Ramses van Zon

SciNet: Lessons Learned from Building a Power-efficient Top-20 System and Data Centre

SciNet, one of seven regional HPC consortia operating under the Compute Canada umbrella, runs Canadas first and third fastest computers (as of June 2010) in a state-of-the-art, highly energy-efficient datacentre with a Power Usage Effectiveness (PUE) design-point of 1.16. Power efficiency, computational bang for the buck and system capability for a handful of flagship science projects were important criteria in choosing the nature of the computers and the data centre itself. Here we outline some of the lessons learned in putting together the systems and the data centre that hosts Canadas fastest computer to date.

Mapping quantum-classical Liouville equation: Projectors and trajectories

The evolution of a mixed quantum-classical system is expressed in the mapping formalism where discrete quantum states are mapped onto oscillator states, resulting in a phase space description of the quantum degrees of freedom. By defining projection operators onto the mapping states corresponding to the physical quantum states, it is shown that the mapping quantum-classical Liouville operator commutes with the projection operator so that the dynamics is confined to the physical space. It is also shown that a trajectory-based solution of this equation can be constructed that requires the simulation of an ensemble of entangled trajectories. An approximation to this evolution equation which retains only the Poisson bracket contribution to the evolution operator does admit a solution in an ensemble of independent trajectories but it is shown that this operator does not commute with the projection operators and the dynamics may take the system outside the physical space. The dynamical instabilities, utility, and domain of validity of this approximate dynamics are discussed. The effects are illustrated by simulations on several quantum systems.

Numerical implementation of the exact dynamics of free rigid bodies

In this paper the exact analytical solution of the motion of a rigid body with arbitrary mass distribution is derived in the absence of forces or torques. The resulting expressions are cast into a form where the dependence of the motion on initial conditions is explicit and the equations governing the orientation of the body involve only real numbers. Based on these results, an efficient method to calculate the location and orientation of the rigid body at arbitrary times is presented. This implementation can be used to verify the accuracy of numerical integration schemes for rigid bodies, to serve as a building block for event-driven discontinuous molecular dynamics simulations of general rigid bodies, and for constructing symplectic integrators for rigid body dynamics.

Discontinuous molecular dynamics for semiflexible and rigid bodies

A general framework for performing event-driven simulations of systems with semiflexible or rigid bodies interacting under impulsive forces is outlined. The method consists of specifying a means of computing the free evolution of constrained motion, evaluating the times at which interactions occur, and determining the consequences of interactions on subsequent motion. Algorithms for computing the times of interaction events and carrying out efficient event-driven simulations are discussed. The semiflexible case and the rigid case differ qualitatively in that the free motion of a rigid body can be computed analytically and need not be integrated numerically.

Symplectic algorithms for simulations of rigid-body systems using the exact solution of free motion

Elegant integration schemes of second and fourth order for simulations of rigid-body systems are presented which treat translational and rotational motion on the same footing. This is made possible by a recent implementation of the exact solution of free rigid-body motion. The two schemes are time reversible and symplectic, and exactly respect conservation principles for both the total linear and the angular momentum vectors. Simulations of simple test systems show that the second-order scheme is stable and conserves all constants of the motion to high precision. Furthermore, the schemes are demonstrated to be more accurate and efficient than existing methods, except for high densities, in which case the second-order scheme performs at least as well, showing their general applicability. Finally, it is demonstrated that the fourth-order scheme is more efficient than the second-order scheme provided the time step is smaller than a system-dependent threshold value.

Event-driven dynamics of rigid bodies interacting via discretized potentials.

A framework for performing event-driven, adaptive time step simulations of systems of rigid bodies interacting under stepped or terraced potentials in which the potential energy is only allowed to have discrete values is outlined. The scheme is based on a discretization of an underlying continuous potential that effectively determines the times at which interaction energies change. As in most event-driven approaches, the method consists of specifying a means of computing the free motion, evaluating the times at which interactions occur, and determining the consequences of interactions on subsequent motion for the terraced potential. The latter two aspects are shown to be simply expressible in terms of the underlying smooth potential. Within this context, algorithms for computing the times of interaction events and carrying out efficient event-driven simulations are discussed. The method is illustrated on a system composed of rigid rods in which the constituents interact via a terraced potential that depends on the relative orientations of the rods.

Constructing smooth potentials of mean force, radial distribution functions, and probability densities from sampled data

In this paper a method of obtaining smooth analytical estimates of probability densities, radial distribution functions, and potentials of mean force from sampled data in a statistically controlled fashion is presented. The approach is general and can be applied to any density of a single random variable. The method outlined here avoids the use of histograms, which require the specification of a physical parameter (bin size) and tend to give noisy results. The technique is an extension of the Berg-Harris method [B. A. Berg and R. C. Harris, Comput. Phys. Commun. 179, 443 (2008)], which is typically inaccurate for radial distribution functions and potentials of mean force due to a nonuniform Jacobian factor. In addition, the standard method often requires a large number of Fourier modes to represent radial distribution functions, which tends to lead to oscillatory fits. It is shown that the issues of poor sampling due to a Jacobian factor can be resolved using a biased resampling scheme, while the requirement of a large number of Fourier modes is mitigated through an automated piecewise construction approach. The method is demonstrated by analyzing the radial distribution functions in an energy-discretized water model. In addition, the fitting procedure is illustrated on three more applications for which the original Berg-Harris method is not suitable, namely, a random variable with a discontinuous probability density, a density with long tails, and the distribution of the first arrival times of a diffusing particle to a sphere, which has both long tails and short-time structure. In all cases, the resampled, piecewise analytical fit outperforms the histogram and the original Berg-Harris method.

Multiple-point and multiple-time correlation functions in a hard-sphere fluid.

A recent mode-coupling theory of higher-order correlation functions is tested on a simple hard-sphere fluid system at intermediate densities. Multiple-point and multiple-time correlation functions of the densities of conserved variables are calculated in the hydrodynamic limit and compared to results obtained from event-based molecular dynamics simulations. It is demonstrated that the mode-coupling theory results are in excellent agreement with the simulation results provided that dissipative couplings are included in the vertices appearing in the theory. In contrast, simplified mode-coupling theories in which the densities obey Gaussian statistics neglect important contributions to both the multiple point and multiple-time correlation functions on all time scales.

Quantum free-energy differences from nonequilibrium path integrals. I. Methods and numerical application.

In this paper, the imaginary-time path-integral representation of the canonical partition function of a quantum system and nonequilibrium work fluctuation relations are combined to yield methods for computing free-energy differences in quantum systems using nonequilibrium processes. The path-integral representation is isomorphic to the configurational partition function of a classical field theory, to which a natural but fictitious Hamiltonian dynamics is associated. It is shown that if this system is prepared in an equilibrium state, after which a control parameter in the fictitious Hamiltonian is changed in a finite time, then formally the Jarzynski nonequilibrium work relation and the Crooks fluctuation relation hold, where work is defined as the change in the energy as given by the fictitious Hamiltonian. Since the energy diverges for the classical field theory in canonical equilibrium, two regularization methods are introduced which limit the number of degrees of freedom to be finite. The numerical applicability of the methods is demonstrated for a quartic double-well potential with varying asymmetry. A general parameter-free smoothing procedure for the work distribution functions is useful in this context.

Discontinuous molecular dynamics for rigid bodies: Applications

Event-driven molecular dynamics simulations are carried out on two rigid-body systems which differ in the symmetry of their molecular mass distributions. First, simulations of methane in which the molecules interact via discontinuous potentials are compared with simulations in which the molecules interact through standard continuous Lennard-Jones potentials. It is shown that under similar conditions of temperature and pressure, the rigid discontinuous molecular dynamics method reproduces the essential dynamical and structural features found in continuous-potential simulations at both gas and liquid densities. Moreover, the discontinuous molecular dynamics approach is demonstrated to be between 3 and 100 times more efficient than the standard molecular dynamics method depending on the specific conditions of the simulation. The rigid discontinuous molecular dynamics method is also applied to a discontinuous-potential model of a liquid composed of rigid benzene molecules, and equilibrium and dynamical properties are shown to be in qualitative agreement with more detailed continuous-potential models of benzene. The few qualitative differences in the angular dynamics of the two models are related to the relatively crude treatment of variations in the discontinuous repulsive interactions as one benzene molecule rotates by another.