Dynamics Of Topological Defects Research Articles

Spiral breakup and defect dynamics in a model for intracellular Ca 2+ dynamics

Pattern formation is investigated near the breakup instability of rotating spiral waves in a model for intracellular Ca 2+ dynamics. Our results show that spiral breakup is strongly dependent on the inactivation parameter of the inositol 1,4,5 triphosphate receptor ion channel. We compute the pulse train instability reponsible for spiral breakup and investigate the influence of the system size. The instability of planar pulse trains is a long wavelength (with respect to the perturbations) Eckhaus instability, that appears upon decrease of the wavelength of the pulse train. Secondly, we study the dynamics of topological defects in the spatiotemporally chaotic state emerging from the spiral breakup. This regime is charcterized by a variance of the number of defects that is considerably smaller than the corresponding mean value. Global characteristics like the creation and annihilation rate and the probability distribution of the number of defects are calculated. The defect transport is characterized by its diffusivity. Most defects move subdiffusively within their lifetime as a consequence of meandering.

A remark on the effective description of topological defects

We subject the methodology used to derive the effective dynamics of topological defects to a critical reappraisal, using the two-dimensional kink as an illustrative example. Special care is taken on how the zero-modes should be handled in order to avoid overcounting of degrees of freedom. This is an issue that has been overlooked in many recent contributions on the derivation of domain wall effective actions. We show that, unless such redundancy is completely removed by means of a sort of gauge fixing, the expression obtained for the effective action will not be consistent. We readdress some earlier calculations over the existence of curvature corrections in the light of the previous discussion and briefly comment on the application of this method to higher dimensional topological defects.

Experimental study of dynamics of topological defects in nematic polymer liquid crystals

The dynamics of stringlike defects and the shrinkage and annihilation processes of individual defect loops in a nematic polymer liquid crystal have been experimentally investigated. In 40<t<500 s, the line density of defects ρ scales as ρ(t)∼t−1, as expected by theory and as found in the experimental studies of some low molecular weight (LMW) liquid crystals. For individual loops, they first change their complex shape into a circle, resulting in a rapid decrease in length, and then the circular loops continuously shrink. Before a complete annihilation, the relation between the radius R(t) of the circular loop and time t0−t, where t0 is the time to annihilation, is described by R2(t)=2Γ(t0−t)(t⩽t0). The kinetic constant Γ determined from this polymer liquid crystal is 0.34∼0.38 μm2/s, much smaller than those (200∼300 μm2/s) obtained in some LMW liquid crystals. The small Γ is ascribed to the high viscosity of this polymer liquid crystal, compared with LMW liquid crystals.

The gauge-invariant dynamic equations for current-carrying plasma-like media with topological defects

A closed set of the gauge-invariant dynamic equations for a current-carrying plasma- like medium with dislocation-type and disclination-type topological defects together with the conditions at strong discontinuities is obtained using the variational principle and discussed. The dislocation and disclination fields, which compensate the non-homogeneity of the action of the gauge group GD SO.3/FT. 3/, are described in the present theory by inexact external differential forms. The set of the Cartan structural equations for these forms has a direct correlation with the continuity equations for topological defects. The integrability conditions for the equations describing the dynamics of topological defects are obtained. It is shown that the integrability condition for the equation for disclination fields is equivalent to the balance equation for the angular momentum of the plasma-like medium together with the magnetic field. This condition is degenerated in the requirement of symmetry of the total stress tensor in the case of lack of topological defects. It is also shown that the total tensor of an energy-momentum of the plasma-like medium and of the magnetic field satisfies the balance equation.

Analytic Scaling Solutions for Cosmic Domain Walls.

A relativistic generalisation of a well-known method for approximating the dynamics of topological defects in condensed matter is constructed, and applied to the evolution of domain walls in a cosmological context. It is shown that there are self-similar ``scaling'' solutions, for which one can in principle calculate many quantities of interest without recourse to numerical simulations. Here, the area density in the scaling regime is calculated in various backgrounds. Remarkably good agreement with numerical simulations is obtained.

Erratum: Dynamics of topological defects and inflation

We study the dynamics of topological defects in the context of ``topological inflation" proposed by Vilenkin and Linde independently. Analysing the time evolution of planar domain walls and of global monopoles, we find that the defects undergo inflationary expansion if $\eta\stackrel{>}{\sim}0.33m_{Pl}$, where $\eta$ is the vacuum expectation value of the Higgs field and $m_{Pl}$ is the Planck mass. This result confirms the estimates by Vilenkin and Linde. The critical value of $\eta$ is independent of the coupling constant $\lambda$ and the initial size of the defect. Even for defects with an initial size much greater than the horizon scale, inflation does not occur at all if $\eta$ is smaller than the critical value. We also examine the effect of gauge fields for static monopole solutions and find that the spacetime with a gauge monopole has an attractive nature, contrary to the spacetime with a global monopole. It suggests that gauge fields affect the onset of inflation.

Dynamics of topological defects and inflation.

We study the dynamics of topological defects in the context of ``topological inflation'' proposed by Vilenkin and Linde independently. Analyzing the time evolution of planar domain walls and of global monopoles, we find that the defects undergo inflationary expansion if \ensuremath{\eta}\ensuremath{\gtrsim}0.33${\mathit{m}}_{\mathrm{Pl}}$, where \ensuremath{\eta} is the vacuum expectation value of the Higgs field and ${\mathit{m}}_{\mathrm{Pl}}$ is the Planck mass. This result confirms the estimates by Vilenkin and Linde. The critical value of \ensuremath{\eta} is independent of the coupling constant \ensuremath{\lambda} and the initial size of the defect. Even for defects with an initial size much greater than the horizon scale, inflation does not occur at all if \ensuremath{\eta} is smaller than the critical value. We also examine the effect of gauge fields for static monopole solutions and find that the spacetime with a gauge monopole has an attractive nature, contrary to the spacetime with a global monopole. It suggests that gauge fields affect the onset of inflation. \textcopyright{} 1996 The American Physical Society.

Finger competition and viscosity contrast in viscous fingering. A topological approach

Abstract Finger competition in rectangular Hele-Shaw cells is discussed with particular focus on topological features of the velocity field. A reduced description of the interface dynamics is provided in terms of the motion of topological defects which obey simple conservation rules. Based on the existence of a topological invariant, a time-dependent quantity S(t) , directly related to the number of topological defects, is proposed as a global characterization of the finger competition. This quantity is not simply related to the interface shape but may be used as a measure of the ‘complexity’ of evolving patterns and to elucidate the conditions for eventual approach to a single finger stationary solution. Some representative numerical simulations for two extreme limits of the viscosity contrast parameter are presented and analyzed in terms of the dynamics of topological defects. Two specific sequences or histories for the creation/annihilation of defects are then identified, corresponding to radically different behavior of competing fingers. With the help of a local vorticity analysis, we conclude that, in the limit of low viscosity contrast, the Saffman-Taylor finger has a drastically reduced but non-vanishing basin of attraction.

Finger competition and viscosity contrast in viscous fingering. A topological approach

Finger competition in rectangular Hele-Shaw cells is discussed with particular focus on topological features of the velocity field. A reduced description of the interface dynamics is provided in terms of the motion of topological defects which obey simple conservation rules. Based on the existence of a topological invariant, a time-dependent quantity S(t), directly related to the number of topological defects, is proposed as a global characterization of the finger competition. This quantity is not simply related to the interface shape but may be used as a measure of the ‘complexity’ of evolving patterns and to elucidate the conditions for eventual approach to a single finger stationary solution. Some representative numerical simulations for two extreme limits of the viscosity contrast parameter are presented and analyzed in terms of the dynamics of topological defects. Two specific sequences or histories for the creation/annihilation of defects are then identified, corresponding to radically different behavior of competing fingers. With the help of a local vorticity analysis, we conclude that, in the limit of low viscosity contrast, the Saffman-Taylor finger has a drastically reduced but non-vanishing basin of attraction.

Non-conserved ordering kinetics

Attempts are made to understand the relationship between the two analytic approaches to non-conserved ordering kinetics. One approach originates from the use of singular perturbation with respect to nonlinearity of the governing field equation. Another one focuses on the dynamics of topological defects. Recent progress in analytical and numerical investigations are briefly reviewed and the origin of complications found in some models of the vector order parameter is tentatively attributed to the interplay between phase and defect dynamics.

Statistical mechanics of magnetic bubble arrays. I. Topology and thermalization.

Bubble domains in thin magnetic garnet films are an experimentally accessible two-dimensional system with well-characterized properties. Bubble arrays can be directly viewed with polarized light using optical microscopy, digital imaging, and computer-video techniques. The structure and dynamics of topological defects can be studied in detail. We present observations of the interactions and dynamics of defects in magnetic bubble arrays. Quasithermal Brownian motion of bubbles in an applied ac magnetic field arises from microscopic substrate roughness. Experimental measurements of diffusive bubble motion are used to estimate the effective temperature, and the effective pinning energy and length scales for microscopic roughness in the garnet film.

Coarsening Dynamics in Nematic Liquid Crystals

Abstract There is considerable interest in the dynamics of topological defects formed during a symmetry breaking phase transition in fields as diverse as condensed matter physics, particle physics and cosmology. Liquid crystals, with their many symmetry breaking phase transitions are ideal materials for such studies. Here we report on light transmission studies of the coarsening of three-dimensional defect tangles in uniaxial nematic liquid crystals subjected to a sudden pressure jump from the isotropic phase to the nematic phase. We have verified that the type 1/2 disclination line density, as a function of time, scales as t−v with v = 1.0 ± 0.1 over four decades in time. The transmitted light intensity exhibits two distinct scaling regimes depending on whether propagation without scattering or diffusive propagation with multiple scattering dominates the light reaching the detector.

Two-Dimensional Melting of Magnetic Bubble Arrays: A Continuous Hexatic-To-Liquid Transition

AbstractArrays of two-dimensional magnetic bubbles in thin garnet films undergo a hexatic-toliquid transition as a function of bubble density controlled by an applied spatially uniform dc bias magnetic field that opposes the magnetization in the bubbles. The phase transition is driven by topological point defects. The bubbles are observed directly using optical microscopy and digital imaging techniques. In the presence of a linear gradient in the dc bias magnetic field the hexatic-to-liquid transition occurs spatially in the direction of the gradient. As the system goes from hexatic to liquid, a continuous decrease in bubble density accompanied by a continuous disordering of the array is observed along the gradient direction. This continuous disordering persists even after the system is allowed to equilibrate for very long periods of time, indicating that the hexatic-to-liquid transition is continuous at equilibrium. Dynamics of topological defects observed in the gradient field correspond to those observed in the uniform field.

Interactions and dynamics of topological defects: Theory and experiments near the onset of weak turbulence.

The dynamics of a single topological defect and the interaction between pairs, including the process of annihilation of defects of opposite topological charge, are studied experimentally and theoretically near the onset of weak turbulence in Williams domains of electroconvecting nematics. The existence of topological defects requires a gauge field theoretical treatment, enriching the commonly used ``amplitude equations.'' The gauge field carries the interaction which is found to be of finite range in agreement with detailed observations.

Spatiotemporal dynamics of oscillatory convection at low Prandtl number: Waves and defects.

A Rayleigh-B\'enard experiment in a moderate-aspect-ratio cell (7\ifmmode\times\else\texttimes\fi{}4\ifmmode\times\else\texttimes\fi{}1) filled with mercury reveals for the first time the spatiotemporal dynamics of the oscillatory regime. The oscillatory instability, first step on the route to turbulence at low Prandtl number, generates propagating waves, which interfere spatially and involves very rich dynamics of topological defects.

Dynamics of topological defects in critical binary fluids, metamagnets and 3He-4He mixtures

Abstract The dynamical theory of topological defects in critical and tricritical systems is presented. Starting with the bulk stochastic equation like the time dependent Ginzburg-Landau model we derive the equation of motion of the topological defects such as interfaces and vortices in a unified way. We are primarily concerned with critical binary fluids, metamagnets and 3 He- 4 He mixtures. The method utilized here is based on the idea originally used by Thiele in his theory of magnetic bubbles.