1,061 research outputs found
Asymptotic analysis of a system of algebraic equations arising in dislocation theory
The system of algebraic equations given by\ud
\ud
\ud
\ud
appears in dislocation theory in models of dislocation pile-ups. Specifically, the case a = 1 corresponds to the simple situation where n dislocations are piled up against a locked dislocation, while the case a = 3 corresponds to n dislocation dipoles piled up against a locked dipole.\ud
\ud
We present a general analysis of systems of this type for a > 0 and n large. In the asymptotic limit n -> ∞, it becomes possible to replace the system of discrete equations with a continuum equation for the particle density. For 0 < a < 2, this takes the form of a singular integral equation, while for a > 2 it is a first-order differential equation. The critical case a = 2 requires special treatment but, up to corrections of logarithmic order, it also leads to a differential equation.\ud
\ud
The continuum approximation is only valid for i not too small nor too close to n. The boundary layers at either end of the pile-up are also analyzed, which requires matching between discrete and continuum approximations to the main problem
Disclination vortices in elastic media
The vortex-like solutions are studied in the framework of the gauge model of
disclinations in elastic continuum. A complete set of model equations with
disclination driven dislocations taken into account is considered. Within the
linear approximation an exact solution for a low-angle wedge disclination is
found to be independent from the coupling constants of the theory. As a result,
no additional dimensional characteristics (like the core radius of the defect)
are involved. The situation changes drastically for 2\pi vortices where two
characteristic lengths, l_\phi and l_W, become of importance. The asymptotical
behaviour of the solutions for both singular and nonsingular 2\pi vortices is
studied. Forces between pairs of vortices are calculated.Comment: 13 pages, published versio
Vortex-glass phases in type-II superconductors
A review is given on the theory of vortex-glass phases in impure type-II
superconductors in an external field. We begin with a brief discussion of the
effects of thermal fluctuations on the spontaneously broken U(1) and
translation symmetries, on the global phase diagram and on the critical
behaviour. Introducing disorder we restrict ourselves to the experimentally
most relevant case of weak uncorrelated randomness which is known to destroy
the long-ranged translational order of the Abrikosov lattice in three
dimensions. Elucidating possible residual glassy ordered phases, we distinguish
betwee positional and phase-coherent vortex glasses. The discussion of elastic
vortex glasses, in two and three dimensions occupy the main part of our review.
In particular, in three dimensions there exists an elastic vortex-glass phase
which still shows quasi-long-range translational order: the `Bragg glass'. It
is shown that this phase is stable with respect to the formation of
dislocations for intermediate fields. Preliminary results suggest that the
Bragg-glass phase may not show phase-coherent vortex-glass order. The latter is
expected to occur in systems with weak disorder only in higher dimensions. We
further demonstrate that the linear resistivity vanishes in the vortex-glass
phase. The vortex-glass transition is studied in detail for a superconducting
film in a parallel field. Finally, we review some recent developments
concerning driven vortex-line lattices moving in a random environment.Comment: 133 pages Latex with figures. accepted for publication in Adv. Phy
Sequential inverse problems Bayesian principles and the\ud logistic map example
Bayesian statistics provides a general framework for solving inverse problems, but is not without interpretation and implementation problems. This paper discusses difficulties arising from the fact that forward models are always in error to some extent. Using a simple example based on the one-dimensional logistic map, we argue that, when implementation problems are minimal, the Bayesian framework is quite adequate. In this paper the Bayesian Filter is shown to be able to recover excellent state estimates in the perfect model scenario (PMS) and to distinguish the PMS from the imperfect model scenario (IMS). Through a quantitative comparison of the way in which the observations are assimilated in both the PMS and the IMS scenarios, we suggest that one can, sometimes, measure the degree of imperfection
Upscaling a model for the thermally-driven motion of screw dislocations
We formulate and study a stochastic model for the thermally-driven motion of
interacting straight screw dislocations in a cylindrical domain with a convex
polygonal cross-section. Motion is modelled as a Markov jump process, where
waiting times for transitions from state to state are assumed to be
exponentially distributed with rates expressed in terms of the potential energy
barrier between the states. Assuming the energy of the system is described by a
discrete lattice model, a precise asymptotic description of the energy barriers
between states is obtained. Through scaling of the various physical constants,
two dimensionless parameters are identified which govern the behaviour of the
resulting stochastic evolution. In an asymptotic regime where these parameters
remain fixed, the process is found to satisfy a Large Deviations Principle. A
sufficiently explicit description of the corresponding rate functional is
obtained such that the most probable path of the dislocation configuration may
be described as the solution of Discrete Dislocation Dynamics with an explicit
anisotropic mobility which depends on the underlying lattice structure.Comment: Major revision, including overhaul of notation, additions to Section
6 on Large Deviations, and resolution of conjecture in original version. 45
pages, 2 figures, 1 tabl
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
The mechanics of a chain or ring of spherical magnets
Strong magnets, such as neodymium-iron-boron magnets, are increasingly being
manufactured as spheres. Because of their dipolar characters, these spheres can
easily be arranged into long chains that exhibit mechanical properties
reminiscent of elastic strings or rods. While simple formulations exist for the
energy of a deformed elastic rod, it is not clear whether or not they are also
appropriate for a chain of spherical magnets. In this paper, we use
discrete-to-continuum asymptotic analysis to derive a continuum model for the
energy of a deformed chain of magnets based on the magnetostatic interactions
between individual spheres. We find that the mechanical properties of a chain
of magnets differ significantly from those of an elastic rod: while both
magnetic chains and elastic rods support bending by change of local curvature,
nonlocal interaction terms also appear in the energy formulation for a magnetic
chain. This continuum model for the energy of a chain of magnets is used to
analyse small deformations of a circular ring of magnets and hence obtain
theoretical predictions for the vibrational modes of a circular ring of
magnets. Surprisingly, despite the contribution of nonlocal energy terms, we
find that the vibrations of a circular ring of magnets are governed by the same
equation that governs the vibrations of a circular elastic ring
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