RMK Darmstadt - 31.01.2020
Sabine Jansen (München) und Dimitrios Tsagkarogiannis (l'Aquila)
Zeit: 15:15–17:45 Uhr
Sabine Jansen, LMU München
Cluster expansions for Gibbs point processes: probabilistic aspects
Gibbs point processes form an important class of models in statistical mechanics, stochastic geometry and spatial statistics. A notorious difficulty is that many quantities cannot be computed explicitly; for example, the intensity measure of a Gibbs point process is a highly non-trivial function of the intensity of the underlying Poisson point process. As a partial way out, mathematical physicists have long worked with perturbation series, called cluster expansions.
The talk explains some probabilistic aspects of cluster expansions, drawing connections with moments, cumulants, and Levy measures, as well as trees, branching processes, and Boolean percolation (S.J., Adv. Appl. Probab., 2019). If time permits, I will present a new if and only if convergence criterion (joint work with Leonid Kolesnikov).
Dimitrios Tsagkarogiannis, Università degli Studi dell´Aquila
Cluster expansions and coarse-graining
One of the main challenges of both theoretical and computational methods in statistical mechanics is the derivation of equations relating thermodynamic quantities in various regimes of the phase diagram. One key result in this direction was a diagrammatic expansion of the pressure developed in the 40's by J. E. Mayer. The rigorous proof of the convergence was only given later in the 60's by Penrose, Groeneveld, Lebowitz, Ruelle and others. Subsequently, other diagrammatic methods have been developed for various thermodynamic quantities leading to what consists now the liquid state theory. These are mainly approximate closures which in conjunction with appropriate computational methods perform well in some cases, leaving however the theoretical understanding of denser regimes still an open problem. Furthermore, despite the impressive increase of computational power over the last decades, treating more complex systems remains a challenging task. For this purpose, the focus has been recently on developing coarse-graining strategies as a natural way to reduce the degrees of freedom. In this talk we give a short overview of these facts and describe the central role of the cluster expansion technique.
- TU Darmstadt S2|04 Raum 213
- Hochschulstr. 8, 64289 Darmstadt
Goethe-Universität Frankfurt am Main, Johannes Gutenberg-Universität Mainz