Axion Fragmentation on the Lattice

We analyze the phenomenon of axion fragmentation when an axion field rolls over many oscillations of a periodic potential. This is particularly relevant for the case of relaxion, in which fragmentation provides the necessary energy dissipation to stop the field evolution. We compare the results of a linear analysis with the ones obtained from a classical lattice simulation, finding an agreement in the stopping time of the zero mode between the two within an $\mathcal{O}(1)$ difference. We finally speculate on the generation of bubbles with different VEVs of the axion field, and discuss their cosmological consequences.