Thermal relaxation for particle systems in interaction with several bosonic heat reservoirs

The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.

Accumulation of complex eigenvalues of an indefinite Sturm--Liouville operator with a shifted Coulomb potential

For a particular family of long-range potentials V, we prove that the eigenvalues of the indefinite Sturm–Liouville operator A = sign(x)(−Δ+V(x)) accumulate to zero asymptotically along specific curves in the complex plane. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators.

Studies on gaussian effective potentials in some models

In classical field theory, the ordinary potential V is an energy density for that state in which the field assumes the value ¢. In quantum field theory, the effective potential is the expectation value of the energy density for which the expectation value of the field is ¢o. As a result, if V has several local minima, it is only the absolute minimum that corresponds to the true ground state of the theory. Perturbation theory remains to this day the main analytical tool in the study of Quantum Field Theory. However, since perturbation theory is unable to uncover the whole rich structure of Quantum Field Theory, it is desirable to have some method which, on one hand, must go beyond both perturbation theory and classical approximation in the points where these fail, and at that time, be sufficiently simple that analytical calculations could be performed in its framework During the last decade a nonperturbative variational method called Gaussian effective potential, has been discussed widely together with several applications. This concept was described as a means of formalizing our intuitive understanding of zero-point fluctuation effects in quantum mechanics in a way that carries over directly to field theory.

Shared information in stationary states at criticality

We consider bipartitions of one-dimensional extended systems whose probability distribution functions describe stationary states of stochastic models. We define estimators of the information shared between the two subsystems. If the correlation length is finite, the estimators stay finite for large system sizes. If the correlation length diverges, so do the estimators. The definition of the estimators is inspired by information theory. We look at several models and compare the behaviors of the estimators in the finite-size scaling limit. Analytical and numerical methods as well as Monte Carlo simulations are used. We show how the finite-size scaling functions change for various phase transitions, including the case where one has conformal invariance.

General unitary SU(1,1) TFD formulation

In this Letter we discuss a generalization for the thermal Bogoliubov transformation in the context of a Hermitian general SU(1,1) transformation generator. The TFD tilde conjugation rules are redefined using an appropriated Tomita-Takesaki modular operator. (C) 2003 Elsevier B.V. All rights reserved.

Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Pseudodifferential analysis in Psi*-algebras on transmission spaces, infinite solving ideal chains and K-theory for conformally compact spaces

The present thesis is a contribution to the theory of algebras of pseudodifferential operators on singular settings. In particular, we focus on the $b$-calculus and the calculus on conformally compact spaces in the sense of Mazzeo and Melrose in connection with the notion of spectral invariant transmission operator algebras. We summarize results given by Gramsch et. al. on the construction of $Psi_0$-and $Psi*$-algebras and the corresponding scales of generalized Sobolev spaces using commutators of certain closed operators and derivations. In the case of a manifold with corners $Z$ we construct a $Psi*$-completion $A_b(Z,{}^bOmega^{1/2})$ of the algebra of zero order $b$-pseudodifferential operators $Psi_{b,cl}(Z, {}^bOmega^{1/2})$ in the corresponding $C*$-closure $B(Z,{}^bOmega^{12})hookrightarrow L(L^2(Z,{}^bOmega^{1/2}))$. The construction will also provide that localised to the (smooth) interior of Z the operators in the $A_b(Z, {}^bOmega^{1/2})$ can be represented as ordinary pseudodifferential operators. In connection with the notion of solvable $C*$-algebras - introduced by Dynin - we calculate the length of the $C*$-closure of $Psi_{b,cl}^0(F,{}^bOmega^{1/2},R^{E(F)})$ in $B(F,{}^bOmega^{1/2}),R^{E(F)})$ by localizing $B(Z, {}^bOmega^{1/2})$ along the boundary face $F$ using the (extended) indical familiy $I^B_{FZ}$. Moreover, we discuss how one can localise a certain solving ideal chain of $B(Z, {}^bOmega^{1/2})$ in neighbourhoods $U_p$ of arbitrary points $pin Z$. This localisation process will recover the singular structure of $U_p$; further, the induced length function $l_p$ is shown to be upper semi-continuous. We give construction methods for $Psi*$- and $C*$-algebras admitting only infinite long solving ideal chains. These algebras will first be realized as unconnected direct sums of (solvable) $C*$-algebras and then refined such that the resulting algebras have arcwise connected spaces of one dimensional representations. In addition, we recall the notion of transmission algebras on manifolds with corners $(Z_i)_{iin N}$ following an idea of Ali Mehmeti, Gramsch et. al. Thereby, we connect the underlying $C^infty$-function spaces using point evaluations in the smooth parts of the $Z_i$ and use generalized Laplacians to generate an appropriate scale of Sobolev spaces. Moreover, it is possible to associate generalized (solving) ideal chains to these algebras, such that to every $ninN$ there exists an ideal chain of length $n$ within the algebra. Finally, we discuss the $K$-theory for algebras of pseudodifferential operators on conformally compact manifolds $X$ and give an index theorem for these operators. In addition, we prove that the Dirac-operator associated to the metric of a conformally compact manifold $X$ is not a Fredholm operator.

Towards Accurate and Efficient Description of Excited States

Thesis (Ph.D.)--University of Washington, 2016-06

Benefits of using a spectral hindcast database for wave power extraction assessment

When considering deployment of wave energy converters at a given site, it is of prime importance from both a technical and an economical point of view to accurately assess the total yearly energy that can be extracted by the given device. Especially, to be considered is the assessment of the efficiency of the device over the widest span of the sea-states spectral bandwidth. Hence, the aim of this study is to assess the biases and errors introduced on extracted power classically computed using spectral data derived from analytical functions such as a JONSWAP spectrum, compared to the power derived using actual wave spectra obtained from a spectral hindcast database.

Local approximations for polarization potentials

We discuss the derivation of an equivalent polarization potential independent of angular momentum l for use in the optical Schrodinger equation that describes the elastic scattering of heavy ions. Three different methods are used for this purpose. Application of our theory to the low energy scattering of light heavy-ion systems at near-barrier energies is made. It is found that the notion of an l-independent polarization potential has some validity but cannot be a good substitute for the l-dependent local equivalent Feshbach polarization potential.

Elongated shape isomers in the (36)Ar nucleus

A recent analysis of the (12)C + (24)Mg scattering [W. Sciani et al., Phys. Rev. C 80, 034319 (2009)] suggests the existence of a hyperdeformed band in the (36)Ar nucleus, completely in line with the predictions of alpha [W. D. M. Rae and A. C. Merchant, Phys. Lett. B279, 207 (1992)] and binary cluster calculations [J. Cseh et al., Phys. Rev. C 70, 034311 (2004)]. Here we review the structural understanding of the superdeformed and the hyperdeformed states of (36)Ar and present new results on the shape isomers as well. Special attention is paid to the clusterization of these states, which indicates the appropriate reaction channels for their formation.

Theoretical study of one- and two-photon absorption spectra of azoaromatic compounds

In this study, the one- and two-photon absorption spectra of seven azoaromatic compounds (five pseudostilbenes-type and two aminoazobenzenes) were theoretically investigated using the density functional theory combined with the response functions formalism. The equilibrium molecular structure of each compound was obtained at three different levels of theory: Hartree-Fock, density functional theory (DFT), and Moller-Plesset 2. The effect of solvent on the equilibrium structure and the electronic transitions of the compounds were investigated using the polarizable continuum model. For the one-photon absorption, the allowed pi ->pi(*) transition energy showed to be dependent on the molecular structures and the effect of solvent, while the n ->pi(*) and pi ->pi(*)(n) transition energies exhibited only a slight dependence. An inversion between the bands corresponding to the pi ->pi(*) and n ->pi(*) states due to the effect of solvent was observed for the pseudostilbene-type compounds. To characterize the allowed two-photon absorption transitions for azoaromatic compounds, the response functions formalism combined with DFT using the hybrid B3LYP and PBE0 functionals and the long-range corrected CAM-B3LYP functional was employed. The theoretical results support the previous findings based on the three-state model. The model takes into account the ground and two electronic excited states and has already been used to describe and interpret the two-photon absorption spectrum of azoaromatic compounds. The highest energy two-photon allowed transition for the pseudostilbene-type compounds shows to be more effectively affected (similar to 20%) by the torsion of the molecular structure than the lowest allowed transition (similar to 10%). In order to elucidate the effect of the solvent on the two-photon absorption spectra, the lowest allowed two-photon transition (dipolar transition) for each compound was analyzed using a two-state approximation and the polarizable continuum model. The results obtained reveal that the effect of solvent increases drastically the two-photon cross-section of the dipolar transition of the pseudostilbene-type compounds. In general, the features of both one- and two-photon absorption spectra of the azoaromatic compounds are well reproduced by the theoretical calculations.

Experiments on stress-triaxiality dependence of material behavior of aluminum alloys

The paper presents and discusses experimental procedures, visual observations and test results considered important to obtain data that can be used in validation of constitutive relations and failure criteria. The aim is to investigate the combined effects of stress intensity, stress-triaxiality and Lode parameter on the material response and failure behavior of aluminum alloys. Smooth and pre-notched tensile and shear specimens were manufactured from both very thin sheets and thicker plates to cover a wide range of stress triaxialities and Lode parameters. In addition, modified Arcan specimens were designed allowing investigation of the effect of sudden changes in stress states and deformation modes on the material behavior. (C) 2009 Elsevier Ltd. All rights reserved.