12,342 research outputs found
Single chain properties of polyelectrolytes in poor solvent
Using molecular dynamics simulations we study the behavior of a dilute
solution of strongly charged polyelectrolytes in poor solvents, where we take
counterions explicitly into account. We focus on the chain conformational
properties under conditions where chain-chain interactions can be neglected,
but the counterion concentration remains finite. We investigate the
conformations with regard to the parameters chain length, Coulomb interaction
strength, and solvent quality, and explore in which regime the competition
between short range hydrophobic interactions and long range Coulomb
interactions leads to pearl-necklace like structures. We observe that large
number and size fluctuations in the pearls and strings lead to only small
direct signatures in experimental observables like the single chain form
factor. Furthermore we do not observe the predicted first order collapse of the
necklace into a globular structure when counterion condensation sets in. We
will also show that the pearl-necklace regime is rather small for strongly
charged polyelectrolytes at finite densities. Even small changes in the charge
fraction of the chain can have a large impact on the conformation due to the
delicate interplay between counterion distribution and chain conformation.Comment: 20 pages, 27 figures, needs jpc.sty (included), to appear in Jour.
Phys. Chem
Continuous and discrete Clebsch variational principles
The Clebsch method provides a unifying approach for deriving variational
principles for continuous and discrete dynamical systems where elements of a
vector space are used to control dynamics on the cotangent bundle of a Lie
group \emph{via} a velocity map. This paper proves a reduction theorem which
states that the canonical variables on the Lie group can be eliminated, if and
only if the velocity map is a Lie algebra action, thereby producing the
Euler-Poincar\'e (EP) equation for the vector space variables. In this case,
the map from the canonical variables on the Lie group to the vector space is
the standard momentum map defined using the diamond operator. We apply the
Clebsch method in examples of the rotating rigid body and the incompressible
Euler equations. Along the way, we explain how singular solutions of the EP
equation for the diffeomorphism group (EPDiff) arise as momentum maps in the
Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch
variational principle is discretised to produce a variational integrator for
the dynamical system. We obtain a discrete map from which the variables on the
cotangent bundle of a Lie group may be eliminated to produce a discrete EP
equation for elements of the vector space. We give an integrator for the
rotating rigid body as an example. We also briefly discuss how to discretise
infinite-dimensional Clebsch systems, so as to produce conservative numerical
methods for fluid dynamics
Finite Size Polyelectrolyte Bundles at Thermodynamic Equilibrium
We present the results of extensive computer simulations performed on
solutions of monodisperse charged rod-like polyelectrolytes in the presence of
trivalent counterions. To overcome energy barriers we used a combination of
parallel tempering and hybrid Monte Carlo techniques. Our results show that for
small values of the electrostatic interaction the solution mostly consists of
dispersed single rods. The potential of mean force between the polyelectrolyte
monomers yields an attractive interaction at short distances. For a range of
larger values of the Bjerrum length, we find finite size polyelectrolyte
bundles at thermodynamic equilibrium. Further increase of the Bjerrum length
eventually leads to phase separation and precipitation. We discuss the origin
of the observed thermodynamic stability of the finite size aggregates
Tilting mutation of weakly symmetric algebras and stable equivalence
We consider tilting mutations of a weakly symmetric algebra at a subset of
simple modules, as recently introduced by T. Aihara. These mutations are
defined as the endomorphism rings of certain tilting complexes of length 1.
Starting from a weakly symmetric algebra A, presented by a quiver with
relations, we give a detailed description of the quiver and relations of the
algebra obtained by mutating at a single loopless vertex of the quiver of A. In
this form the mutation procedure appears similar to, although significantly
more complicated than, the mutation procedure of Derksen, Weyman and Zelevinsky
for quivers with potentials. By definition, weakly symmetric algebras connected
by a sequence of tilting mutations are derived equivalent, and hence stably
equivalent. The second aim of this article is to study these stable
equivalences via a result of Okuyama describing the images of the simple
modules. As an application we answer a question of Asashiba on the derived
Picard groups of a class of self-injective algebras of finite representation
type. We conclude by introducing a mutation procedure for maximal systems of
orthogonal bricks in a triangulated category, which is motivated by the effect
that a tilting mutation has on the set of simple modules in the stable
category.Comment: Description and proof of mutated algebra made more rigorous (Prop.
3.1 and 4.2). Okuyama's Lemma incorporated: Theorem 4.1 is now Corollary 5.1,
and proof is omitted. To appear in Algebras and Representation Theor
Two-component {CH} system: Inverse Scattering, Peakons and Geometry
An inverse scattering transform method corresponding to a Riemann-Hilbert
problem is formulated for CH2, the two-component generalization of the
Camassa-Holm (CH) equation. As an illustration of the method, the multi -
soliton solutions corresponding to the reflectionless potentials are
constructed in terms of the scattering data for CH2.Comment: 22 pages, 3 figures, draft, please send comment
The Hamiltonian structure and Euler-Poincar\'{e} formulation of the Vlasov-Maxwell and gyrokinetic systems
We present a new variational principle for the gyrokinetic system, similar to
the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in
the Eulerian frame and based on constrained variations of the phase space fluid
velocity and particle distribution function. Using a Legendre transform, we
explicitly derive the field theoretic Hamiltonian structure of the system. This
is carried out with a modified Dirac theory of constraints, which is used to
construct meaningful brackets from those obtained directly from
Euler-Poincar\'{e} theory. Possible applications of these formulations include
continuum geometric integration techniques, large-eddy simulation models and
Casimir type stability methods.
[1] H. Cendra et. al., Journal of Mathematical Physics 39, 3138 (1998)Comment: 36 pages, 1 figur
Lagrangian Reduction, the Euler--Poincar\'{e} Equations, and Semidirect Products
There is a well developed and useful theory of Hamiltonian reduction for
semidirect products, which applies to examples such as the heavy top,
compressible fluids and MHD, which are governed by Lie-Poisson type equations.
In this paper we study the Lagrangian analogue of this process and link it with
the general theory of Lagrangian reduction; that is the reduction of
variational principles. These reduced variational principles are interesting in
their own right since they involve constraints on the allowed variations,
analogous to what one finds in the theory of nonholonomic systems with the
Lagrange d'Alembert principle. In addition, the abstract theorems about
circulation, what we call the Kelvin-Noether theorem, are given.Comment: To appear in the AMS Arnold Volume II, LATeX2e 30 pages, no figure
Quasi-conservation laws for compressible 3D Navier-Stokes flow
We formulate the quasi-Lagrangian fluid transport dynamics of mass density
and the projection q=\bom\cdot\nabla\rho of the vorticity \bom onto
the density gradient, as determined by the 3D compressible Navier-Stokes
equations for an ideal gas, although the results apply for an arbitrary
equation of state. It turns out that the quasi-Lagrangian transport of
cannot cross a level set of . That is, in this formulation, level sets of
(isopychnals) are impermeable to the transport of the projection .Comment: 2 page note, to appear in Phys Rev
Lattice Models of Quantum Gravity
Standard Regge Calculus provides an interesting method to explore quantum
gravity in a non-perturbative fashion but turns out to be a CPU-time demanding
enterprise. One therefore seeks for suitable approximations which retain most
of its universal features. The -Regge model could be such a desired
simplification. Here the quadratic edge lengths of the simplicial complexes
are restricted to only two possible values , with
, in close analogy to the ancestor of all lattice theories, the
Ising model. To test whether this simpler model still contains the essential
qualities of the standard Regge Calculus, we study both models in two
dimensions and determine several observables on the same lattice size. In order
to compare expectation values, e.g. of the average curvature or the Liouville
field susceptibility, we employ in both models the same functional integration
measure. The phase structure is under current investigation using mean field
theory and numerical simulation.Comment: 4 pages, 1 figure
The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles
The gradient of potential vorticity (PV) is an important quantity because of
the way PV (denoted as ) tends to accumulate locally in the oceans and
atmospheres. Recent analysis by the authors has shown that the vector quantity
\bdB = \bnabla q\times \bnabla\theta for the three-dimensional incompressible
rotating Euler equations evolves according to the same stretching equation as
for \bom the vorticity and \bB, the magnetic field in magnetohydrodynamics
(MHD). The \bdB-vector therefore acts like the vorticity \bom in Euler's
equations and the \bB-field in MHD. For example, it allows various analogies,
such as stretching dynamics, helicity, superhelicity and cross helicity. In
addition, using quaternionic analysis, the dynamics of the \bdB-vector
naturally allow the construction of an orthonormal frame attached to fluid
particles\,; this is designated as a quaternion frame. The alignment dynamics
of this frame are particularly relevant to the three-axis rotations that
particles undergo as they traverse regions of a flow when the PV gradient
\bnabla q is large.Comment: Dedicated to Raymond Hide on the occasion of his 80th birthda
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