22 research outputs found
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
Density Matrix Renormalization for Model Reduction in Nonlinear Dynamics
We present a novel approach for model reduction of nonlinear dynamical
systems based on proper orthogonal decomposition (POD). Our method, derived
from Density Matrix Renormalization Group (DMRG), provides a significant
reduction in computational effort for the calculation of the reduced system,
compared to a POD. The efficiency of the algorithm is tested on the one
dimensional Burgers equations and a one dimensional equation of the Fisher type
as nonlinear model systems.Comment: 12 pages, 12 figure
Normalized STEAM-based diffusion tensor imaging provides a robust assessment of muscle tears in football players: preliminary results of a new approach to evaluate muscle injuries
Objectives: To assess acute muscle tears in professional football players by diffusion tensor imaging (DTI) and evaluate the impact of normalization of data.
Methods: Eight football players with acute lower limb muscle tears were examined. DTI metrics of the injured muscle and corresponding healthy contralateral muscle and of ROIs drawn in muscle tears (ROItear) in the corresponding healthy contralateral muscle (ROIhc_t) in a healthy area ipsilateral to the injury (ROIhi) and in a corresponding contralateral area (ROIhc_i) were compared. The same comparison was performed for ratios of the injured (ROItear/ROIhi) and contralateral sides (ROIhc_t/ROIhc_i). ANOVA, Bonferroni corrected post-hoc and Students t-tests were used.
Results: Analyses of the entire muscle did not show any differences (p>0.05 each) except for axial diffusivity (AD; p=0.048). ROItear showed higher mean diffusivity (MD) and AD than ROIhc_t (p<0.05). Fractional anisotropy (FA) was lower in ROItear than in ROIhi and ROIhc_t (p<0.05). Radial diffusivity (RD) was higher in ROItear than in any other ROI (p<0.05). Ratios revealed higher MD and RD and lower FA and reduced number and length of fibre tracts on the injured side (p<0.05 each).
Conclusions: DTI allowed a robust assessment of muscle tears in athletes especially after normalization to healthy muscle tissue.
Key Points
STEAM-based DTI allows the investigation of muscle tears affecting professional football players.
Fractional anisotropy and mean diffusivity differ between injured and healthy muscle areas.
Only normalized data show differences of fibre tracking metrics in muscle tears.
The normalization of DTI-metrics enables a more robust characterization of muscle tears.(VLID)475075
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
Non-Universal Quasi-Long Range Order in the Glassy Phase of Impure Superconductors
The structural correlation functions of a weakly disordered Abrikosov lattice
are calculated for the first time in a systematic 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
non-universal exponent \bar\eta_{\bf G} which depends on the ratio of the
renormalized elastic constants \kappa =\tilde c_{66}/\tilde c_{11} of the flux
line (FL) lattice. Our calculations show clearly three distinct scaling regimes
corresponding to the Larkin, the manifold and the asymptotic Bragg glass
regime. On a wide range of intermediate length scales the FL displacement
correlation function increases as a power law with twice of the manifold
roughness exponent \zeta_{rm}(\kappa), which is also non-universal. Our
results, in particular the \kappa-dependence 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: 4 pages, 3 figure
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 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
which depends on the ratio of the renormalized elastic constants
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 , 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
-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