Numerical solution of eigenvalue problems for Hamiltonian systems

This paper discusses the numerical solution of eigenvalue problems for Hamiltonian systems of ordinary differential equations. Two new codes are presented which incorporate the algorithms described here; to the best of the author’s knowledge, these are the first codes capable of solving numerically such general eigenvalue problems. One of these implements a new new method of solving a differential equation whose solution is a unitary matrix. Both codes are fully documented and are written inPfort-verifiedFortran 77, and will be available in netlib/aicm/sl11f and netlib/aicm/sl12f.     

Level shift operators for open quantum systems

Level shift operators describe the second-order displacement of eigenvalues under perturbation. They play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures. We exhibit intrinsic properties of level shift operators, properties which stem from the structure of open quantum systems at positive temperatures and which are common to all such systems. They determine the geometry of resonances bifurcating from eigenvalues of positive temperature Hamiltonians and they relate the Gibbs state, the kernel of level shift operators, and zero energy resonances. We show that degeneracy of energy levels of the small part of the open quantum system causes the Fermi Golden Rule Condition to be violated and we analyze ergodic properties of such systems.     

Master integrals for massless three-loop form factors: One-loop and two-loop insertions

The three-loop form factors in massless QCD can be expressed as a linear combination of master integrals. Besides a number of master integrals which factorise into products of one-loop and two-loop integrals, one finds 16 genuine three-loop integrals. Of these, six have the form of a bubble insertion inside a one-loop or two-loop vertex integral. We compute all master integrals with these insertion topologies.     

“A priori” estimates, uniqueness and existence of positive solutions of Yamabe type equations on complete manifolds

We study the asymptotic behaviour of non-negative solutions of Yamabe type equations on a complete Riemannian manifold. Then we provide a comparison result, based on a form of the weak maximum principle at infinity, which together with the “a priori” estimates previously obtained, yields uniqueness under very general Ricci assumptions. The paper ends with an existence result and an application to the non-compact Yamabe problem.     

Probing microscopic chaotic dynamics by observing macroscopic transport processes

We derive the Kramers equation, namely, the Fokker-Planck equation for an oscillator, from a completely deterministic picture. The oscillator is coupled to a “booster”, i.e., a deterministic system in a fully chaotic state, wherein diffusion is derived from the sensitive dependence of chaos on initial conditions and friction is a consequence of the linear response of the booster to the action exerted on it by the oscillator. To deal with the Hamiltonian nature of the system of interest and of its coupling to the booster, we extend the earlier theoretical derivation of macroscopic transport coefficients from deterministic dynamics. We show that the frequency of the oscillator can be tuned to the microscopic frequencies of the booster without affecting the canonical nature of the “macroscopic” statistics. The theoretical predictions are supported by numerical simulations.     

Theory of Non-Equilibrium Stationary States as a Theory of Resonances

We study a small quantum system (e.g., a simplified model for an atom or molecule) interacting with two bosonic or fermionic reservoirs (say, photon or phonon fields). We show that the combined system has a family of stationary states parametrized by two numbers, T ₁ and T ₂ (‘reservoir temperatures’). If T ₁ ≠ T ₂, then these states are non-equilibrium stationary states (NESS). In the latter case we show that they have nonvanishing heat fluxes and positive entropy production and are dynamically asymptotically stable. The latter means that the evolution with an initial condition, normal with respect to any state where the reservoirs are in equilibria at temperatures T ₁ and T ₂, converges to the corresponding NESS. Our results are valid for the temperatures satisfying the bound min (T ₁,T ₂) > g ^{2 + α}, where g is the coupling constant and 0 < α < 1 is a power related to the infra-red behaviour of the coupling functions. Submitted: March 20, 2006. Revised: March 19, 2007. Accepted: May 11, 2007. Marco Merkli: Partly supported by an NSERC PDF, the Institute of Theoretical Physics of ETH Zürich, Switzerland, the Departments of Mathematics of McGill University and the University of Toronto, Canada. Matthias Mück: Supported by DAAD under grant HSP III. Israel Michael Sigal: Supported by NSERC under grant NA7901.     

Determination of glycols by HPLC with refractive index detection

A simple and fast HPLC method for the determination of glycols is described. It is characterized by a reversed-phase separation using water as eluent and a refractive index detection. The method was applied to investigate the biodegradation of glycols in a laboratory activated sludge plant and to determine the content of glycols or alcohols in detergents. The detection limit is 4 mg/l ethylene glycol or propylene glycol in an aqueous sample.     

Analysis of the crystallization kinetics in poly(ethylene oxide) by calorimetric measurements

Summary Differential scanning calorimetric measurements in the early stage of isothermal crystal growth of polyethylene oxide are analysed in the light of irreversible thermodynamics. An accurate evaluation of the equilibrium melting temperature is done by fitting the thermograms obtained at different undercoolings and referring to the activation energy values already known from the literature. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.