Kinematic analysis of Bricard’s mechanism

We show how the tools of computational algebra can be used to analyze the configuration space of multibody systems. One advantage of this approach is that the mobility can be computed without using the Jacobian of the system. As an example, we treat thoroughly the well-known Bricard’s mechanism, but the same methods can be applied to a wide class of rigid multibody systems. It turns out that the configuration space of Bricard’s system is a smooth closed curve, which can be explicitly parametrized. Our computations also yield a new formulation of constraints which is better than the original one from the point of view of numerical simulations.     

An Efficient Geometric Integrator for Thermostatted Anti-/Ferromagnetic Models

(Anti)-/ferromagnetic Heisenberg spin models arise from discretization of Landau–Lifshitz models in micromagnetic modelling. In many applications it is essential to study the behavior of the system at a fixed temperature. A formulation for thermostatted spin dynamics was given by Bulgac and Kusnetsov, which incorporates a complicated nonlinear dissipation/driving term while preserving spin length. It is essential to properly model this term in simulation, and simplified schemes give poor numerical performance, e.g., requiring an excessively small timestep for stable integration. In this paper we present an efficient, structure-preserving method for thermostatted spin dynamics.     

On the numerical solution of involutive ordinary differential systems: enhanced linear algebra

^** Email: jukka.tuomela{at}joensuu.fi^*** Corresponding author. Email: arponen{at}maths.warwick.ac.uk^**** Email: villesamuli.normi{at}joensuu.fi We analyse some Runge–Kutta type methods for computing1D integral manifolds, i.e. solutions to ordin-ary differentialequations and differential-algebraic equations. We show thatwe can compute the solutions which respect all the constraintsof the problem reliably and reasonably quickly. Moreover, weshow that the so-called impasse points are regular points inour approach and hence require no special attention.     

Electron density and positron trapping at a vacancy

The redistribution of valence electrons at a vacancy in aluminium is calculated in the Thomas-Fermi approximation. This result is then used in evaluating the annihilation characteristics of positrons trapped at vacancies. A good agreement with experiments is established.     

Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems

The exp S formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using two different sets of amplitudes one either recovers the normal exp S method or obtains a new scheme called the extended exp S method. The same functional can be used also to obtain the average values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time dependence to the cluster amplitudes. This allows the calculation of both nonlinear dynamical behaviour and of dynamical linear response and Green's functions. Practical approximation schemes are considered. In a SUB n approximation the m-body amplitudes are restricted to the order $\text{m ? n}$ and the energy functional is a finite-order multinomial in the amplitudes to be variationally determined. It is shown that the solution corresponds to summing well-defined subsets of Goldstone diagrams. These subsets are conveniently specificed in terms of tree structures, the normal or extended generalized time ordering g.t.o. trees. The extended exp S method is in the SUB n approximation able to sum, in addition to the normal SUB n diagrams, a set which contains m-body cluster amplitudes of arbitrarily high order (m > n) in the ordinary sense. The article also discusses how the SUB n truncation schemes must be modified to be able to treat a system with a strong repulsive core in the two-body interaction. The method is formulated for the general cases of Bose and Fermi systems which may or may not conserve total particle number. It is shown that the simplest approximation, SUB 1, in the extended exp S method agrees with the mean field theory, which is the coherent-state approximation in the boson case or the Hartree-Fock approximation in the fermion case. It is argued that the extended exp S method already in low-order approximations can realistically treat a great variety of diverse many-body problems, even including systems which may undergo ground-state phase transitions. A few applications are described in more detail. The Bose liquid is treated in the extended SUB 2 approximation. It is shown that the ground-state results in the uniform limit are exact and agree with the hypernetted-chain approximation. The modifications due to hard-core interactions and the non-linear equations of motion are also discussed in this case. For Fermi systems it is shown that the supercondictive phase transition of the BCS model Hamiltonian and the deformation phase transition of the Lipkin model are properly obtained by the extended exp S method in a low-order approximation.     

Internal structure of neutron stars

The equation of state and the structure and composition of neutron star matter are investigated in the density region 3.1 × 10¹¹−2 × 10¹⁵g/cm³. Below the density 3.1 × 10¹¹ g/cm³ the matter is a solid consisting of neutron-rich nuclei in a degenerate electron gas. At 3.1 × 10¹¹g/cm³ neutrons start to drip out of the nuclei; as the density increases, the lattice spacing continuously decreases while the geometrical size of the nuclei only slightly increases, until at about 15 × 10¹³g/cm³ the nuclei disappear by coalescing into a homogeneous liquid in an almost continuous phase transition. The maximum proton number per nucleus is 40, which is obtained between the densities 1−2.5 × 10¹³g/cm³; after that the proton number decreases until at the solid-to-liquid phase boundary it is about 20. In the liquid-core region, muons appear at the density 20.5 × 10¹³g/cm³.     

On the Numerical Solution of Involutive Ordinary Differential Systems: Higher Order Methods

We analyse some Taylor and Runge—Kutta type methods for computing one-dimensional integral manifolds, i.e. solutions to ODEs and DAEs. The distribution defining the solutions is taken to be defined only on the relevant manifold and hence all the intermediate points occuring in the computations are projected orthogonally to the manifold. We analyse the order of such methods, and somewhat surprisingly there does not appear any new order conditions for the Runge—Kutta methods in our context, at least up to order 4. The analysis shows that some terms appearing in the error expansions can be quite naturally expressed in terms of standard notions of Riemannian geometry. The numerical examples show that the methods work reliably and moreover produce qualitatively correct results for Hamiltonian systems although the methods are not symplectic.This revised version was published online in October 2005 with corrections to the Cover Date.     

Independent-cluster methods as mappings of quantum theory into classical mechanics

Summary The algebraic structures of theconfiguration interaction, normal coupled cluster, andextended coupled cluster methods are reviewed and developed. These methods are pointed out to perform a mapping of the quantum mechanical problem into a classical phase space, where in each case the classical canonical coordinates have characteristically different cluster and locality properties. Special focus is given to the extended coupled cluster method (ECCM), which alone is based on an entirely additively separable coordinate system. The general principles are formulated for systems with both bosonic and fermionic degrees of freedom, allowing both commutative and anticommutative (Grassmann) cluster amplitudes. The properties of the classical images are briefly discussed. It is proposed that phase spaces may exist which are fixed points of quantization.Based on a talk given at theWorkshop on Coupled-Cluster Theory at the Interface of Atomic Physics and Quantum Chemistry, Harvard-Smithsonian Institute for Theoretical Atomic and Molecular Physics, Cambridge, MA, August 7–11, 1990     

Continuous-time monitoring of Landau-Zener interference in a cooper-pair box

Landau-Zener (LZ) tunneling can occur with a certain probability when crossing energy levels of a quantum two-level system are swept across the minimum energy separation. Here we present experimental evidence of quantum interference effects in solid-state LZ tunneling. We used a Cooper-pair box qubit where the LZ tunneling occurs at the charge degeneracy. By employing a weak nondemolition monitoring, we observe interference between consecutive LZ-tunneling events; we find that the average level occupancies depend on the dynamical phase. The system's unusually strong linear response is explained by interband relaxation. Our interferometer can be used as a high-resolution Mach-Zehnder-type detector for phase and charge.