Home > Archives > Additional Archives > Simon J. Cox   Simon J. Cox Department of Electronics and Computer Science University of Southampton, Southampton, SO17 1BJ, UK sc@ecs.soton.ac.uk http://www.hpcc.ecs.soton.ac.uk  O(N) Methods for Particle-Particle Simulations and the Perils of Handling Long-Range Forces We present results from simulations of the vortex state in layered high temperature superconductors. A number of problems arise when long-range forces, such as those governed by Bessel functions, are used in particle-particle simulations. If a simple cut-off for the interaction is used, the system may find an equilibrium configuration at zero temperature that is not a regular lattice yet has an energy lower than the theoretically predicted minimum for the physical system. We demonstrate two methods to overcome these problems in Hybrid Monte Carlo and molecular dynamics simulations. The first uses a smoothed potential to truncate the interaction in a single unit cell: this is appropriate for phenomenological characterisations, but may be applied to any potential. The second is a new method for summing the unmodified potential in an infinitely tiled periodic system, which is in excess of 20,000 times faster than previous nave methods which add periodic images in shells of increasing radius: this is suitable for quantitative studies. We show that numerical experiments which do not handle the long-range force carefully may give misleading results: both of our proposed methods overcome these problems. Finally we present a new fast multipole algorithm suitable for computing the energy inside the unit cell of a system of N particles in O(N) time, where the inter-particle energy is governed by a Bessel function, Kn.
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