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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.