Generating Generalized Distributions from Dynamical Simulation

We present a general molecular-dynamics simulation scheme, based on the Nose' thermostat, for sampling according to arbitrary phase space distributions. We formulate numerical methods based on both Nose'-Hoover and Nose'-Poincare' thermostats for two specific classes of distributions; namely, those that are functions of the system Hamiltonian and those for which position and momentum are statistically independent. As an example, we propose a generalized variable temperature distribution that designed to accelerate sampling in molecular systems.Comment: 10 pages, 3 figure

A molecular-dynamics algorithm for mixed hard-core/continuous potentials

We present a new molecular-dynamics algorithm for integrating the equations of motion for a system of particles interacting with mixed continuous/impulsive forces. This method, which we call Impulsive Verlet, is constructed using operator splitting techniques similar to those that have been used successfully to generate a variety molecular-dynamics integrators. In numerical experiments, the Impulsive Verlet method is shown to be superior to previous methods with respect to stability and energy conservation in long simulations.Comment: 18 pages, 6 postscript figures, uses rotate.st

The adaptive Verlet method

This is the published version, also available here: http://dx.doi.org/10.1137/S1064827595284658.We discuss the integration of autonomous Hamiltonian systems via dynamical rescaling of the vector field (reparameterization of time). Appropriate rescalings (e.g., based on normalization of the vector field or on minimum particle separation in an N-body problem) do not alter the time-reversal symmetry of the flow, and it is desirable to maintain this symmetry under discretization. For standard form mechanical systems without rescaling, this can be achieved by using the explicit leapfrog--Verlet method; we show that explicit time-reversible integration of the reparameterized equations is also possible if the parameterization depends on positions or velocities only. For general rescalings, a scalar nonlinear equation must be solved at each step, but only one force evaluation is needed. The new method also conserves the angular momentum for an N-body problem. The use of reversible schemes, together with a step control based on normalization of the vector field (arclength reparameterization), is demonstrated in several numerical experiments, including a double pendulum, the Kepler problem, and a three-body problem

Generalized dynamical thermostating technique

This is the publisher's version, also available electronically from http://journals.aps.org/pre/abstract/10.1103/PhysRevE.68.016704.We demonstrate that the Nosé method for constant-temperature molecular-dynamics simulation [Mol. Phys. 52, 255 (1984)] can be substantially generalized by the addition of auxiliary variables to encompass an infinite variety of Hamiltonian thermostats. Such thermostats can be used to enhance ergodicity in systems, such as the one-dimensional harmonic oscillator or certain molecular systems, for which the standard Nosé-Hoover methods fail to reproduce converged canonical distributions. In this respect the method is similar in spirit to the method of Nosé-Hoover chains, but is both more general and Hamiltonian in structure (which allows for the use of efficient symplectic integration schemes). In particular, we show that, within the generalized Nosé formalism outlined herein, any Hamiltonian system can be thermostated with any other, including a copy of itself. This gives one an enormous flexibility in choosing the form of the thermostating bath. Numerical experiments are included in which a harmonic oscillator is thermostated with a collection of noninteracting harmonic oscillators as well as by a soft billiard system

Semi-geostrophic particle motion and exponentially accurate normal forms

We give an exponentially-accurate normal form for a Lagrangian particle moving in a rotating shallow-water system in the semi-geostrophic limit, which describes the motion in the region of an exponentially-accurate slow manifold (a region of phase space for which dynamics on the fast scale are exponentially small in the Rossby number). The result extends to numerical solutions of this problem via backward error analysis, and extends to the Hamiltonian Particle-Mesh (HPM) method for the shallow-water equations where the result shows that HPM stays close to balance for exponentially-long times in the semi-geostrophic limit. We show how this result is related to the variational asymptotics approach of [Oliver, 2005]; the difference being that on the Hamiltonian side it is possible to obtain strong bounds on the growth of fast motion away from (but near to) the slow manifold

Two-dimensional mobile breather scattering in a hexagonal crystal lattice

We describe, for the first time, the full 2D scattering of long-lived breathers in a model hexagonal lattice of atoms. The chosen system, representing an idealized model of mica, combines a Lennard-Jones interatomic potential with an "egg-box" harmonic potential well surface. We investigate the dependence of breather properties on the ratio of the well depths associated to the interaction and on-site potentials. High values of this ratio lead to large spatial displacements in adjacent chains of atoms and thus enhance the two dimensional character of the quasi-one-dimensional breather solutions. This effect is further investigated during breather-breather collisions by following the constrained energy density function in time for a set of randomly excited mobile breather solutions. Certain collisions lead to 60$^\circ$ scattering, and collisions of mobile and stationary breathers can generate a rich variety of states.Comment: 4 pages, 5 figure

A symplectic method for rigid-body molecular simulation

This is the publisher's version, also available electronically from http://scitation.aip.org/content/aip/journal/jcp/107/7/10.1063/1.474596.Rigid-body molecular dynamics simulations typically are performed in a quaternion representation. The nonseparable form of the Hamiltonian in quaternions prevents the use of a standard leapfrog (Verlet) integrator, so nonsymplectic Runge–Kutta, multistep, or extrapolation methods are generally used. This is unfortunate since symplectic methods like Verlet exhibit superior energy conservation in long-time integrations. In this article, we describe an alternative method, which we call RSHAKE (for rotation-SHAKE), in which the entire rotation matrix is evolved (using the scheme of McLachlan and Scovel [J. Nonlin. Sci. 16 233 (1995)]) in tandem with the particle positions. We employ a fast approximate Newton solver to preserve the orthogonality of the rotation matrix. We test our method on a system of soft-sphere dipoles and compare with quaternion evolution using a 4th-order predictor–corrector integrator. Although the short-time error of the quaternion algorithm is smaller for fixed time step than that for RSHAKE, the quaternion scheme exhibits an energy drift which is not observed in simulations with RSHAKE, hence a fixed energy tolerance can be achieved by using a larger time step. The superiority of RSHAKE increases with system size

Generating generalized distributions from dynamical simulation

This is the publisher's version, also available electronically from http://scitation.aip.org/content/aip/journal/jcp/118/13/10.1063/1.1557413.We present a general molecular-dynamics simulation scheme, based on the Nosé thermostat, for sampling from arbitrary phase space distributions. We formulate numerical methods based on both Nosé–Hoover and Nosé–Poincaré thermostats for two specific classes of distributions; namely, those that are functions of the system Hamiltonian and those for which position and momentum are statistically independent. As an example, we propose a generalized variable temperature distribution that is designed to accelerate sampling in molecular systems