Auswahl der wissenschaftlichen Literatur zum Thema „Equation de Langevin généralisé“

En chimie moléculaire, l'approximation de Born-Oppenheimer permet d'utiliser le concept de surface d'énergie potentielle à l'échelle atomique. Les intersections coniques (ICs) correspondent aux points où ces surfaces sont dégénérées, et où cette approximation n'est plus valable; les concepts de la dynamique classique y voient leur application réduite et les transitions non-adiabatiques entre ces surfaces deviennent importantes. Les ICs sont particulièrement d'intérêt en photochimie, puisque elles sont responsables de la relaxation électronique non-radiative du premier état excité vers l'état fondamental. Cette thèse étudie deux aspects théoriques différents de la dynamique associée aux ICs. Dans un premier temps, l'effet de la dynamique du solvant sur la dynamique autour des ICs est abordé. En particulier, nous présentons l'influence d'un solvant polaire sur la réaction photo-isomérisationd'un modèle d'une base de Schiff protonée. L'attention est centrée sur l'introduction d'un terme de friction dépendante du temps dans la description dynamique des coordonnées internes de la molécule et celle du solvant. Nous démontrons que la prise en compte des dissipations est nécessaire à une description correcte de la dynamique du modèle, et que l'échelle de temps du mouvement du solvant détermine la vitesse de relaxation de l'état excité ainsi que le rapport isomérique des produits. Dans une seconde partie, nous nous focalisons sur la formulation d'une description simple, et utilisable dans les applications, pour la dynamique autour des ICs et plus particulièrement, pour décrire la vitesse de relaxation non-adiabatique au voisinage de l'une IC. Un modèle pour le mécanisme de cette relaxation est introduit, en utilisant une description de l'IC comme un double cône, des trajectoires classiques et l'expression de Landau-Zener pour la probabilité de transition non-adiabatique. Deux géométries différentes pour le cône sont considérées. Le cas le plus simple est le double cône vertical circulaire puisqu'une formulation analytique de la constante de vitesse est rendue possible par des propriétés des trajectoires dues à la symétrie du problème. Nous présentons dans un second temps le cas d'un double cône asymétrique penché, pour lequel une constante de vitesse analytique n'a pas été obtenue, mais où les concepts précédemment introduits dans le modèle sont utilisés pour étudier la variation de la vitesse de relaxation en fonction des conditions initiales et de la géométrie de l'IC. Les résultats du modèle sont comparés avec des simulations utilisant la méthode de surface hopping, avec laquelle le modèle partage un certain nombre de points communs.

Thesis (Ph.D.)--Tufts University, 2004.<br>Adviser: Larry H. Ford. Submitted to the Dept. of Physics. Includes bibliographical references (leaves 117-120). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

In this thesis we concentrate on instances of the GLE which can be represented in a Markovian form in an extended phase space. We extend previous results on the geometric ergodicity of this class of GLEs using Lyapunov techniques, which allows us to conclude ergodicity for a large class of GLEs relevant to molecular dynamics applications. The main body of this thesis concerns the numerical discretisation of the GLE in the extended phase space representation. We generalise numerical discretisation schemes which have been previously proposed for the underdamped Langevin equation and which are based on a decomposition of the vector field into a Hamiltonian part and a linear SDE. Certain desirable properties regarding the accuracy of configurational averages of these schemes are inherited in the GLE context. We also rigorously prove geometric ergodicity on bounded domains by showing that a uniform minorisation condition and a uniform Lyapunov condition are satisfied for sufficiently small timestep size. We show that the discretisation schemes which we propose behave consistently in the white noise and overdamped limits, hence we provide a family of universal integrators for Langevin dynamics. Finally, we consider multiple-time stepping schemes making use of a decomposition of the fluctuation-dissipation term into a reversible and non-reversible part. These schemes are designed to efficiently integrate instances of the GLE whose Markovian representation involves a high number of auxiliary variables or a configuration dependent fluctuation-dissipation term. We also consider an application of dynamics based on the GLE in the context of large scale Bayesian inference as an extension of previously proposed adaptive thermostat methods. In these methods the gradient of the log posterior density is only evaluated on a subset (minibatch) of the whole dataset, which is randomly selected at each timestep. Incorporating a memory kernel in the adaptive thermostat formulation ensures that time-correlated gradient noise is dissipated in accordance with the fluctuation-dissipation theorem. This allows us to relax the requirement of using i.i.d. minibatches, and explore a variety of minibatch sampling approaches.

12 month embargo; First Online: 11 July 2016<br>We study damped geodesic motion of a particle of mass m on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as , its solutions converge to solutions of a limiting equation which includes a noise-induced drift term. A very special case of the main result presents Brownian motion on the manifold as a limit of inertial systems.

Thesis (Ph. D.)--University of Oregon, 2006.<br>Typescript. Includes vita and abstract. Includes bibliographical references (leaves 90-99). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemistry, 2009.<br>Includes bibliographical references (leaves 127-137).<br>In this dissertation, we discuss two methods developed during my PhD study to simulate electron transfer systems. The first method, the semi-classical approximation, is derived from the stationary phase approximation to the path integral in the spin-coherent representation. The resulting equation of motion is a classical-like ordinary differential equation subject to a two-ended boundary condition. The boundary value problem is solved using the "near real trajectory" algorithm. This method is applied to three scattering problems to compute the transmission and reflection probabilities. The strength and weakness of this approach is investigated in details. The second approach is based on the generalized Langevin equation, in which the quantum transitions of electronic states are condensed into a linear regression equation. The memory kernel in the regression equation is computed using a second perturbation expansion. The perturbation is optimized to achieve the best convergence of the second order expansion. This procedure results in a tow-hop Langevin equation, the THLE. Results from a spin-boson system validate the THLE in a wide range of parameter regimes. Lastly, we tested the feasibility of using Monte Carlo sampling to compute the memory kernel from the spin-boson system and proposed a smoothing technique to reduce the number of sampling points.<br>by XiaoGeng Song.<br>Ph.D.

The Magnus effect, i.e., dependence of the trajectory of a body on its rotation, is widely used in sport, science and technical applications. Recently we have shown [1, 2] that due to the Magnus force the single-domain ferromagnetic particles, which are suspended in a viscous fluid and subjected to a harmonic driving force and a non-uniformly rotating magnetic field, can perform drift in a preferred direction. This result has been obtained within the deterministic approach when thermal fluctuations, leading to translational and rotational Brownian motion of particles, are ignored. Our estimations show [2] that it is possible for relatively large particles (> 102 nm). Therefore, to study the drift phenomenon for smaller particles, it is necessary to account for these fluctuations in basic equations.

Made available in DSpace on 2014-06-11T19:25:34Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-02-21Bitstream added on 2014-06-13T19:12:26Z : No. of bitstreams: 1 attanasio_f_me_ift.pdf: 793752 bytes, checksum: 490b63eed4bdd7ec83984c78ac824d6d (MD5)<br>Nesta Dissertação apresentamos um estudo numéerico em uma dimensão espacial da equação de Ginzburg-Landau-Langevin (GLL), com ênfase na aplicabilidade de um método de perturbação estocástico e na mecânica estatística de defeitos topológicos em modelos de campos escalares reais. Revisamos brevemente conceitos de mecânica estatística de sistemas em equilíbrio e próximos a ele e apresentamos como a equação de GLL pode ser usada em sistemas que exibem transições de fase, na quantização estocástica e no estudo da interação de estruturas coerentes com fônons de origem térmica. Também apresentamos um método perturbativo, denominado teoria de perturbação no ruído (TPR), adequado para situações onde a intensidade do ruído estocástico é fraca. Através de simulações numéricas, investigamos a restauração de uma simetria 'Z IND. 2' quebrada, a aplicabilidade da TPR em uma dimensão e efeitos de temperatura finita numa solução topológica do tipo kink - onde apresentamos novos resultados sobre defeitos de dois kinks<br>In this Dissertation we present a numerical study of the GinzburgLandau-Langevin (GLL) equation in one spatial dimension, with emphasis on the applicability of a stochastic perturbative method and the statistical mechanics of topological defect structures in field-theoretic models of real scalar fields. We briefly review concepts of equilibrium and near-equilibrium statistical mechanics and present how the GLL equation can be used in systems that exhibit phase transitions, in stochastic quantization and in the study of the interaction of coherent structures with thermal phonons. We also present a perturbative method, named noise perturbation theory (NPT), suitable for situations where the stochastic noise intensity is weak. Through numerical simulations we investigate the restoration of a broken 'Z IND. 2' symmetry, the applicability of the NPT in one dimension and finite temperature effects on a topological kink solution - where we present new results on two-kink defects