Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Florian Conrad
Universität Bielefeld
On a tagged particle process in continuum with singular interaction potential
We consider the dynamics of a tagged particle in a (moving) environment consisting of infinitely many
particles interacting by a physically realistic pair potential like e.g. the Lennard-Jones potential. The dynamics
of the environment as well as the coupled dynamics of the tagged particle and the environment
have recently been constructed using Dirichlet form methods by T. Fattler and M. Grothaus in the sense of
martingale solutions for the corresponding generators. We derive several results to strengthen the relation
between the path (ξt)t≥0 of the tagged particle and the environment (γt)t≥0 seen from the tagged particle
(with γt being a locally finite subset the d-dimensional Euclidean space). As a consequence, this better
understanding of the tagged particle process shows that a general method by de Masi, Ferrari, Goldstein
and Wick (1989) can be directly applied for proving that under diffusive scaling the displacement of the
tagged particle converges to a Brownian motion scaled by some diffusion matrix.
(Joint work with Torben Fattler, Martin Grothaus and Yuri Kondratiev.)
Literatur
De Masi, A., Ferrari, P.A., Goldstein, S. and Wick, W.D. (1989): An invariance principle for reversible
Markov processes. Applications to random motions in random environments. J. Stat. Phys., 55, 787-855.
Fattler, T. and Grothaus, M. (2011). Tagged particle process in continuum with singular interactions. Infinite
Dimensional Analysis, Quantum Probability and Related Topics, 14, 105-136.
Torben Fattler
Technische Universität Kaiserslautern
The dynamical wetting model in (1+1)-dimension
We consider a dynamical ∇φ interface model (also known as Ginzburg–Landau dynamics) on a onedimensional
lattice with reflection and an additional pinning effect on a hard wall for a large class of
interaction potentials. The pinning effect on the hard wall is modeled by an additional self-potential. In
particular, we make use of a δ -pinning potential, but also the relation to a square well pinning potential is
discussed. The dynamics under consideration realizes the so-called dynamical wetting model. It describes
the motion of an interface resulting from wetting of a solid surface by a fluid. For the construction of the
underlying stochastic process we apply Dirichlet form techniques in combination with Wentzell boundary
conditions. Finally, we use the obtained results as ingredients for the study of fluctuations of the interface
near the hard wall.
Literatur
Deuschel, J.-D., Giacomin, G., and Zambotti, L. (2005). Scaling limits of equilibrium wetting models in
(1+1)-dimension Probab. Theory Relat. Fields, 132, 471 - 500.
Funaki, T., and Olla, S. (2001). Fluctuations for ∇φ interface model on a wall Stochastic Process. Appl.,
94(1), 1 - 27.
Giacomin, G., Olla, S., and Spohn, H. (2001). Equilibrium fluctuations for ∇ϕ interface model Ann. Probab.,
29(3), 1138 - 1172.
Vogt, H., and Voigt, J. (2003). Wentzell boundary conditions in the context of Dirichlet forms Adv. Differential
Equations, 8(7), 821 - 842.
Zambotti, L. (2004). Fluctuations for a ∇φ interface model with repulsion from a wall. Probab. Theory
Relat. Fields, 129, 315 - 339.
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