Dichtematrix-Renormierung, angewandt auf nichtlineare dynamische Systeme

Bogner T. Density matrix renormalisation applied to nonlinear dynamical systems. Bielefeld (Germany): Bielefeld University; 2007.In dieser Dissertation wird die effektive numerische Beschreibung nichtlinearer dynamischer Systeme untersucht. Systeme dieser Art tauchen praktisch überall auf, wo zeitabhängige Größen quantitativ untersucht werden, d.h. in fast allen Bereichen der Physik, aber auch in der Biologie, Ökonomie oder Mathematik. Ziel ist die Bestimmung reduzierter Modelle, deren Phasenraum eine signifikant reduzierte Dimensionalität aufweist. Dies wird erreicht durch Benutzung von Konzepten aus der Dichtematrix-Renormierung. In dieser Arbeit werden drei neue Anwendungen vorgeschlagen. Zuerst wird eine Dichtematrix-Renormierungsmethode zur Berechnung einer Schur-Zerlegung vorgestellt. Verglichen mit bereits existierenden Arbeiten liegt der Vorteil dieses Ansatzes in der Möglichkeit, auch für nicht-normale Operatoren orthonormale Basen von sukzessive invarianten Unterräumen zu bestimmen. Der Algorithmus wird dann angewandt auf Gittermodelle stochastischer Systeme, wobei als Beispiele ein Reaktions-Diffusions- und ein Oberflächenablagerungs-Modell dienen. Als Nächstes wird ein Dichtematrix-Renormierungsansatz für die orthogonale Zerlegung (proper orthogonal decomposition) entwickelt. Diese Zerlegung erlaubt die Bestimmung relevanter linearer Unterräume auch für nichtlineare Systeme. Durch die Verwendung der Dichtematrix-Renormierung werden alle Berechnungen nur für kleine Untersysteme durchgeführt. Dabei werden diskretisierte partielle Differentialgleichungen, d.h. die Diffusionsgleichung, die Burgers-Gleichung und eine nichtlineare Diffusionsgleichung als numerische Beispiele betrachtet. Schließlich wird das vorige Konzept auf höherdimensionale Probleme in Form eines Variationsverfahrens erweitert. Dies Verfahren wird dann an den zweidimensionalen Navier-Stokes-Gleichungen erprobt.In this work the effective numerical description of nonlinear dynamical systems is investigated. Such systems arise in most fields of physics, as well as in mathematics, biology, economy and essentially in all problems for which a quantitative description of a time evolution is considered. The aim is to find reduced models with a phase space of significantly reduced dimensionality. This is achieved by the use of concepts from density matrix renormalisation. Three new applications are proposed in this work. First, a density matrix renormalisation method for calculating a Schur decomposition is introduced. The advantage of this approach, compared to existing work, is the possibility to obtain orthonormal bases for successively invariant subspaces even if the generator of evolution is not normal. The algorithm is applied to lattice models for stochastic systems, namely a reaction diffusion and a surface deposition model. Next, a density matrix renormalisation approach to the proper orthogonal decomposition is developed. This allows the determination of relevant linear subspaces even for nonlinear systems. Due to the use of density matrix renormalisation concepts, all calculations are done on small subsystems. Here discretised partial differential equations, i.e. the diffusion equation, the Burgers equation and a nonlinear diffusion equation are considered as numerical examples. Finally, the previous concept is extended to higher dimensional problems in a variational form. This method is then applied to the two-dimensional, incompressible Navier-Stokes equations as testing ground

Molecular recognition in a lattice model: An enumeration study

We investigate the mechanisms underlying selective molecular recognition of single heteropolymers at chemically structured planar surfaces. To this end, we study systems with two-letter (HP) lattice heteropolymers by exact enumeration techniques. Selectivity for a particular surface is defined by an adsorption energy criterium. We analyze the distributions of selective sequences and the role of mutations. A particularly important factor for molecular recognition is the small-scale structure on the polymers.Comment: revised version with additional plot

Delocalization in Coupled Luttinger Liquids with Impurities

We study effects of quenched disorder on coupled two-dimensional arrays of Luttinger liquids (LL) as a model for stripes in high-T_c compounds. In the framework of a renormalization-group analysis, we find that weak inter-LL charge-density-wave couplings are always irrelevant as opposed to the pure system. By varying either disorder strength, intra- or inter-LL interactions, the system can undergo a delocalization transition between an insulator and a novel strongly anisotropic metallic state with LL-like transport. This state is characterized by short-ranged charge-density-wave order, the superconducting order is quasi long-ranged along the stripes and short-ranged in the transversal direction.Comment: 6 pages, 5 figures, substantially extended and revised versio

Is there a Glass Transition in Planar Vortex Systems?

The criteria for the existence of a glass transition in a planar vortex array with quenched disorder are studied. Applying a replica Bethe ansatz, we obtain for self-avoiding vortices the exact quenched average free energy and effective stiffness which is found to be in excellent agreement with recent numerical results for the related random bond dimer model [1]. Including a repulsive vortex interaction and a finite vortex persistence length \xi, we find that for \xi \to 0 the system is at {\em all} temperatures in a glassy phase; a glass transition exists only for finite \xi. Our results indicate that planar vortex arrays in superconducting films are glassy at presumably all temperatures.Comment: 4 pages, 1 figur

Test of Replica Theory: Thermodynamics of 2D Model Systems with Quenched Disorder

We study the statistics of thermodynamic quantities in two related systems with quenched disorder: A (1+1)-dimensional planar lattice of elastic lines in a random potential and the 2-dimensional random bond dimer model. The first system is examined by a replica-symmetric Bethe ansatz (RBA) while the latter is studied numerically by a polynomial algorithm which circumvents slow glassy dynamics. We establish a mapping of the two models which allows for a detailed comparison of RBA predictions and simulations. Over a wide range of disorder strength, the effective lattice stiffness and cumulants of various thermodynamic quantities in both approaches are found to agree excellently. Our comparison provides, for the first time, a detailed quantitative confirmation of the replica approach and renders the planar line lattice a unique testing ground for concepts in random systems.Comment: 16 pages, 14 figure

Nonuniversal Correlations and Crossover Effects in the Bragg-Glass Phase of Impure Superconductors

The structural correlation functions of a weakly disordered Abrikosov lattice are calculated in a functional RG-expansion in $d=4-\epsilon$ dimensions. It is shown, that in the asymptotic limit the Abrikosov lattice exhibits still quasi-long-range translational order described by a {\it nonuniversal} exponent $\eta_{\bf G}$ which depends on the ratio of the renormalized elastic constants $\kappa ={c}_{66}/ {c}_{11}$ of the flux line (FL) lattice. Our calculations clearly demonstrate three distinct scaling regimes corresponding to the Larkin, the random manifold and the asymptotic Bragg-glass regime. On a wide range of {\it intermediate} length scales the FL displacement correlation function increases as a power law with twice the manifold roughness exponent $\zeta_{\rm RM}(\kappa)$, which is also {\it nonuniversal}. Correlation functions in the asymptotic regime are calculated in their full anisotropic dependencies and various order parameters are examined. Our results, in particular the $\kappa$-dependency of the exponents, are in variance with those of the variational treatment with replica symmetry breaking which allows in principle an experimental discrimination between the two approaches.Comment: 17 pages, 10 figure

Developing and analyzing idealized models for molecular recognition

Behringer H, Bogner T, Polotsky A, Degenhard A, Schmid F. Developing and analyzing idealized models for molecular recognition. Journal of Biotechnology. 2007;129(2):268-278.We study equilibrium aspects of molecular recognition of two biomolecules using idealized model systems and methods from statistical physics. Starting from the basic experimental findings we demonstrate exemplarily how an idealized coarse-grained model for the investigation of molecular recognition of two biomolecules can be developed. In addition we provide details regarding two model systems for the recognition of a flexible and a rigid biomolecule respectively, the latter taking into account conformational changes. We focus particularly on the interplay and influence of the correlations of the residue distributions of the biomolecules on the recognition process. (c) 2007 Elsevier B.V. All rights reserved