2,075 research outputs found
Lie-Poincare' transformations and a reduction criterion in Landau theory
In the Landau theory of phase transitions one considers an effective
potential whose symmetry group and degree depend on the system
under consideration; generally speaking, is the most general
-invariant polynomial of degree . When such a turns out to be too
complicate for a direct analysis, it is essential to be able to drop
unessential terms, i.e. to apply a simplifying criterion. Criteria based on
singularity theory exist and have a rigorous foundation, but are often very
difficult to apply in practice. Here we consider a simplifying criterion (as
stated by Gufan) and rigorously justify it on the basis of classical
Lie-Poincar\'e theory as far as one deals with fixed values of the control
parameter(s) in the Landau potential; when one considers a range of values, in
particular near a phase transition, the criterion has to be accordingly
partially modified, as we discuss. We consider some specific cases of group
as examples, and study in detail the application to the
Sergienko-Gufan-Urazhdin model for highly piezoelectric perovskites.Comment: 32 pages, no figures. To appear in Annals of Physic
Asymptotic symmetries in an optical lattice
It was recently remarked by Lutz [{\it Phys. Rev. A} {\bf 67} (2003),
051402(R)] that the equation for the marginal Wigner distribution in an optical
lattice admits a scale-free distribution corresponding to Tsallis statistics.
Here we show that this distribution is invariant under an asymptotic symmetry
of the equation, hence that this scale-free behavior can be understood in terms
of symmetry analysis
Lambda and mu-symmetries
Lambda-symmetries of ODEs were introduced by Muriel and Romero, and discussed
by C. Muriel in her talk at SPT2001. Here we provide a geometrical
characterization of lambda-prolongations, and a generalization of these -- and
of lambda-symmetries -- to PDEs and systems thereof
Poincar\'e-like approach to Landau theory. II. Simplifying the Landau-deGennes potential for nematic liquid crystals
In a previous paper we have discussed how the Landau potential (entering in
Landau theory of phase transitions) can be simplified using the Poincar\'e
normalization procedure. Here we apply this approach to the Landau-deGennes
functional for the isotropic-nematic transitions, and transitions between
different nematic phases, in liquid crystals. {We give special attention to
applying our method in the region near the main transition point, showing in
full detail how this can be done via a suitable simple modification of our
Poincar\'e-like method. We also consider the question if biaxial phases can
branch directly off the fully symmetric state; some partial results in this
direction are presented
Twisted symmetries and integrable systems
Symmetry properties are at the basis of integrability. In recent years, it
appeared that so called "twisted symmetries" are as effective as standard
symmetries in many respects (integrating ODEs, finding special solutions to
PDEs). Here we discuss how twisted symmetries can be used to detect
integrability of Lagrangian systems which are not integrable via standard
symmetries
Solitons in Yakushevich-like models of DNA dynamics with improved intrapair potential
The Yakushevich (Y) model provides a very simple pictures of DNA torsion
dynamics, yet yields remarkably correct predictions on certain physical
characteristics of the dynamics. In the standard Y model, the interaction
between bases of a pair is modelled by a harmonic potential, which becomes
anharmonic when described in terms of the rotation angles; here we substitute
to this different types of improved potentials, providing a more physical
description of the H-bond mediated interactions between the bases. We focus in
particular on soliton solutions; the Y model predicts the correct size of the
nonlinear excitations supposed to model the ``transcription bubbles'', and this
is essentially unchanged with the improved potential. Other features of soliton
dynamics, in particular curvature of soliton field configurations and the
Peierls-Nabarro barrier, are instead significantly changed
- …