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17
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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Cited by 4 (1 self)
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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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Cited by 3 (1 self)
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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
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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Cited by 1 (1 self)
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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.
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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Cited by 1 (0 self)
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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
An introductory overview of actionderived molecular dynamics for multiple timescale
, 2003
"... simulations ..."
From Computational Biophysics to Systems Biology (CBSB08),
"... Proceedings of the NIC Workshop 2008, ..."
A Tutorial
"... Permission to make digital or hard copies of portions of this work for personal or classroom use is granted provided that the copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise requires pri ..."
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Permission to make digital or hard copies of portions of this work for personal or classroom use is granted provided that the copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise requires prior specific permission by the publisher mentioned above.
Molecular Dynamics and Stochastic Simulations of Surface Diffusion
, 2007
"... Despite numerous advances in experimental methodologies capable of addressing the various phenomenon occurring on metal surfaces, atomic scale resolution of the microscopic dynamics remains elusive for most systems. Computational models of the processes may serve as an alternative tool to fill this ..."
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Despite numerous advances in experimental methodologies capable of addressing the various phenomenon occurring on metal surfaces, atomic scale resolution of the microscopic dynamics remains elusive for most systems. Computational models of the processes may serve as an alternative tool to fill this void. To this end, parallel molecular dynamics simulations of selfdiffusion on metal surfaces have been developed and employed to address microscopic details of the system. However these simulations are not without their limitations and prove to be computationally impractical for a variety of chemically relevant systems, particularly for diffusive events occurring in the low temperature regime. To circumvent this difficulty, a corresponding coarsegrained representation of the surface is also developed resulting in a reduction of the required computational effort by several orders of magnitude, and this description becomes all the more advantageous with increasing system size and complexity. This representation provides a convenient framework to address fundamental aspects of diffusion in nonequilibrium environments and an interesting mechanism for directing diffusive motion along the surface is explored. In the ensuing discussion, additional topics including transition state theory in noisy systems and the construction of a checking function for protein structure validation are outlined. For decades the former has served as a cornerstone for estimates of chemical reaction rates. However, in complex environments transition state theory most always provides only an upper bound for the true rate. An alternative approach is described that may alleviate some of the difficulties associated with this problem. Finally, one of the grand challenges facing the computational sciences is to develop methods capable of reconstructing protein structure based solely on readilyavailable sequence information. Herein a checking function is developed that may prove useful for addressing whether a particular proposed structure is a viable possibility.
Bridging Scale Methods for Computational Nanotechnology ∗
, 2004
"... The rapid advances in nanotechnology, nanomaterials and nanomechanics offer huge potentials in national defense, homeland security, and private industry. An emphasis on nanoscale entities will make our manufacturing technologies and infrastructure more sustainable in terms of reduced energy ..."
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The rapid advances in nanotechnology, nanomaterials and nanomechanics offer huge potentials in national defense, homeland security, and private industry. An emphasis on nanoscale entities will make our manufacturing technologies and infrastructure more sustainable in terms of reduced energy
MODELING: APPLICATIONS IN MATERIALS SCIENCE AND CHEMISTRY AND ADVANCES IN SCALABILITY BY
, 2007
"... Effective and efficient multiscale modeling is essential to advance both the science and synthesis in a wide array of fields such as physics, chemistry, materials science, biology, biotechnology and pharmacology. This study investigates the efficacy and potential of using genetic algorithms for mult ..."
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Effective and efficient multiscale modeling is essential to advance both the science and synthesis in a wide array of fields such as physics, chemistry, materials science, biology, biotechnology and pharmacology. This study investigates the efficacy and potential of using genetic algorithms for multiscale materials modeling and addresses some of the challenges involved in designing competent algorithms that solve hard problems quickly, reliably and accurately. In particular, this thesis demonstrates the use of genetic algorithms (GAs) and genetic programming (GP) in multiscale modeling with the help of two nontrivial case studies in materials science and chemistry. The first case study explores the utility of genetic programming (GP) in multitimescaling alloy kinetics simulations. In essence, GP is used to bridge molecular dynamics and kinetic Monte Carlo methods to span ordersofmagnitude in simulation time. Specifically, GP is used to regress symbolically an inline barrier function from a limited set of molecular dynamics simulations to enable kinetic Monte Carlo that simulate seconds of real time. Results on a nontrivial example of vacancyassisted migration on a surface of a facecentered cubic (fcc) CopperCobalt (CuxCo1−x) alloy show that GP predicts all barriers with 0.1 % error from calculations for less than 3 % of active