Energy-entropy-momentum time integration methods for coupled smooth dissipative problems

Esta tesis aborda la formulación, análisis e implementación de métodos numéricos de integración temporal para la solución de sistemas disipativos suaves de dimensión finita o infinita de manera que su estructura continua sea conservada. Se entiende por dichos sistemas aquellos que involucran acoplamiento termo-mecánico y/o efectos disipativos internos modelados por variables internas que siguen leyes continuas, de modo que su evolución es considerada suave. La dinámica de estos sistemas está gobernada por las leyes de la termodinámica y simetrías, las cuales constituyen la estructura que se pretende conservar de forma discreta. Para ello, los sistemas disipativos se describen geométricamente mediante estructuras metriplécticas que identifican claramente las partes reversible e irreversible de la evolución del sistema. Así, usando una de estas estructuras conocida por las siglas (en inglés) de GENERIC, la estructura disipativa de los sistemas es identificada del mismo modo que lo es la Hamiltoniana para sistemas conservativos. Con esto, métodos (EEM) con precisión de segundo orden que conservan la energía, producen entropía y conservan los impulsos lineal y angular son formulados mediante el uso del operador derivada discreta introducido para asegurar la conservación de la Hamiltoniana y las simetrías de sistemas conservativos. Siguiendo estas directrices, se formulan dos tipos de métodos EEM basados en el uso de la temperatura o de la entropía como variable de estado termodinámica, lo que presenta importantes implicaciones que se discuten a lo largo de esta tesis. Entre las cuales cabe destacar que las condiciones de contorno de Dirichlet son naturalmente impuestas con la formulación basada en la temperatura. Por último, se validan dichos métodos y se comprueban sus mejores prestaciones en términos de la estabilidad y robustez en comparación con métodos estándar. This dissertation is concerned with the formulation, analysis and implementation of structure-preserving time integration methods for the solution of the initial(-boundary) value problems describing the dynamics of smooth dissipative systems, either finite- or infinite-dimensional ones. Such systems are understood as those involving thermo-mechanical coupling and/or internal dissipative effects modeled by internal state variables considered to be smooth in the sense that their evolutions follow continuos laws. The dynamics of such systems are ruled by the laws of thermodynamics and symmetries which constitutes the structure meant to be preserved in the numerical setting. For that, dissipative systems are geometrically described by metriplectic structures which clearly identify the reversible and irreversible parts of their dynamical evolution. In particular, the framework known by the acronym GENERIC is used to reveal the systems' dissipative structure in the same way as the Hamiltonian is for conserving systems. Given that, energy-preserving, entropy-producing and momentum-preserving (EEM) second-order accurate methods are formulated using the discrete derivative operator that enabled the formulation of Energy-Momentum methods ensuring the preservation of the Hamiltonian and symmetries for conservative systems. Following these guidelines, two kind of EEM methods are formulated in terms of entropy and temperature as a thermodynamical state variable, involving important implications discussed throughout the dissertation. Remarkably, the formulation in temperature becomes central to accommodate Dirichlet boundary conditions. EEM methods are finally validated and proved to exhibit enhanced numerical stability and robustness properties compared to standard ones.

First-order equivalent to Einstein-Hilbert Lagrangian

A first-order Lagrangian L ∇ variationally equivalent to the second-order Einstein- Hilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by L ∇ is proved to be regular and its Hamiltonian formulation is studied, including its covariant Hamiltonian attached to ∇ .

Comparing the Luttinger–Kohn–Pikus–Bir and the Empiric K·P Hamiltonians in quantum dot intermediate band solar cells manufactured in zincblende semiconductors

The calculation of the energy spectrum and absorption coefficients of quantum dot nanostructured intermediate band solar cells using the Empiric K·P Hamiltonian method and its agreement with experimental data are summarized. The well established Luttinger Kohn Hamiltonian modified by Pikus and Bir for strained material, such as quantum dot arrays, is presented using a simplified strain field that allows for square band offsets. The energy spectrum and absorption coefficients are calculated with this new Hamiltonian. With the approximations made the energy spectrum results to be exactly the same but the absorption coefficient fits experiments less accurately. The computer time using the latter Hamiltonian is much longer than the former one.

Mössbauer and electron paramagnetic resonance studies of chloroperoxidase following mechanism-based inactivation with allylbenzene

We have used Mössbauer and electron paramagnetic resonance (EPR) spectroscopy to study a heme-N-alkylated derivative of chloroperoxidase (CPO) prepared by mechanism-based inactivation with allylbenzene and hydrogen peroxide. The freshly prepared inactivated enzyme (“green CPO”) displayed a nearly pure low-spin ferric EPR signal with g = 1.94, 2.15, 2.31. The Mössbauer spectrum of the same species recorded at 4.2 K showed magnetic hyperfine splittings, which could be simulated in terms of a spin Hamiltonian with a complete set of hyperfine parameters in the slow spin fluctuation limit. The EPR spectrum of green CPO was simulated using a three-term crystal field model including g-strain. The best-fit parameters implied a very strong octahedral field in which the three 2T2 levels of the (3d)5 configuration in green CPO were lowest in energy, followed by a quartet. In native CPO, the 6A1 states follow the 2T2 ground state doublet. The alkene-mediated inactivation of CPO is spontaneously reversible. Warming of a sample of green CPO to 22°C for increasing times before freezing revealed slow conversion of the novel EPR species to two further spin S = ½ ferric species. One of these species displayed g = 1.82, 2.25, 2.60 indistinguishable from native CPO. By subtracting spectral components due to native and green CPO, a third species with g = 1.86, 2.24, 2.50 could be generated. The EPR spectrum of this “quasi-native CPO,” which appears at intermediate times during the reactivation, was simulated using best-fit parameters similar to those used for native CPO.

Stability of nonrelativistic quantum mechanical matter coupled to the (ultraviolet cutoff) radiation field

We announce a proof of H-stability for the quantized radiation field, with ultraviolet cutoff, coupled to arbitrarily many non-relativistic quantized electrons and static nuclei. Our result holds for arbitrary atomic numbers and fine structure constant. We also announce bounds for the energy of many electrons and nuclei in a classical vector potential and for the eigenvalue sum of a one-electron Pauli Hamiltonian with magnetic field.

The two-dimensional IR nonlinear spectroscopy of a cyclic penta-peptide in relation to its three-dimensional structure

A form of two-dimensional (2D) vibrational spectroscopy, which uses two ultrafast IR laser pulses, is used to examine the structure of a cyclic penta-peptide in solution. Spectrally resolved cross peaks occur in the off-diagonal region of the 2D IR spectrum of the amide I region, analogous to those in 2D NMR spectroscopy. These cross peaks measure the coupling between the different amide groups in the structure. Their intensities and polarizations relate directly to the three-dimensional structure of the peptide. With the help of a model coupling Hamiltonian, supplemented by density functional calculations, the spectra of this penta-peptide can be regenerated from the known solution phase structure. This 2D-IR measurement, with an intrinsic time resolution of less than 1 ps, could be used in all time regimes of interest in biology.

Isotropic isotopy and symplectic null sets

Capacity is an important numerical invariant of symplectic manifolds. This paper studies when a subset of a symplectic manifold is null, i.e., can be removed without affecting the ambient capacity. After examples of open null sets and codimension-2 non-null sets, geometric techniques are developed to perturb any isotopy of a loop to a hamiltonian flow; it follows that sets of dimension 0 and 1 are null. For isotropic sets of higher dimensions, obstructions to the perturbation are found in homotopy groups of the orthogonal groups.

Dynamical hierarchy in transition states: Why and how does a system climb over the mountain?

How a reacting system climbs through a transition state during the course of a reaction has been an intriguing subject for decades. Here we present and quantify a technique to identify and characterize local invariances about the transition state of an N-particle Hamiltonian system, using Lie canonical perturbation theory combined with microcanonical molecular dynamics simulation. We show that at least three distinct energy regimes of dynamical behavior occur in the region of the transition state, distinguished by the extent of their local dynamical invariance and regularity. Isomerization of a six-atom Lennard–Jones cluster illustrates this: up to energies high enough to make the system manifestly chaotic, approximate invariants of motion associated with a reaction coordinate in phase space imply a many-body dividing hypersurface in phase space that is free of recrossings even in a sea of chaos. The method makes it possible to visualize the stable and unstable invariant manifolds leading to and from the transition state, i.e., the reaction path in phase space, and how this regularity turns to chaos with increasing total energy of the system. This, in turn, illuminates a new type of phase space bottleneck in the region of a transition state that emerges as the total energy and mode coupling increase, which keeps a reacting system increasingly trapped in that region.

Conformation, energy, and folding ability of selected amino acid sequences.

Evolutionary selection of sequences is studied with a knowledge-based Hamiltonian to find the design principle for folding to a model protein structure. With sequences selected by naive energy minimization, the model structure tends to be unstable and the folding ability is low. Sequences with high folding ability have only the low-lying energy minimum but also an energy landscape which is similar to that found for the native sequence over a wide region of the conformation space. Though there is a large fluctuation in foldable sequences, the hydrophobicity pattern and the glycine locations are preserved among them. Implications of the design principle for the molecular mechanism of folding are discussed.

Critical behavior of su(1|1) supersymmetric spin chains with long-range interactions

We introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain.

Anisotropic exchange interaction induced by a single photon in semiconductor microcavities

We investigate coupling of localized spins in a semiconductor quantum dot embedded in a microcavity. The lowest cavity mode and the quantum dot exciton are coupled and close in energy, forming a polariton. The fermions forming the exciton interact with localized spins via exchange. Exact diagonalization of a Hamiltonian in which photons, spins, and excitons are treated quantum mechanically shows that a single polariton induces a sizable indirect anisotropic exchange interaction between spins. At sufficiently low temperatures strong ferromagnetic correlations show up without an appreciable increase in exciton population. In the case of a (Cd,Mn)Te quantum dot, Mn-Mn ferromagnetic coupling is still significant at 1 K: spin-spin correlation around 3 for exciton occupation smaller than 0.3. We find that the interaction mediated by photon-polaritons is 10 times stronger than the one induced by a classical field for equal Rabi splitting.

Single-exciton spectroscopy of semimagnetic quantum dots

A photoexcited II-VI semiconductor quantum dots doped with a few Mn spins is considered. The effects of spin-exciton interactions and the resulting multispin correlations on the photoluminescence are calculated by numerical diagonalization of the Hamiltonian, including exchange interaction between electrons, holes, and Mn spins, as well as spin-orbit interaction. The results provide a unified description of recent experiments on the photoluminesnce of dots with one and many Mn atoms as well as optically induced ferromagnetism in semimagnetic quantum dots.

Single-exciton spectroscopy of single Mn doped InAs quantum dots

The optical spectroscopy of a single InAs quantum dot doped with a single Mn atom is studied using a model Hamiltonian that includes the exchange interactions between the spins of the quantum dot electron-hole pair, the Mn atom, and the acceptor hole. Our model permits linking the photoluminescence spectra to the Mn spin states after photon emission. We focus on the relation between the charge state of the Mn, A0 or A−, and the different spectra which result through either band-to-band or band-to-acceptor transitions. We consider both neutral and negatively charged dots. Our model is able to account for recent experimental results on single Mn doped InAs photoluminescence spectra and can be used to account for future experiments in GaAs quantum dots. Similarities and differences with the case of single Mn doped CdTe quantum dots are discussed.

Spin dynamics of current-driven single magnetic adatoms and molecules

A scanning tunneling microscope can probe the inelastic spin excitations of a single magnetic atom in a surface via spin-flip assisted tunneling in which transport electrons exchange spin and energy with the atomic spin. If the inelastic transport time, defined as the average time elapsed between two inelastic spin flip events, is shorter than the atom spin-relaxation time, the scanning tunnel microscope (STM) current can drive the spin out of equilibrium. Here we model this process using rate equations and a model Hamiltonian that describes successfully spin-flip-assisted tunneling experiments, including a single Mn atom, a Mn dimer, and Fe Phthalocyanine molecules. When the STM current is not spin polarized, the nonequilibrium spin dynamics of the magnetic atom results in nonmonotonic dI/dV curves. In the case of spin-polarized STM current, the spin orientation of the magnetic atom can be controlled parallel or antiparallel to the magnetic moment of the tip. Thus, spin-polarized STM tips can be used both to probe and to control the magnetic moment of a single atom.

Cotunneling theory of atomic spin inelastic electron tunneling spectroscopy

We propose cotunneling as the microscopic mechanism that makes possible inelastic electron tunneling spectroscopy of magnetic atoms in surfaces for a wide range of systems, including single magnetic adatoms, molecules, and molecular stacks. We describe electronic transport between the scanning tip and the conducting surface through the magnetic system (MS) with a generalized Anderson model, without making use of effective spin models. Transport and spin dynamics are described with an effective cotunneling Hamiltonian in which the correlations in the magnetic system are calculated exactly and the coupling to the electrodes is included up to second order in the tip MS and MS substrate. In the adequate limit our approach is equivalent to the phenomenological Kondo exchange model that successfully describes the experiments. We apply our method to study in detail inelastic transport in two systems, stacks of cobalt phthalocyanines and a single Mn atom on Cu2N. Our method accounts for both the large contribution of the inelastic spin exchange events to the conductance and the observed conductance asymmetry.