Werner Pesch

Core structure and low-energy spectrum of isolated vortex lines in clean superconductors atT ≪T c

We present analytic solutions of the Eilenberger equations for the low-frequency Greens functions corresponding to those quasiparticles whose trajectories pass near the center of an isolated vortex. Using these results we find that for type II superconductors in the temperature rangeTc≫T≫Tc2/εF the order parameter and the supercurrent near the vortex center increase over a lengthξ1∼ξBCST/Tc (ξBCS=BCS coherence length) and that the density of states at the Fermi surface isN0 2π3ξBCS2/3 In (ξBCS/ξ1). The results can be reproduced with the Bogoliubov equations for the elementary excitations. It is shown that this peculiar behavior is connected with the low-lying bound states in the vortex core. ForTťc2/εF one hasξ1∼kF−1.

Mechanisms of extensive spatiotemporal chaos in Rayleigh-Benard convection

Spatially extended dynamical systems exhibit complex behaviour in both space and time—spatiotemporal chaos. Analysis of dynamical quantities (such as fractal dimensions and Lyapunov exponents) has provided insights into low-dimensional systems; but it has proven more difficult to understand spatiotemporal chaos in high-dimensional systems, despite abundant data describing its statistical properties. Initial attempts have been made to extend the dynamical approach to higher-dimensional systems, demonstrating numerically that the spatiotemporal chaos in several simple models is extensive (the number of dynamical degrees of freedom scales with the system volume). Here we report a computational investigation of a phenomenon found in nature, ‘spiral defect’ chaos in Rayleigh–Bénard convection, in which we find that the spatiotemporal chaos in this state is extensive and characterized by about a hundred dynamical degrees of freedom. By studying the detailed space–time evolution of the dynamical degrees of freedom, we find that the mechanism for the generation of chaotic disorder is spatially and temporally localized to events associated with the creation and annihilation of defects.

Structure and dynamics of dislocations in anisotropic pattern-forming systems

Abstract The motion of dislocations in convective roll patterns provides an important wavevector selection mechanism. In this work the structure and velocity of dislocations is calculated near threshold using amplitude equations appropriate for systems with an axial anisotropy. The fact that then the roll pattern has a preferred direction leads to characteristic differences to isotropic systems like Rayleigh-Benard convection in simple fluids. Furthermore the nucleation process of dislocation pairs is discussed by analyzing the threshold solution that describes the nucleation barrier.

Invited Lecture. New results on the electrohydrodynamic instability in nematics

Abstract We present theoretical results on the threshold and near-threshold behaviour of electrohydrodynamic convection of planarly aligned nematics under D.C. and A.C. driving. We use the general three dimensional description and include the flexoelectric effect. The experimentally established threshold behaviour is captured in many cases quantitatively, an exception being the extended travelling patterns. Slightly above threshold the observed undulated rolls pose some problems. Defectmediated turbulence can presumably be explained by mean-flow effects.

Rayleigh–Bénard convection in the presence of spatial temperature modulations

Motivated by recent experiments, we study a rich variation of the familiar Rayleigh–Benard convection (RBC), where the temperature at the lower boundary varies sinusoidally about a mean value. As usual the Rayleigh number R measures the average temperature gradient, while the additional spatial modulation is characterized by a (small) amplitude δ m and a wavevector q m . Our analysis relies on precise numerical solutions of suitably adapted Oberbeck–Boussinesq equations (OBE). In the absence of forcing (δ m = 0), convection rolls with wavenumber q c bifurcate only for R above the critical Rayleigh number R c . In contrast, for δ m ≠0, convection is unavoidable for any finite R ; in the most simple case in the form of ‘forced rolls’ with wavevector q m . According to our first comprehensive stability diagram of these forced rolls in the q m – R plane, they develop instabilities against resonant oblique modes at R ≲ R c in a wide range of q m / q c . Only for q m in the vicinity of q c , the forced rolls remain stable up to fairly large R > R c . Direct numerical simulations of the OBE support and extend the findings of the stability analysis. Moreover, we are in line with the experimental results and also with some earlier theoretical results on this problem, based on asymptotic expansions in the limit δ m → 0 and R → R c . It is satisfying that in many cases the numerical results can be directly interpreted in terms of suitably constructed amplitude and generalized Swift–Hohenberg equations.

Nonlinear analysis of spatial structures in two-dimensional anisotropic pattern forming systems

Motivated in particular by recent experiments on convective instabilities in nematic liquid crystals we examine the possible stationary patterns in anisotropic quasi-two-dimensional systems. The generalized SH-model we use exhibits the typical Lifshitz point, which separates the regions of normal and of oblique rolls at threshold. Above threshold the two-dimensional wavenumber regions of stable roll solutions take on interesting shapes in the vicinity of the Lifshitz point. Undulated (wavy) rolls also exist metastably in a narrow parameter range. We derive envelope equations which show that this scenario is general near threshold. Our results suggest experimental investigation, especially in the neighborhood of the Lifshitz point.

Electrohydrodynamic Instabilities in Nematic Liquid Crystals

We discuss various aspects of the progress in the understanding of electroconvec-tion in nematic layers achieved during the last 12 years.

Interface properties of the three-state potts model in two dimensions

Interface properties, in particular the interface free energy and the interface profile of the three-state Potts model in two dimensions are studied using Monte Carlo techniques and a generalized version of the method of Müller-Hartmann and Zittartz. The role of the third state in characterizing the interface between the two other states is elucidated.

Flexoelectricity and pattern formation in nematic liquid crystals.

We present in this paper a detailed analysis of the flexoelectric instability of a planar nematic layer in the presence of an alternating electric field (frequency ω), which leads to stripe patterns (flexodomains) in the plane of the layer. This equilibrium transition is governed by the free energy of the nematic, which describes the elasticity with respect to the orientational degrees of freedom supplemented by an electric part. Surprisingly the limit ω→0 is highly singular. In distinct contrast to the dc case, where the patterns are stationary and time independent, they appear at finite, small ω periodically in time as sudden bursts. Flexodomains are in competition with the intensively studied electrohydrodynamic instability in nematics, which presents a nonequilibrium dissipative transition. It will be demonstrated that ω is a very convenient control parameter to tune between flexodomains and convection patterns, which are clearly distinguished by the orientation of their stripes.

Direct transition to electroconvection in a homeotropic nematic liquid crystal

We present an experimental and theoretical investigation of a variant of electroconvection using an unusual nematic liquid crystal in an isotropic configuration (homeotropic alignment). The significance of the system is a direct transition to the convecting state due to the negative conductivity anisotropy and positive dielectric anisotropy. We observe at onset rolls or squares depending on the frequency and amplitude of the applied ac voltage with a strong signature of the zigzag instability. Good agreement with calculations based on the underlying hydrodynamic theory is found. We also construct an extended Swift-Hohenberg model which allows us to capture complex patterns like squares with a quasiperiodic modulation.