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25
Wigner’s dynamical transition state theory in phase space: Classical and quantum
 Nonlinearity
, 2008
"... We develop Wigner’s approach to a dynamical transition state theory in phase space in both the classical and quantum mechanical settings. The key to our development is the construction of a normal form for describing the dynamics locally in the neighborhood of a specific type of saddle point that go ..."
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We develop Wigner’s approach to a dynamical transition state theory in phase space in both the classical and quantum mechanical settings. The key to our development is the construction of a normal form for describing the dynamics locally in the neighborhood of a specific type of saddle point that governs the evolution from reactants to products in high dimensional systems. In the classical case this is just the standard PoincaréBirkhoff normal form. In the quantum case we develop a version of the PoincaréBirkhoff normal form for quantum systems and a new algorithm for computing this quantum normal form that follows the same steps as the algorithm for computing the classical normal form. The classical normal form allows us to discover and compute phase space structures that govern reaction dynamics. From this knowledge we are able to provide a direct construction of an energy dependent dividing surface in phase space having the properties that trajectories do not locally “recross ” the surface and the directional flux across the surface is minimal. Using this, we are able to give a formula for the directional flux that goes beyond the harmonic approximation. We relate this construction to the fluxflux autocorrelation function which is a standard ingredient in the expression for the reaction rate in the chemistry community. We also give a classical mechanical interpretation of the activated complex as a normally hyperbolic invariant manifold (NHIM), and further describe the NHIM in terms of a foliation by invariant tori. The quantum normal form allows us to understand the quantum mechanical significance of the classical phase space structures and quantities governing reaction dynamics. In particular,
DIFFUSION MAPS, REDUCTION COORDINATES AND LOW DIMENSIONAL REPRESENTATION OF STOCHASTIC SYSTEMS
"... The concise representation of complex high dimensional stochastic systems via a few reduced coordinates is an important problem in computational physics, chemistry and biology. In this paper we use the first few eigenfunctions of the backward FokkerPlanck diffusion operator as a coarse grained low ..."
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The concise representation of complex high dimensional stochastic systems via a few reduced coordinates is an important problem in computational physics, chemistry and biology. In this paper we use the first few eigenfunctions of the backward FokkerPlanck diffusion operator as a coarse grained low dimensional representation for the long term evolution of a stochastic system, and show that they are optimal under a certain mean squared error criterion. We denote the mapping from physical space to these eigenfunctions as the diffusion map. While in high dimensional systems these eigenfunctions are difficult to compute numerically by conventional methods such as finite differences or finite elements, we describe a simple computational datadriven method to approximate them from a large set of simulated data. Our method is based on defining an appropriately weighted graph on the set of simulated data, and computing the first few eigenvectors and eigenvalues of the corresponding random walk matrix on this graph. Thus, our algorithm incorporates the local geometry and density at each point into a global picture that merges in a natural way data from different simulation runs. Furthermore, we describe lifting and restriction operators between the diffusion map space and the original space. These operators facilitate the description of the coarsegrained dynamics, possibly in the form of a lowdimensional effective free energy surface parameterized by the diffusion map reduction coordinates. They also enable a systematic exploration of such effective free energy surfaces through the design of additional “intelligently biased ” computational experiments. We conclude by demonstrating our method on a few examples. Key words. Diffusion maps, dimensional reduction, stochastic dynamical systems, Fokker Planck operator, metastable states, normalized graph Laplacian. AMS subject classifications. 60H10, 60J60, 62M05
An introductory overview of actionderived molecular dynamics for multiple timescale simulations
, 2004
"... ..."
General Utility Lattice Program Version 3.4
"... Introduction & background The General Utility Lattice Program (GULP) is designed to perform a variety of tasks based on force field methods. The original code was written to facilitate the fitting of interatomic potentials to both energy surfaces and empirical data. However, it has expanded now ..."
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Introduction & background The General Utility Lattice Program (GULP) is designed to perform a variety of tasks based on force field methods. The original code was written to facilitate the fitting of interatomic potentials to both energy surfaces and empirical data. However, it has expanded now to be a general purpose code for the modelling of condensed
Geometrical Models of the Phase Space Structures Governing Reaction Dynamics
, 2009
"... Hamiltonian dynamical systems possessing equilibria of saddle × centre × · · · × centre stability type display reactiontype dynamics for energies close to the energy of such equilibria; entrance and exit from certain regions of the phase space is only possible via narrow bottlenecks created by ..."
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Hamiltonian dynamical systems possessing equilibria of saddle × centre × · · · × centre stability type display reactiontype dynamics for energies close to the energy of such equilibria; entrance and exit from certain regions of the phase space is only possible via narrow bottlenecks created by the influence of the equilibrium points. In this paper we provide a thorough pedagogical description of the phase space structures that are responsible for controlling transport in these problems. Of central importance is the existence of a Normally Hyperbolic Invariant Manifold (NHIM), whose stable and unstable manifolds have sufficient dimensionality to act as separatrices, partitioning energy surfaces into regions of qualitatively distinct behaviour. This NHIM forms the natural (dynamical) equator of a (spherical) dividing surface which locally divides an energy surface into two components (‘reactants ’ and ‘products’), one on either side of the bottleneck. This dividing surface has all the desired properties sought for in transition state theory where reaction rates are computed from the flux through a dividing surface. In fact, the dividing surface that we construct is crossed exactly once by reactive trajectories, and not crossed by nonreactive trajectories, and related to these properties, minimizes the flux upon variation of the dividing surface. We discuss three presentations of the energy surface and the phase space structures contained in it for
25 Tflop/s multibillionatom molecular dynamics simulations and visualization/analysis on BlueGene/L
 In Proceedings of IEEE/ACM Supercomputing ’05
, 2005
"... We demonstrate the excellent performance and scalability of a classical molecular dynamics code, SPaSM, on the IBM BlueGene/L supercomputer at LLNL. Simulations involving up to 160 billion atoms (micronsize cubic samples) on 65,536 processors are reported, consistently achieving 24.4–25.5 Tflop/s f ..."
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We demonstrate the excellent performance and scalability of a classical molecular dynamics code, SPaSM, on the IBM BlueGene/L supercomputer at LLNL. Simulations involving up to 160 billion atoms (micronsize cubic samples) on 65,536 processors are reported, consistently achieving 24.4–25.5 Tflop/s for the commonly used LennardJones 612 pairwise interaction potential. Two extended production simulations (one lasting 8 hours and the other 13 hours wallclock time) of the shock compression and release of porous copper using a more realistic manybody potential are also reported, demonstrating the capability for sustained runs including onthefly parallel analysis and visualization of such massive data sets. This opens up the exciting new possibility of using atomistic simulations at micron length scales to directly bridge to mesoscale and continuumlevel models.
Colloquium: Failure of molecules, bones, and the Earth itself
, 2010
"... Materials fail by recurring rupture and shearing of interatomic bonds at microscopic, molecular scales, leading to disintegration of matter at macroscale and a loss of function. In this Colloquium, the stateoftheart of investigations on failure mechanisms in materials are reviewed, in particular ..."
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Materials fail by recurring rupture and shearing of interatomic bonds at microscopic, molecular scales, leading to disintegration of matter at macroscale and a loss of function. In this Colloquium, the stateoftheart of investigations on failure mechanisms in materials are reviewed, in particular focusing on atomistic origin of deformation and fracture and relationships between molecular mechanics and macroscale behavior. Simple examples of fracture phenomena are used to illustrate the significance and impact of material failure on our daily lives. Based on case studies, mechanisms of failure of a wide range of materials are discussed, ranging from tectonic plates to rupture of single molecules, and an explanation on how atomistic simulation can be used to complement experimental studies and theory to provide a novel viewpoint in the analysis of complex systems is provided. Biological protein materials are used to illustrate how extraordinary properties are achieved through the utilization of intricate structures where the interplay of weak and strong chemical bonds, size and confinement effects, and hierarchical features play a fundamental role. This leads to a discussion of how even the most robust biological material systems fail, leading to diseases that arise from structural and mechanical alterations at molecular, cellular, and tissue levels. New research directions in the field of materials failure and materials science are discussed and the impact of improving the current