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 Poincaré-Birkhoff normal form. In the quantum case we develop a version of the Poincaré-Birkhoff 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 “re-cross ” 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 flux-flux 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,

Hamiltonian dynamical systems possessing equilibria of saddle × centre × · · · × centre stability type display reaction-type 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

Abstract. We consider the nucleation of one-dimensional stochastic Cahn-Hilliard dynamics with the standard double well potential. We design the string method for computing the most probable transition path in the zero temperature limit based on large deviation theory. We derive the nucleation rate formula for the stochastic Cahn-Hilliard dynamics through finite dimensional discretization. We also discuss the algorithmic issues for calculating the nucleation rate, especially the high dimensional sampling for computing the determinant ratios. Key words. Cahn-Hilliard equation, large deviation theory, nucleation rate, string method, Metropolis-Hastings algorithm. AMS subject classifications. 60H35, 65Z05, 82C26. 1.

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Although first-order rate constants are basic ingredients of physical chemistry, biochemistry and systems modeling, their innermost nature is derived from complex physical chemistry mechanisms. The present study suggests that equivalent conclusions can be more straightly obtained from simple statistics. The different facets of kinetic constants are first classified and clarified with respect to time and energy and the equivalences between traditional flux rate and modern probabilistic modeling are summarized. Then, a naive but rigorous approach is proposed to concretely perceive how the Arrhenius law naturally emerges from the geometric distribution. It appears that (1) the distribution in time of chemical events as well as (2) their mean frequency, are both dictated by randomness only and as such, are accurately described by time-based and spatial exponential processes respectively.

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The success of Transition State Theory (TST) in describing the rates of chemical reactions has been less-than-perfect in solution (and sometimes even in the gas phase) because conventional dividing surfaces are only approximately free of recrossings between reactants and products. Recent advances in dynamical systems theory have helped to identify the interconnected manifolds —“superhighways” — leading from reactants to products. The existence of these manifolds has been proven rigorously, and explicit algorithms are available for their calculation. We now show that these extended structures can be used to obtain reaction rates directly in dissipative systems. We also suggest a treatment for the substantially more general case in which the molecular solvent is fully specified by the positions of all its atoms. Specifically, we can construct effective solvent configurations for which the exact TST manifolds can be constructed and used to sample the rates of an open system.

Contribution à la modélisation explicite et à la représentation des données de composants industriels: application au modèle PLIB Directeurs de Thèse: Guy Pierra, Yamine Ait-Ameur

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 self-diffusion 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 coarse-grained 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 readily-available sequence information. Herein a checking function is developed that may prove useful for addressing whether a particular proposed structure is a viable possibility.