a few other special (non-integer) dimensions. We explain the origin of these facts, and discuss more generally the structure of singularities governing the asymptotic behavior of the ideal gas virial expansion.

cond-mat.stat-mech↗
arXiv2013

Automatic code generator for higher order integrators

Some explicit algorithms for higher order symplectic integration of a large class of Hamilton's equations have recently been discussed by Mushtaq \emph{et. al}. Here we present a Python program for automatic numerical implementation of these algorithms for a given Hamiltonian, both for double precision and multiprecision computations. We provide examples of how to use this program, and illustrate behaviour of both the code generator and the generated solver module(s).

math.NA↗
arXiv2013

Higher order splitting methods with modified integrators for a class of Hamiltonian systems

We discuss systematic extensions of the standard (St{ö}rmer-Verlet) splitting method for differential equations of Hamiltonian mechanics, with relative accuracy of order $τ^2$ for a timestep of length $τ$, to higher orders in $τ$. We present some splitting schemes, with all intermediate timesteps real and positive, which increase the relative accuracy to order $τ^{N}$ (for N=4, 6, and 8) for a large class of Hamiltonian systems.

math.NA↗
arXiv2012

Generating Very-High-Precision Frobenius Series with Apriori Estimates of Coefficients

The Frobenius method can be used to compute solutions of ordinary linear differential equations by generalized power series. Each series converges in a circle which at least extends to the nearest singular point; hence exponentially fast inside the circle. This makes this method well suited for very-high-precision solutions of such equations. It is useful for this purpose to have prior knowledge of the behaviour of the series. We show that the magnitude of its coefficients can be apriori predicted to surprisingly high accuracy, employing a Legendre transformation of the WKB approximated solutions of the equation.

math-ph↗
arXiv2012

Quantum loop expansion to high orders, extended Borel summation, and comparison with exact results

We compare predictions of the quantum loop expansion to (essentially) infinite orders with (essentially) exact results in a simple quantum mechanical model.We find that there are exponentially small corrections to the loop expansion, which cannot be explained by any obvious "instanton" type corrections. It is not the mathematical occurence of exponential corrections, but their seemingly lack of any physical origin, which we find surprising and puzzling.

math-ph↗
arXiv2012

Estimating Coefficients of Frobenius Series by Legendre Transform and WKB Approximation

The Frobenius method can be used to represent solutions of ordinary differential equations by (generalized) power series. It is useful to have prior knowledge of the coefficients of this series. In this contribution we demonstrate that the magnitude of the coefficients can be predicted to surprisingly high accuracy by a Legendre transformation of WKB approximated solutions to the differential equations.

math-ph↗
arXiv2012

High precision series solution of differential equations: Ordinary and regular singular point of second order ODEs

A subroutine for very-high-precision numerical solution of a class of ordinary differential equations is provided. For given evaluation point and equation parameters the memory requirement scales linearly with precision $P$, and the number of algebraic operations scales roughly linearly with $P$ when $P$ becomes sufficiently large. We discuss results from extensive tests of the code, and how one for a given evaluation point and equation parameters may estimate precision loss and computing time in advance.

math-ph↗
arXiv2012

Systematic Improvement of Splitting Methods for the Hamilton Equations

We show how the standard (St{ö}rmer-Verlet) splitting method for differential equations of Hamiltonian mechanics (with accuracy of order $τ^2$ for a timestep of length $τ$) can be improved in a systematic manner without using the composition method. We give the explicit expressions which increase the accuracy to order $τ^8$, and demonstrate that the method work on a simple anharmonic oscillator.

math.NA↗
arXiv2010

Very-high-precision solutions of a class of Schr{ö}dinger equations

We investigate a method to solve a class of Schr{ö}dinger equation eigenvalue problems numerically to very high precision $P$ (from thousands to a million of decimals). The memory requirement, and the number of high precision algebraic operations, of the method scale essentially linearly with $P$ when only eigenvalues are computed. However, since the algorithms for multiplying high precision numbers scale at a rate between $P^{1.6}$ and $P\,\log P\,\log\log P$, the time requirement of our method increases somewhat faster than $P^2$.

math-ph↗
arXiv2008

Collapse of the Quantum Wavefunction

We show using a realistic Hamiltonian-type model that definite outcomes of quantum measurements may emerge from quantum evolution of pure states, i.e quantum dynamics provides a deterministic collapse of the wavefunction in a quantum measurement process. The relaxation of the wavefunction into a pointer state with classical properties is driven by the interaction with an environment. The destruction of superpositions, i.e. choosing a preferred attraction basin and thereby a preferred pointer state, is caused by a tiny nonlinearity in the macroscopic measurement apparatus. In more details, we numerically studied the many-body quantum dynamics of a closed Universe consisting of a system spin measured by a ferromagnet embedded in a spin-glass environment. The nonlinear term is the self-induced magnetic field of the ferromagnet. The statistics for the outcomes of this quantum measurement process depends on the size of the attraction basins in the measurement apparatus and are in accordance to Born's rule.

quant-ph↗
arXiv1996

The third virial coefficient of anyons revisited

We use the method of solving the three-anyon problem developed in our earlier publication to evaluate numerically the third virial coefficient of free anyons. In order to improve precision, we explicitly correct for truncation effects. The present calculation is about three orders of magnitude more precise than the previous Monte Carlo calculation and indicates the presence of a term $a sin^4 πν$ with a very small coefficient $a \simeq -1.65 10^{-5}$.

hep-th↗
arXiv1995

Anyon trajectories and the systematics of the three-anyon spectrum

We develop the concept of trajectories in anyon spectra, i.e., the continuous dependence of energy levels on the kinetic angular momentum. It provides a more economical and unified description, since each trajectory contains an infinite number of points corresponding to the same statistics. For a system of non-interacting anyons in a harmonic potential, each trajectory consists of two infinite straight line segments, in general connected by a nonlinear piece. We give the systematics of the three-anyon trajectories. The trajectories in general cross each other at the bosonic/fermionic points. We use the (semi-empirical) rule that all such crossings are true crossings, i.e.\ the order of the trajectories with respect to energy is opposite to the left and to the right of a crossing.

hep-th↗
arXiv1995

Using Conservation Laws to Solve Toda Field Theories

We investigate the question of how the knowledge of sufficiently many local conservation laws for a model can be utilized to solve the model. We show that for models where the conservation laws can be written in one-sided forms, like $\barpartial Q_s = 0$, the problem can always be reduced to solving a closed system of ordinary differential equations. We investigate the $A_1$, $A_2$, and $B_2$ Toda field theories in considerable detail from this viewpoint. One of our findings is that there is in each case a transformation group intrinsic to the model. This group is built on a specific real form of the Lie algebra used to label the Toda field theory. It is the group of field transformations which leaves the conserved densities invariant.

hep-th↗
arXiv1994

Solution of the Three--Anyon Problem

We solve, by separation of variables, the problem of three anyons with a harmonic oscillator potential. The anyonic symmetry conditions from cyclic permutations are separable in our coordinates. The conditions from two-particle transpositions are not separable, but can be expressed as reflection symmetry conditions on the wave function and its normal derivative on the boundary of a circle. Thus the problem becomes one-dimensional. We solve this problem numerically by discretization. $N$-point discretization with very small $N$ is often a good first approximation, on the other hand convergence as $N\to\infty$ is sometimes very slow.

hep-th↗