Interplay of fast and slow dynamics in rare transition pathways: The disk-to-slab transition in the 2d Ising model

Rare transitions between long-lived stable states are often analyzed in terms of free energy landscapes computed as functions of a few collective variables. Here, using transitions between geometric phases as example, we demonstrate that the effective dynamics of a system along these variables are an essential ingredient in the description of rare events and that the static perspective provided by the free energy alone may be misleading. In particular, we investigate the disk-to-slab transition in the two-dimensional Ising model starting with a calculation of a two-dimensional free energy landscape and the distribution of committor probabilities. While at first sight it appears that the committor is incompatible with the free energy, they can be reconciled with each other using a two-dimensional Smoluchowski equation that combines the free energy landscape with state dependent diffusion coefficients. These results illustrate that dynamical information is not only required to calculate rate constants but that neglecting dynamics may also lead to an inaccurate understanding of the mechanism of a given process.