All fields of physics - be it nuclear, atomic and molecular, solid state, or optical - offer examples of systems which are strongly influenced by the environment of the actual system under investigation. The scope of what is called "the environment" may vary, i.e., how far from the system of interest an interaction between the two does persist. Typically, however, it is much larger than the open system itself. Hence, a fully quantum mechanical treatment of the combined system without approximations and without limitations of the type of system is currently out of reach. With the single assumption of the environment to consist of an internally thermalized set of infinitely many harmonic oscillators, the seminal work of Stockburger and Grabert [Chem. Phys., 268:249-256, 2001] introduced an open system description that captures the environmental influence by means of a stochastic driving of the reduced system. The resulting stochastic Liouville-von Neumann equation describes the full non-Markovian dynamics without explicit memory but instead accounts for it implicitly through the correlations of the complex-valued noise forces. The present thesis provides a first application of the Stockburger-Grabert stochastic Liouville-von Neumann equation to the computation of the dynamics of anharmonic, continuous open systems. In particular, it is demonstrated that trajectory based propagators allow for the construction of a numerically stable propagation scheme. With this approach it becomes possible to achieve the tremendous increase of the noise sample count necessary to stochastically converge the results when investigating such systems with continuous variables. After a test against available analytic results for the dissipative harmonic oscillator, the approach is subsequently applied to the analysis of two different realistic, physical systems. As a first example, the dynamics of a dissipative molecular oscillator is investigated. Long time propagation - until thermalization is reached - is shown to be possible with the presented approach. The properties of the thermalized density are determined and they are ascertained to be independent of the system's initial state. Furthermore, the dependence on the bath's temperature and coupling strength is analyzed and it is demonstrated how a change of the bath parameters can be used to tune the system from the dissociative to the bound regime. A second investigation is conducted for a dissipative tunneling scenario in which a wave packet impinges on a barrier. The dependence of the transmission probability on the initial state's kinetic energy as well as the bath's temperature and coupling strength is computed. For both systems, a comparison with the high-temperature Markovian quantum Brownian limit is performed. The importance of a full non-Markovian treatment is demonstrated as deviations are shown to exist between the two descriptions both in the low temperature cases where they are expected and in some of the high temperature cases where their appearance might not be anticipated as easily.

abstract: The tools developed for the use of investigating dynamical systems have provided critical understanding to a wide range of physical phenomena. Here these tools are used to gain further insight into scalar transport, and how it is affected by mixing. The aim of this research is to investigate the efficiency of several different partitioning methods which demarcate flow fields into dynamically distinct regions, and the correlation of finite-time statistics from the advection-diffusion equation to these regions. For autonomous systems, invariant manifold theory can be used to separate the system into dynamically distinct regions. Despite there being no equivalent method for nonautonomous systems, a similar analysis can be done. Systems with general time dependencies must resort to using finite-time transport barriers for partitioning; these barriers are the edges of Lagrangian coherent structures (LCS), the analog to the stable and unstable manifolds of invariant manifold theory. Using the coherent structures of a flow to analyze the statistics of trapping, flight, and residence times, the signature of anomalous diffusion are obtained. This research also investigates the use of linear models for approximating the elements of the covariance matrix of nonlinear flows, and then applying the covariance matrix approximation over coherent regions. The first and second-order moments can be used to fully describe an ensemble evolution in linear systems, however there is no direct method for nonlinear systems. The problem is only compounded by the fact that the moments for nonlinear flows typically don't have analytic representations, therefore direct numerical simulations would be needed to obtain the moments throughout the domain. To circumvent these many computations, the nonlinear system is approximated as many linear systems for which analytic expressions for the moments exist. The parameters introduced in the linear models are obtained locally from the nonlinear deformation tensor. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2018

All fields of physics - be it nuclear, atomic and molecular, solid state, or optical - offer examples of systems which are strongly influenced by the environment of the actual system under investigation. The scope of what is called "the environment" may vary, i.e., how far from the system of interest an interaction between the two does persist. Typically, however, it is much larger than the open system itself. Hence, a fully quantum mechanical treatment of the combined system without approximations and without limitations of the type of system is currently out of reach. With the single assumption of the environment to consist of an internally thermalized set of infinitely many harmonic oscillators, the seminal work of Stockburger and Grabert [Chem. Phys., 268:249-256, 2001] introduced an open system description that captures the environmental influence by means of a stochastic driving of the reduced system. The resulting stochastic Liouville-von Neumann equation describes the full non-Markovian dynamics without explicit memory but instead accounts for it implicitly through the correlations of the complex-valued noise forces. The present thesis provides a first application of the Stockburger-Grabert stochastic Liouville-von Neumann equation to the computation of the dynamics of anharmonic, continuous open systems. In particular, it is demonstrated that trajectory based propagators allow for the construction of a numerically stable propagation scheme. With this approach it becomes possible to achieve the tremendous increase of the noise sample count necessary to stochastically converge the results when investigating such systems with continuous variables. After a test against available analytic results for the dissipative harmonic oscillator, the approach is subsequently applied to the analysis of two different realistic, physical systems. As a first example, the dynamics of a dissipative molecular oscillator is investigated. Long time propagation - until thermalization is reached - is shown to be possible with the presented approach. The properties of the thermalized density are determined and they are ascertained to be independent of the system's initial state. Furthermore, the dependence on the bath's temperature and coupling strength is analyzed and it is demonstrated how a change of the bath parameters can be used to tune the system from the dissociative to the bound regime. A second investigation is conducted for a dissipative tunneling scenario in which a wave packet impinges on a barrier. The dependence of the transmission probability on the initial state's kinetic energy as well as the bath's temperature and coupling strength is computed. For both systems, a comparison with the high-temperature Markovian quantum Brownian limit is performed. The importance of a full non-Markovian treatment is demonstrated as deviations are shown to exist between the two descriptions both in the low temperature cases where they are expected and in some of the high temperature cases where their appearance might not be anticipated as easily.:1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Theory of Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Influence Functional Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Quantum Brownian Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Stochastic Unraveling of the Influence Functional . . . . . . . . . . . . . . . 20 2.4 Improved Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Modified Dynamic Response . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.2 Guide Trajectory Transformation . . . . . . . . . . . . . . . . . . . . 24 2.5 Obtaining Properly Correlated Stochastic Samples from Filtered White Noise 24 3 Unified Stochastic Trajectory Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1 Semiclassical Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Guide Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.2 Real Coherent State Center Coordinates . . . . . . . . . . . . . . . . 31 3.1.3 Propagation Scheme Including Stochastic Forces . . . . . . . . . . . 32 3.2 Stochastic Bohmian Mechanics with Complex Action . . . . . . . . . . . . . 33 3.2.1 Hydrodynamic Formulation of Bohmian Mechanics . . . . . . . . . . 33 3.2.2 Bohmian Mechanics with Complex Action . . . . . . . . . . . . . . . 34 3.2.3 Stochastic BOMCA Trajectories . . . . . . . . . . . . . . . . . . . . 38 3.3 Noise Distribution Preserving Removal of Adverse Samples . . . . . . . . . . 39 4 Dissipative Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1 Reservoir Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Harmonic Oscillator Analytic Expectation Values . . . . . . . . . . . . . . . 42 4.2.1 Ohmic Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.2 Drude Regularized Bath . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Sampling Strategies and Analytic Comparison . . . . . . . . . . . . . . . . . 44 4.4 Limits of the Markovian Approximation . . . . . . . . . . . . . . . . . . . . 45 5 Dissipative Vibrational Dynamics of Diatomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1 Molecular Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Anharmonic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Transient Non-Markovian Effects . . . . . . . . . . . . . . . . . . . . . . . . 53 5.4 Trapping by Dissipation and Thermalization . . . . . . . . . . . . . . . . . . 53 6 Tunneling with Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.1 Eckart Barrier Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2 Dissipative Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.3 Investigation of Markovianity . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Appendix A Conventions for Constants, Reservoir Kernels, and Influence Phases 69 Appendix B Stochastic Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 B.1 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 71 B.2 Position Verlet Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 B.3 Runge-Kutta Fourth Order Scheme . . . . . . . . . . . . . . . . . . . . . . . 73 Appendix CMorse Oscillator Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Appendix DPrerequisites of a Successful Stochastic Propagation . . . . . . . . . . . . . . 79 D.1 Hubbard-Stratonovich Transformation . . . . . . . . . . . . . . . . . . . . . 79 D.2 Kernels of the Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 D.2.1 Quadratic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 D.2.2 Quartic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 D.2.3 Strictly Ohmic Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 89 D.3 Guide Trajectory Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 90 D.3.1 Quadratic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 D.3.2 Quartic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 D.3.3 Strictly Ohmic Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 92 D.4 Computation of Matrix Elements and Expectation Values . . . . . . . . . . 92 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97