Estudio de la estructura de autofunciones de sistemas caóticos en una base de funciones de scar

El problema central que aborda esta tesis doctoral es el estudio de la correspondencia entre la mecanica clasica y la mecanica cuantica en sistemas hamiltonianos clasicamente caoticos, tema que se enmarca dentro del llamado caos cuantico. En concreto, en este trabajo proponemos un nuevo y efectivo metodo para calcular las autofunciones de sistemas caoticos usando una base de funciones de scar. Dichas funciones de scar juegan un papel fundamental en el estudio de las manifestaciones cuanticas del caos, ya que se trata de funciones de onda semiclasicas con una dispersion muy peque~na y localizadas a lo largo de las variedades invariantes de las orbitas periodicas inestables del sistema que conforman la estructura organizativa del caos clasico. El metodo de calculo desarrollado se ha denominado Metodo de Gram- Schmidt Selectivo (MGSS), dado que construye la base haciendo uso del metodo de Gram{Schmidt convencional pero teniendo en cuenta, ademas, la dispersi on de las funciones de scar y la longitud de la orbita periodica a lo largo de la cual se localizan. El MGSS nos ha permitido calcular con gran precision las 2400 autofunciones con menor energa de un oscilador cuartico altamente caotico con dos grados de libertad acoplados, as como autofunciones muy excitadas en una ventana de energa, utilizando en ambos casos una base mucho mas eciente que las descritas en la literatura. Ademas, hemos empleado el MGSS para calcular las autofunciones del sistema molecular LiNC/LiCN, que presenta un espacio de fases con zonas de regularidad y con regiones en las que el movimiento es altamente caotico; en este ultimo sistema, hemos calculado de forma muy eciente las autofunciones asociadas a las primeras 66 energas. Finalmente, hemos propuesto un metodo perturbativo para calcular velocidades de reaccion en sistemas abiertos descritos por potenciales anarmonicos. Con este metodo, hemos calculado la velocidad de reaccion de distintos potenciales de uno y dos grados de libertad, as como la velocidad de isomerizacion del sistema molecular LiNC/LiCN, tanto sometido a un ruido blanco (sin correlaciones) como en presencia de un ba~no de atomos de argon, lo que constituye un entorno con correlaciones. El metodo desarrollado es independiente de la supercie divisoria y nos ha permitido obtener correcciones analticas a la famosa formula de Kramers, lo que posibilita el calculo exacto de velocidades de reaccion en potenciales anarmonicos que interaccionan con el entorno

Communication: Transition State Theory for dissipative systems without a dividing surface

Transition state theory is a central cornerstone in reaction dynamics. Its key step is the identification of a dividing surface that is crossed only once by all reactive trajectories. This assumption is often badly violated, especially when the reactive system is coupled to an environment. The calculations made in this way then overestimate the reaction rate and the results depend critically on the choice of the dividing surface. In this Communication, we study the phase space of a stochastically driven system close to an energetic barrier in order to identify the geometric structure unambiguously determining the reactive trajectories, which is then incorporated in a simple rate formula for reactions in condensed phase that is both independent of the dividing surface and exact

Towards AC-induced optimum control of dynamical localization

It is shown that dynamical localization (quantum suppression of classical diffusion) in the context of ultracold atoms in periodically shaken optical lattices subjected to time-periodic modulations having equidistant zeros depends on the impulse transmitted by the external mod- ulation over half-period rather than on the modulation amplitude. This result provides a useful principle for optimally controlling dynamical localization in general periodic systems, which is capable of experimental realization

Dynamic localization in walking droplets?

In this paper we explore the possibility of localization in the dynamics of walking droplets on a vertically vibrated liquid surface in the presence of a chaotic external potential and the limit of low memory, i.e. considering the effect of the wave generated only in the last bounce. Taking into account the similarity existing between this macroscopic system and de Broglie?s pilot wave theory this study aims at establishing manifestations of chaos in the later theory

Rate calculation in two dimensional barriers with colored noise

The identification of an optimal dividing surface that is free of recrossings is the most important re- quirement for transition state theory to be exact. This task is particularly difficult in the presence of non-Markovian friction, i.e., colored noise forces. In this paper, we report a novel geometric method that circumvents the recross- ing problem and is able to (i) identify reactive trajectories exactly, and (ii) compute reaction rates in a system with two degrees of freedom driven by non-Markovian friction. The extension of our method to higher dimensional systems is also discussed

Quantization Scheme for the Experiments with "Walking Droplets"

In this paper we explore the interest and feasibility of quantizing the macroscopic surface wave generated in the dynamics of walking droplets on a vertically vibrated liquid surface in the limit of high memory of the droplet trajectory, where an astonishing similarity with the quantum behavior has been experimentally observed

Bohr-Sommerfeld-like quantization in the theory of walking droplets

Recent experiments have shown that self-propelled millimetric walking droplets bouncing on a vibrating liquid surface exhibit phenomena, such as interference or tunneling, that so far were thought to be possible only in the microscopic realm. Here we present calculations showing that the surface wave satisfies, in the long-memory limit, a Bohr-Sommerfeld quantization-like relation. This strongly suggest the possibility of a novel fundamental type of quantization in these experiments, which can simultaneously explain their emulation of the quantum behavior and, more importantly, shed light into some of the interpretational difficulties of the standard quantum theory

La geometría de la reactividad química

La aplicación de sofisticadas técnicas matemáticas, que derivan de la Mecánica Celeste, a la Teoría del Estado de Transición ha permitido resolver recientemente de forma exacta el conocido como problema del recruzamiento, mediante una reformulación de la teoría en el entorno geométrico adecuado. Sophisticated mathematical techniques, borrowed from Celestial Mechanics, have recently improved Transition State Theory, completely eliminating the so-called recrossing problem by recasting the theory in the right geometrical setting

La geometría de la reactividad química

La aplicación de sofisticadas técnicas matemáticas, que derivan de la Mecánica Celeste, a la Teoría del Estado de Transición ha permitido resolver recientemente de forma exacta el conocido como problema del recruzamiento, mediante una reformulación de la teoría en el entorno geométrico adecuado. Sophisticated mathematical techniques, borrowed from Celestial Mechanics, have recently improved Transition State Theory, completely eliminating the so-called recrossing problem by recasting the theory in the right geometrical setting

Unveiling the chaotic structure in molecular systems using Lagrangian descriptors

The phase space structure of a dynamical system determines, among other issues, its regular and chaotic bahavior, and the several indicators have been developed in order to correctly detect, characterize and analyze it