Scaling Law for Irreversible Entropy Production in Critical Systems

We examine the Jarzynski equality for a quenching process across the critical point of second-order phase transitions, where absolute irreversibility and the effect of finite-sampling of the initial equilibrium distribution arise in a single setup with equal significance. We consider the Ising model as a prototypical example for spontaneous symmetry breaking and take into account the finite sampling issue by introducing a tolerance parameter. The initially ordered spins become disordered by quenching the ferromagnetic coupling constant. For a sudden quench, the deviation from the Jarzynski equality evaluated from the ideal ensemble average could, in principle, depend on the reduced coupling constant ϵ₀ of the initial state and the system size L. We find that, instead of depending on ϵ₀ and L separately, this deviation exhibits a scaling behavior through a universal combination of ϵ₀ and L for a given tolerance parameter, inherited from the critical scaling laws of second-order phase transitions. A similar scaling law can be obtained for the finite-speed quench as well within the Kibble-Zurek mechanism.