1,552 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.
Bit flipping and time to recover
We call `bits' a sequence of devices indexed by positive integers, where
every device can be in two states: (idle) and (active). Start from the
`ground state' of the system when all bits are in -state. In our first
Binary Flipping (BF) model, the evolution of the system is the following: at
each time step choose one bit from a given distribution on the
integers independently of anything else, then flip the state of this bit to the
opposite. In our second Damaged Bits (DB) model a `damaged' state is added:
each selected idling bit changes to active, but selecting an active bit changes
its state to damaged in which it then stays forever.
In both models we analyse the recurrence of the system's ground state when no
bits are active. We present sufficient conditions for both BF and DB models to
show recurrent or transient behaviour, depending on the properties of
. We provide a bound for fractional moments of the return time to
the ground state for the BF model, and prove a Central Limit Theorem for the
number of active bits for both models
On well-posedness of variational models of charged drops
Electrified liquids are well known to be prone to a variety of interfacial
instabilities that result in the onset of apparent interfacial singularities
and liquid fragmentation. In the case of electrically conducting liquids, one
of the basic models describing the equilibrium interfacial configurations and
the onset of instability assumes the liquid to be equipotential and interprets
those configurations as local minimizers of the energy consisting of the sum of
the surface energy and the electrostatic energy. Here we show that,
surprisingly, this classical geometric variational model is mathematically
ill-posed irrespectively of the degree to which the liquid is electrified.
Specifically, we demonstrate that an isolated spherical droplet is never a
local minimizer, no matter how small is the total charge on the droplet, since
the energy can always be lowered by a smooth, arbitrarily small distortion of
the droplet's surface. This is in sharp contrast with the experimental
observations that a critical amount of charge is needed in order to destabilize
a spherical droplet. We discuss several possible regularization mechanisms for
the considered free boundary problem and argue that well-posedness can be
restored by the inclusion of the entropic effects resulting in finite screening
of free charges.Comment: 18 pages, 2 figure
One-dimensional Neel walls under applied external fields
We present a detailed analysis of one-dimensional N\'eel walls in thin
uniaxial ferromagnetic films in the presence of an in-plane applied external
field in the direction normal to the easy axis. Within the reduced
one-dimensional thin film model, we formulate a non-local variational problem
whose minimizers are given by one-dimensional N\'eel wall profiles. We prove
existence, uniqueness (up to translations and reflections), regularity, strict
monotonicity and the precise asymptotics of the decay of the minimizers in the
considered variational problem
Asymptotic properties of ground states of scalar field equations with a vanishing parameter
We study the leading order behaviour of positive solutions of the equation
-\Delta u +\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\R^N, where , and when is a small parameter. We give a complete
characterization of all possible asymptotic regimes as a function of ,
and . The behavior of solutions depends sensitively on whether is less,
equal or bigger than the critical Sobolev exponent . For
the solution asymptotically coincides with the solution of the
equation in which the last term is absent. For the solution
asymptotically coincides with the solution of the equation with
. In the most delicate case the asymptotic behaviour
of the solutions is given by a particular solution of the critical
Emden--Fowler equation, whose choice depends on in a nontrivial
way
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