Ergodicity Breaking and Scaling Relations for Finite-Time First-Order Phase Transition

Hysteresis and metastable states are typical features associated with ergodicity breaking in the first-order phase transition. We explore the scaling relations of nonequilibrium thermodynamics in finite-time first-order phase transitions. Using the Curie-Weiss model as an example, for large systems we find the excess work scales as $v^{2\mathrm{/}3}$ when the magnetic field is quenched at a finite rate $v$ across the phase transition. We further reveal a crossover in the scaling of the excess work from $v^{2\mathrm{/}3}$ to $v$ when downsizing the system. Our study elucidates the interplay between the finite-time dynamics and the finite-size effect, which leads to different scaling behaviors of the excess work with or without ergodicity breaking.