205 research outputs found
A quantitative approximation scheme for the traveling wave solutions in the Hodgkin-Huxley model
We introduce an approximation scheme for the Hodgkin-Huxley model of nerve
conductance which allows to calculate both the speed of the traveling pulses
and their shape in quantitative agreement with the solutions of the model. We
demonstrate that the reduced problem for the front of the traveling pulse
admits a unique solution. We obtain an explicit analytical expression for the
speed of the pulses which is valid with good accuracy in a wide range of the
parameters.Comment: 22 pages (Latex), 9 figures (postscript
Theory of domain patterns in systems with long-range interactions of Coulomb type
We develop a theory of the domain patterns in systems with competing
short-range attractive interactions and long range repulsive Coulomb
interactions. We take an energetic approach, in which patterns are considered
as critical points of a mean-field free energy functional. Close to the
microphase separation transition, this functional takes on a universal form,
allowing to treat a number of diverse physical situations within a unified
framework. We use asymptotic analysis to study domain patterns with sharp
interfaces. We derived an interfacial representation of the pattern's free
energy which remains valid in the fluctuating system, with a suitable
renormalization of the Coulomb interaction's coupling constant. We also derived
integrodifferential equations describing the stationary domain patterns of
arbitrary shapes and their thermodynamic stability, coming from the first and
second variation of the interfacial free energy. We showed that the length
scale of a stable domain pattern must obey a certain scaling law with the
strength of the Coulomb interaction. We analyzed existence and stability of
localized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patterns
in two dimensions. We showed that these patterns are metastable in certain
ranges of the parameters and that they can undergo morphological instabilities
leading to the formation of more complex patterns. We discuss nucleation of the
domain patterns by thermal fluctuations and pattern formation scenarios for
various thermal quenches. We argue that self-induced disorder is an intrinsic
property of the domain patterns in the systems under consideration.Comment: 59 pages (RevTeX 4), 9 figures (postscript), to be published in the
Phys. Rev.
Instabilities and disorder of the domain patterns in the systems with competing interactions
The dynamics of the domains is studied in a two-dimensional model of the
microphase separation of diblock copolymers in the vicinity of the transition.
A criterion for the validity of the mean field theory is derived. It is shown
that at certain temperatures the ordered hexagonal pattern becomes unstable
with respect to the two types of instabilities: the radially-nonsymmetric
distortions of the domains and the repumping of the order parameter between the
neighbors. Both these instabilities may lead to the transformation of the
regular hexagonal pattern into a disordered pattern.Comment: ReVTeX, 4 pages, 3 figures (postscript); submitted to Phys. Rev. Let
Front propagation in geometric and phase field models of stratified media
We study front propagation problems for forced mean curvature flows and their
phase field variants that take place in stratified media, i.e., heterogeneous
media whose characteristics do not vary in one direction. We consider phase
change fronts in infinite cylinders whose axis coincides with the symmetry axis
of the medium. Using the recently developed variational approaches, we provide
a convergence result relating asymptotic in time front propagation in the
diffuse interface case to that in the sharp interface case, for suitably
balanced nonlinearities of Allen-Cahn type. The result is established by using
arguments in the spirit of -convergence, to obtain a correspondence
between the minimizers of an exponentially weighted Ginzburg-Landau type
functional and the minimizers of an exponentially weighted area type
functional. These minimizers yield the fastest traveling waves invading a given
stable equilibrium in the respective models and determine the asymptotic
propagation speeds for front-like initial data. We further show that
generically these fronts are the exponentially stable global attractors for
this kind of initial data and give sufficient conditions under which complete
phase change occurs via the formation of the considered fronts
Bit storage by domain walls in ferromagnetic nanorings
We propose a design for the magnetic memory cell which allows an efficient
storage, recording, and readout of information on the basis of thin film
ferromagnetic nanorings. The information bit is represented by the polarity of
a stable 360 domain wall introduced into the ring. Switching between
the two magnetization states is achieved by the current applied to a wire
passing through the ring, whereby the domain wall splits into two
charged walls, which then move to the opposite extreme of the ring
to recombine into a wall of the opposite polarity
Reduced energies for thin ferromagnetic films with perpendicular anisotropy
We derive four reduced two-dimensional models that describe, at
different spatial scales, the micromagnetics of ultrathin
ferromagnetic materials of finite spatial extent featuring
perpendicular magnetic anisotropy and interfacial
Dzyaloshinskii-Moriya interaction. Starting with a microscopic model
that regularizes the stray field near the material's lateral edges,
we carry out an asymptotic analysis of the energy by means of
-convergence. Depending on the scaling assumptions on the
size of the material domain vs. the strength of dipolar interaction,
we obtain a hierarchy of the limit energies that exhibit
progressively stronger stray field effects of the material
edges. These limit energies feature, respectively, a renormalization
of the out-of-plane anisotropy, an additional local boundary penalty
term forcing out-of-plane alignment of the magnetization at the
edge, a pinned magnetization at the edge, and, finally, a pinned
magnetization and an additional field-like term that blows up at the
edge, as the sample's lateral size is increased. The pinning of the
magnetization at the edge restores the topological protection and
enables the existence of magnetic skyrmions in bounded samples.Comment: 29 pages, 1 figur
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