Zero-temperature Potts Model Research Articles

Non-self-similar grain growth by zero-temperature Potts model

Self-similar coarsening, a well-researched aspect of normal grain growth, is a key component of grain growth theories and visible in mesoscopic simulations of ideal grain growth. One particular simulation algorithm is the Potts model, which can be used to perform a thorough investigation of the temporal evolution of polycrystalline microstructures. Used with carefully selected parameters, the Potts model recreates normal grain growth quite accurately. One specific feature necessary to reach this goal is the simulation temperature, i.e. simulation thermal energy. It prevents grain boundaries from aligning along the underlying simulation lattice. Nevertheless, there are still today many researchers who implement classical zero-temperature Potts model simulations—often for reasons of simplification and/or modeling speed. In the current study, we show in detail the negative effect of setting the simulation temperature to zero. As a result, a unique type of non-self-similar coarsening is observed, where, e.g. the von Neumann–Mullins-relation shows unexpected time-dependent behavior. Hence, this study can be understood as a blueprint on why not to use a zero-temperature Potts model.

A note on recognizing an old friend in a new place: list coloring and the zero-temperature Potts model

Here we observe that list coloring in graph theory coincides with the zero-temperature antiferromagnetic Potts model with an external field. We give a list coloring polynomial that equals the partition function in this case. This is analogous to the connection between the chromatic polynomial and the zero-temperature, zero-field, antiferromagnetic Potts model. The subsequent cross fertilization yields immediate results for the Potts model and suggests new research directions in list coloring.

Capillarity-driven interface dynamics: Application to grain growth phenomenon

Abstract The dynamics of interfaces separating two regions with small and large grain sizes are studied by both computer simulations and a theoretical approach in the framework of a zero-temperature Potts model. Flat and circular interfaces are considered. It is shown that the interface is advancing in the direction of smaller grains with dynamics predicted by a theory based on Neumann's law for the circular interface and energy considerations for the flat interface. In both geometries, the simulation results are correctly described by these analytical models.