Disentangling Growth and Decay of Domains During Phase Ordering

Using Monte Carlo simulations we study the phase ordering dynamics of a $\mathit{\text{multi}}$-species system modeled via the prototype $q$-state Potts model. In such a $\mathit{\text{multi}}$-species system, we identify a spin states or species as the $\mathit{\text{winner}}$ if it has survived as the majority in the final state, otherwise we mark them as $\mathit{\text{loser}}$. We disentangle the time ($t$) dependence of the domain length of the $\mathit{\text{winner}}$ from $\mathit{\text{losers}}$, rather than monitoring the average domain length obtained by treating all spin states or species alike. The kinetics of domain growth of the $\mathit{\text{winner}}$ at a finite temperature in space dimension $d=2$ reveal that the expected Lifshitz-Cahn-Allen scaling law $\sim t^{1\mathrm{/}2}$ can be observed with no early-time corrections, even for system sizes much smaller than what is traditionally used. Up to a certain period, all the others species, i.e., the $\mathit{\text{losers}}$, also show a growth that, however, is dependent on the total number of species, and slower than the expected $\sim t^{1\mathrm{/}2}$ growth. Afterwards, the domains of the $\mathit{\text{losers}}$ start decaying with time, for which our data strongly suggest the behavior $\sim t^{-z}$, where $z=2$ is the dynamical exponent for nonconserved dynamics. We also demonstrate that this new approach of looking into the kinetics also provides new insights for the special case of phase ordering at zero temperature, both in $d=2$ and $d=3$.