Modelling and analysis of phase transitions

# seminar: Periodic mimimizers in atomistic systems

speaker: Florian Theil (Warwick Mathematics Institute)

abstract: Many materials have a crystalline phase at low temperatures. The simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type $E(\{y\})=\sum\limits{1\leq x y(x)-y(x') )$ where $y(x) \in \R2$ is the position of particle $x$ and $V(r) \in \R$ is the pair-interaction energy of two particles which are placed at distance $r$. We focus on the zero temperature case and show rigorously that under suitable assumptions on the potential $V$ which are compatible with the growth behavior of the Lennard-Jones potential the ground state energy per particle converges to an explicit constant $\Estar$: $\lim_{N \to \infty} \tfrac{1}{N}\min_{y:\{1\ldots N\} \to \R2} E(\{y\}) = \Estar,$ where $\Estar \in \R$ is the minimum of a simple function on $[0,\infty)$. Furthermore, if suitable Dirichlet- or periodic boundary conditions are used, then the minimizers form a triangular lattice. To the best knowledge of the author this is the first result in the literature where periodicity of ground states is established for a physically relevant model which is invariant under the Euclidean symmetry group consisting of rotations and translations.

timetable:
Sat 21 Jan, 10:05 - 10:50, Sala Conferenze Centro De Giorgi
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