John W. Perram

Abstract The effective interactions of ions, dipoles and higher-order multipoles under periodic boundary conditions are calculated where the array of periodic replications forms an infinite sphere surrounded by a vacuum. Discrepancies between the results of different methods of calculation are resolved and some shape-dependent effects are discussed briefly. In a simulation under these periodic boundary conditions, the net Hamiltonian contains a positive term proportional to the square of the net dipole moment of the configuration. Surrounding the infinite sphere by a continuum of dielectric constant ε.' changes this positive term, the coefficient being zero as ε' ->∞ . We report on the simulation of a dense fluid of hard spheres with embedded point dipoles; simulations are made for different values of showing how the Kirkwood gr-factor and the long-range part of hA (r) depend on ε' in a finite simulation. We show how this dependence on ε' nonetheless leads to a dielectric constant for the system that is independent of ε . In particular, the Clausius-Mosotti and Kirkwood formulae for the dielectric constant e of the system give consistent ε values.

We present a constructive tool for generation and orbital stabilization of periodic solutions for underactuated nonlinear systems. Our method can be applied to any mechanical system with a number of independent actuators smaller than the number of degrees of freedom by one. The synthesized feedback control law is nonlinear and time-dependent. It is derived from a feedback structure that explicitly uses the general or full integral of the systems zero dynamics. The control law generates a periodic solution and makes it exponentially orbitally stable.

Abstract Closed formulae for both real and reciprocal space parts of cutoff errors in the Ewald summation method in cubic periodic boundary conditions are derived. Such estimates are useful in tuning parameters in molecular simulations. Errors in both the electrostatic energy and forces are considered. The estimates apply to a disordered configuration of point charges and, with some limitations, also to point-charge molecular models. The accuracy of our estimates is tested and confirmed using simulated configurations of two systems (molten salt and diethylether) under a variety of conditions.

Current supercomputers and the impending availability of large scale parallel machines makes possible the study by molecular dynamics of a number of fundamental problems in condensed matter science hitherto beyond our scope because of the enormous computing time involved. This is because a realistic model of such systems contains one or two orders of magnitude more particles than the systems studied to date. Moreover, the intermolecular forces between these particles will usually include contributions from distributions of permanent electric charges, so that the usual assumption of short-ranged forces cannot be made. We give a list of typical problems in this class and attack the problem of improving the performance of molecular dynamics algorithms to take advantage of these new architectures. We use the Jacobi theta function transformation to derive rapidly computable forms for the energy of and forces between large assemblies of N particles interacting in periodic boundary conditions as the sum of real space pair-pair interactions and one particle sums in reciprocal Fourier space. By suitable choice of the separation constant controlling the relative overheads of the two contributions, we show that the total overhead grows as N 3/2. We present an experimental investigation of the N dependence of the computing overhead performed on a Siemens VP200 vector processor. The advantages of the algorithm for parallel computation are also discussed.