406 research outputs found
Kinetics of Ordering in Fluctuation-Driven First-Order Transitions: Simulations and Dynamical Renormalization
Many systems where interactions compete with each other or with constraints
are well described by a model first introduced by Brazovskii. Such systems
include block copolymers, alloys with modulated phases, Rayleigh-Benard Cells
and type-I superconductors. The hallmark of this model is that the fluctuation
spectrum is isotropic and has a minimum at a nonzero wave vector represented by
the surface of a d-dimensional hyper-sphere. It was shown by Brazovskii that
the fluctuations change the free energy structure from a to a
form with the disordered state metastable for all quench depths.
The transition from the disordered to the periodic, lamellar structure changes
from second order to first order and suggests that the dynamics is governed by
nucleation. Using numerical simulations we have confirmed that the equilibrium
free energy function is indeed of a form. A study of the dynamics,
however, shows that, following a deep quench, the dynamics is described by
unstable growth rather than nucleation. A dynamical calculation, based on a
generalization of the Brazovskii calculations shows that the disordered state
can remain unstable for a long time following the quench.Comment: 18 pages, 15 figures submitted to PR
Density mismatch in thin diblock copolymer films
Thin films of diblock copolymer subject to gravitational field are simulated
by means of a cell dynamical system model. The difference in density of the two
sides of the molecule and the presence of the field causes the formation of
lamellar patterns with orientation parallel to the confining walls even when
they are neutral. The concentration profile of those films is analyzed in the
weak segregation regime and a functional form for the profile is proposed.Comment: 9 pages and 8 figures. Needs EPSF macros. Submitted to PR
Striped periodic minimizers of a two-dimensional model for martensitic phase transitions
In this paper we consider a simplified two-dimensional scalar model for the
formation of mesoscopic domain patterns in martensitic shape-memory alloys at
the interface between a region occupied by the parent (austenite) phase and a
region occupied by the product (martensite) phase, which can occur in two
variants (twins). The model, first proposed by Kohn and Mueller, is defined by
the following functional: where
is periodic in and almost everywhere.
Conti proved that if then the minimal specific
energy scales like ,
as . In the regime , we improve Conti's results, by computing exactly the
minimal energy and by proving that minimizers are periodic one-dimensional
sawtooth functions.Comment: 29 pages, 3 figure
Froth-like minimizers of a non local free energy functional with competing interactions
We investigate the ground and low energy states of a one dimensional non
local free energy functional describing at a mean field level a spin system
with both ferromagnetic and antiferromagnetic interactions. In particular, the
antiferromagnetic interaction is assumed to have a range much larger than the
ferromagnetic one. The competition between these two effects is expected to
lead to the spontaneous emergence of a regular alternation of long intervals on
which the spin profile is magnetized either up or down, with an oscillation
scale intermediate between the range of the ferromagnetic and that of the
antiferromagnetic interaction. In this sense, the optimal or quasi-optimal
profiles are "froth-like": if seen on the scale of the antiferromagnetic
potential they look neutral, but if seen at the microscope they actually
consist of big bubbles of two different phases alternating among each other. In
this paper we prove the validity of this picture, we compute the oscillation
scale of the quasi-optimal profiles and we quantify their distance in norm from
a reference periodic profile. The proof consists of two main steps: we first
coarse grain the system on a scale intermediate between the range of the
ferromagnetic potential and the expected optimal oscillation scale; in this way
we reduce the original functional to an effective "sharp interface" one. Next,
we study the latter by reflection positivity methods, which require as a key
ingredient the exact locality of the short range term. Our proof has the
conceptual interest of combining coarse graining with reflection positivity
methods, an idea that is presumably useful in much more general contexts than
the one studied here.Comment: 38 pages, 2 figure
Weak Segregation Theory and Non-Conventional Morphologies in the Ternary ABC Triblock Copolymers
The Leibler weak segregation theory in molten diblock copolymers is
generalized with due regard for the 2nd shell harmonics contributions defined
in the paper and the phase diagrams are built for the linear and miktoarm
ternary ABC triblock copolymers. The symmetric linear copolymers with the
middle block non-selective with respect to the side ones are shown to undergo
the continuous ODT not only into the lamellar phase but also into various
non-conventional cubic phases (depending on the middle block composition it
could be the simple cubic, face-centered cubic or non-centrosymmetric phase
revealing the symmetry of space group No.214 first predicted to appear in
molten block copolymers). For asymmetric linear ABC copolymers a region of
compositions is found where the weakly segregated gyroid (double gyroid) phase
exists between the planar hexagonal and lamellar or one of the non-conventional
cubic phases up to the very critical point. In contrast, the miktoarm ABC block
copolymers with one of its arm non-selective with respect to the two others are
shown to reveal a pronounced tendency towards strong segregation, which is
preceded by increase of stability of the conventional BCC phase and a peculiar
weakly segregated BCC phase (BCC3), where the dominant harmonics belong to the
3rd co-ordination sphere of the reciprocal lattice. The validity region of the
developed theory is discussed and outlined in the composition triangles both
for linear and miktoarm copolymers.Comment: 61 pages, 12 figure
The nature of slow dynamics in a minimal model of frustration-limited domains
We present simulation results for the dynamics of a schematic model based on
the frustration-limited domain picture of glass-forming liquids. These results
are compared with approximate theoretical predictions analogous to those
commonly used for supercooled liquid dynamics. Although model relaxation times
increase by several orders of magnitude in a non-Arrhenius manner as a
microphase separation transition is approached, the slow relaxation is in many
ways dissimilar to that of a liquid. In particular, structural relaxation is
nearly exponential in time at each wave vector, indicating that the mode
coupling effects dominating liquid relaxation are comparatively weak within
this model. Relaxation properties of the model are instead well reproduced by
the simplest dynamical extension of a static Hartree approximation. This
approach is qualitatively accurate even for temperatures at which the mode
coupling approximation predicts loss of ergodicity. These results suggest that
the thermodynamically disordered phase of such a minimal model poorly
caricatures the slow dynamics of a liquid near its glass transition
Susceptibility amplitude ratio for generic competing systems
We calculate the susceptibility amplitude ratio near a generic higher
character Lifshitz point up to one-loop order. We employ a renormalization
group treatment with independent scaling transformations associated to the
various inequivalent subspaces in the anisotropic case in order to compute the
ratio above and below the critical temperature and demonstrate its
universality. Furthermore, the isotropic results with only one type of
competition axes have also been shown to be universal. We describe how the
simpler situations of -axial Lifshitz points as well as ordinary
(noncompeting) systems can be retrieved from the present framework.Comment: 20 pages, no figure
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
Search for direct production of charginos and neutralinos in events with three leptons and missing transverse momentum in √s = 7 TeV pp collisions with the ATLAS detector
A search for the direct production of charginos and neutralinos in final states with three electrons or muons and missing transverse momentum is presented. The analysis is based on 4.7 fb−1 of proton–proton collision data delivered by the Large Hadron Collider and recorded with the ATLAS detector. Observations are consistent with Standard Model expectations in three signal regions that are either depleted or enriched in Z-boson decays. Upper limits at 95% confidence level are set in R-parity conserving phenomenological minimal supersymmetric models and in simplified models, significantly extending previous results
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