Coercivity Panorama of Dynamic Hysteresis

We study the stochastic $\phi^{4}$ model under periodic driving by an external field $H$ at different scales of driving rate $v_{H}$, where the noise strength $\sigma$ quantifies the deviation of the system size from the thermodynamic limit. For large systems with small $\sigma$, we find the coercivity $H_{c}=H(\langle \phi \rangle=0)$ sequentially exhibits distinct behaviors with increasing $v_{H}$: $v_{H}$-scaling increase from zero, stable plateau ($v_H^{0}$), $v_H^{1/2}$-scaling increase, and abrupt decline to disappearance. The $H_{c}$-plateau reflects the competition between thermodynamic and quasi-static limits, namely, $lim_{\sigma \to 0} lim_{v_{H} \to 0} H_{c} = 0$, and $lim_{v_{H} \to 0} lim_{ \sigma \to 0} H_{c}=H^{*}$. Here, $H^{*}$ is exactly the first-order phase transition (FOPT) point. In the post-plateau slow-driving regime, $H_{c}-H^{*}$ scales with $v_H^{2/3}$. Moreover, we reveal a finite-size scaling for the coercivity plateau $H_P$ as $(H^*-H_P)\sim \sigma^{4/3}$ by utilizing renormalization-group theory. These predicted scaling relations are demonstrated in magnetic hysteresis obtained with the Curie-Weiss model. Our work provides a panoramic view of the finite-time evolution of the stochastic $\phi^4$ model, bridges dynamics of FOPT and dynamic phase transition, and offers new insights into finite-time/finite-size effect interplay in non-equilibrium thermodynamics.

arXiv