13 research outputs found
Finite element method for epitaxial growth with thermodynamic boundary conditions
We develop an adaptive finite element method for island dynamics in epitaxial growth. We study a step-flow model, which consists of an adatom (adsorbed atom) diffusion equation on terraces of different height, thermodynamic boundary conditions on terrace boundaries including anisotropic line tension, and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional (possibly anisotropic) "surface" diffusion of edge-adatoms along the step-edges. The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the boundary evolution. A penalty method is used in order to incorporate the boundary conditions. The evolution of the terrace boundaries includes both the weighted/anisotropic mean curvature flow and the weighted/anisotropic surface diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements
Return Radius and volume of recrystallized material in Ostwald Ripening
Within the framework of the LSW theory of Ostwald ripening the amount of
volume of the second (solid) phase that is newly formed by recrystallization is
investigated. It is shown, that in the late stage, the portion of the newly
generated volume formed within an interval from time to is a certain
function of and an explicit expression of this volume is given. To
achieve this, we introduce the notion of the {\it return radius} ,
which is the unique radius of a particle at time such that this particle
has -- after growing and shrinking -- the same radius at time . We derive a
formula for the return radius which later on is used to obtain the newly formed
volume. Moreover, formulas for the growth rate of the return radius and the
recrystallized material at time are derived
Morphological stability of electromigration-driven vacancy islands
The electromigration-induced shape evolution of two-dimensional vacancy
islands on a crystal surface is studied using a continuum approach. We consider
the regime where mass transport is restricted to terrace diffusion in the
interior of the island. In the limit of fast attachment/detachment kinetics a
circle translating at constant velocity is a stationary solution of the
problem. In contrast to earlier work [O. Pierre-Louis and T.L. Einstein, Phys.
Rev. B 62, 13697 (2000)] we show that the circular solution remains linearly
stable for arbitrarily large driving forces. The numerical solution of the full
nonlinear problem nevertheless reveals a fingering instability at the trailing
end of the island, which develops from finite amplitude perturbations and
eventually leads to pinch-off. Relaxing the condition of instantaneous
attachment/detachment kinetics, we obtain non-circular elongated stationary
shapes in an analytic approximation which compares favorably to the full
numerical solution.Comment: 12 page
A variational formulation of anisotropic geometric evolution equations in higher dimensions
Accepted versio
Gitter-Quantenfeldtheorien mit Quantensymmetrie
Cover and Contents
Introduction
1. DHR-superselection theory
2. Quantum groups as symmetry algebras
3. Lattice models and amplified DHR-theory
4. Overview and summary of results
Chapter 1. Diagonal crossed products by duals of quantum groups
1. Coactions and crossed products
2. Two-sided coactions and diagonal crossed products
3. Generating matrices
4. Quantum group spin chains and lattice current algebras
Chapter 2. Diagonal crossed products by duals of quasi-quantum groups
1. Quasi-quantum groups
2. Coactions of quasi-quantum groups
3. Two-sided coactions
4. Left and right diagonal crossed products
5. Generating matrices
6. Proofs
Chapter 3. Generalization to weak quasi-quantum groups
1. Weak quasi-quantum groups
2. Diagonal crossed products
Chapter 4. The quantum double D(G)
1. D(G) as a quasi-bialgebra and D(G)-coactions
2. The quasitriangular quasi-Hopf structure
3. The twisted double of a finite group
4. The monodromy algebra
Chapter 5. Quantum group spin chains and lattice current algebras
1. Two-sided crossed products
2. Quantum group spin chains
3. Lattice current algebras
4. Representation theory
5. Proofs
Appendix A. Representation theoretic interpretation
Appendix B. Graphical calculus
1. Basic definitions
2. The antipode image of the R-matrix
3. The antipode in the quantum double D(G)
4. Graphical description of the diagonal crossed product
Conclusions and outlook
Bibliography
Curriculum VitaeIn low dimensional quantum field theories the global (gauge) symmetry can in
general not be described by an ordinary group but by some more general
algebraic object such as quantum groups or generalizations thereof. In this
thesis we construct 1+1 - dimensional lattice quantum field theories -
socalled quantum group spin chains and lattice current algebras - whose global
symmetry is given by some quantum group at roots of unity.
The main problem in constructing these models stems from the fact that the
semisimple quotients of quantum groups at roots of unity are no longer
coassiciative and have to be described by weak quasi-quantum groups. To solve
this problem we introduce a new mathematical construction, the so-called
diagonal crossed product of an algebra M with the dual of a quantum group G.
We give a natural generalization of this construction to the case where G is a
quasi-Hopf algebra in the sense of Drinfeld and, more generally, also in the
sense of Mack and Schomerus (i.e., where the coproduct is non-unital). In
these cases our diagonal crossed product will still be an associative algebra,
even though the analogue of an ordinary crossed product in general is not well
defined as an associative algebra.
In the case M = G we obtain an explicit definition of the quantum double D(G)
for (weak) quasi-Hopf algebras G. We prove that D(G) is itself a (weak) quasi-
triangular quasi-Hopf algebra and we give explicit formulas for the coproduct,
the antipode and the R-matrix. Moreover we show that any diagonal crossed
product naturally admits a two-sided D(G)-coaction.
We then apply our formalism to construct quantum spin chains and lattice
current algebras based on a weak quasi-Hopf algebra as iterated diagonal
crossed products. This contains the important cases of truncated quantum
groups at roots of unity. Both lattice models admit the quantum double D(G) as
a localized cosymmetry. We investigate the representation theory of these
models. In particular we show that irreducible representations of lattice
current algebras (based on a semisimple weak quasi Hopf algebra G) are in one-
to-one correspondence with the irreducible representations of the quantum
double D(G).Quantenfeldtheorien in niederen Raum-Zeit-Dimensionen (kleiner als 4) zeigen
das PhĂ€nomen der Quantensymmetrie, d.h. die globale (Eich-)Symmetrie lĂ€Ăt sich
nicht mehr durch eine Gruppe beschreiben, sondern man benötigt allgemeinere
allgebraische Objekte wie zum Beispiel Quantengruppen. In dieser Arbeit werden
1+1 dimensionale Gitterquantenfeldtheorien - sogenannte
Quantengruppenspinketten und Gitterstromalgebren - konstuiert, deren globale
Symmetrie durch Quantengruppen an den Einheitswurzeln gegeben ist.
Die Hauptschwierigkeit bei dieser Konstruktion rĂŒhrt von der Tatsache her, daĂ
die halbeinfachen Quotienten von Quantengruppen an den Einheitswurzeln nicht
mehr koassoziativ sind. Sie besitzen die Struktur einer schwachen Quasi-
Quantengruppe. Wir fĂŒhren deswegen eine neue mathematische Konstruktion, das
sogenannte diagonale verschrÀnkte Produkt einer Algebra M mit der dualen einer
Quantengruppe G ein. Diese Konstruktion lĂ€Ăt sich auf natĂŒrliche Weise auf
Quasi-Hopfalgebren im Sinne von Drinfeld oder noch allgemeiner im Sinne von
Mack and Schomerus (i.e. mit nicht unitalem Koprodukt) verallgemeinern. In
diesen FÀllen ergibt unser diagonales verschrÀnktes Produkt immer noch eine
assoziative Algebra obwohl eine entsprechende Verallgemeinerung des
gewöhnlichen verschrÀnkten Produkts im allgemeinen zu keiner wohldefinierten
assoziativen Algebra fĂŒhrt.
Der Fall M = G fĂŒhrt zu einer Definition des Quantendoppels D(G) einer
(schwachen) Quasi-Hopfalgebra G. Wir zeigen, daĂ D(G) selbst eine schwache
quasitriangulĂ€re Quasi-Hopfalgebra ist. Wir geben explizite Formeln fĂŒr das
Koprodukt, fĂŒr die Antipode und fĂŒr die R-Matrix an. AuĂerdem zeigen wir, daĂ
jedes diagonale verschrĂ€nkte Produkt auf natĂŒrliche Weise eine zweiseitige
D(G)-Kowirkung besitzt.
Dann benutzen wir unseren Formalismus zur Konstruktion von Quantenspinketten
und Gitterstromalgebren als iterierte diagonale verschrÀnkte Produkte und
erreichen so unser Ziel der Konstruktion dieser Modelle an den
Einheitswurzeln. Auf beiden Gittermodellen wirkt das Quantendoppel als
lokalisierte Kosymmetrie. Zum SchluĂ untersuchen wir die Darstellungstheorie
der konstruierten Quantenketten. Insbesondere zeigen wir, daĂ die irreduziblen
Darstellungen einer Gitterstromalgebra eineindeutig den irreduziblen
Darstellungen des Quantendoppels D(G) der zugrundegelegten schwachen Quasi-
Hopfalgebra G zugeordnet werden können
Diagonal Crossed Products by Duals of Quasi-Quantum Groups
A two-sided coaction ffi : M! G\Omega M\Omega G of a Hopf algebra (G; \Delta; ffl; S) on an associative algebra M is an algebra map of the form ffi = (\Omega id M ) ffi ae = (id M\Omega ae) ffi , where (; ae) is a commuting pair of left and right G-coactions on M, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra G on M by / and . , respectively, we define the diagonal crossed product M ./ G to be the algebra generated by M and G with relations given by 'm = (' (1) . m / S \Gamma1 (' (3) )) ' (2) ; m 2 M; ' 2 G : We give a natural generalization of this construction to the case where G is a quasi--Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e., where the coproduct \Delta is non-unital). In these cases our diagonal crossed product will still be an associative algebra structure on M\Omega G extending M j M\Omega 1, even though the analogue of an ordinary crossed ..
A Level Set Approach to Anisotropic Flows with Curvature Regularization
Modeling and simulation of faceting effects on surfaces are topics of growing importance in modern nanotechnology. Such effects pose various theoretical and computational challenges, since they are caused by non-convex surface energies, which lead to ill-posed evolution equations for the surfaces. In order to overcome the illposedness, regularization of the energy by a curvature-dependent term has become a standard approach, which seems to be related to the actual physics, too. The use of curvature-dependent energies yields higher order partial differential equations for surface variables, whose numerical solution is a very challenging task. In this paper we investigate the numerical simulation of anisotropic growth with curvature-dependent energy by level set methods, which yield flexible and robust surface representations. We consider the two dominating growth modes, namely attachment-detachment kinetics and surface diffusion. The level set formulations are given in terms of metric gradient flows, which are discretized by finite element methods in space and in a semi-implicit way as local variational problems in time. Finally, the constructed level set methods are applied to the simulation of faceting of embedded surfaces and thin films