Finite-time and finite-size scalings of coercivity in dynamic hysteresis

The coercivity landscape for characterizing hysteresis in interacting systems across multiple timescales is proposed by Chen et al. in a companion paper [Phys. Rev. Lett. 136, 117102 (2026)10.1103/5rg8-52gl]. For the stochastic ϕ^{4} model under periodic driving of rate v_{H}, the coercivity landscape H_{c}(v_{H}) exhibits plateau features at a characteristic rate v_{P} with the corresponding coercivity H_{P}. Below this plateau (v_{H}v_{P}), scaling in the fast-driving regime, H_{c}∼v_{H}^{1/2}, is completely different from that, H_{c}-H_{P}∼(v_{H}-v_{P})^{2/3}, in the postplateau slow-driving regime. The emergence of the plateau with a finite-size scaling reflects the competition between the thermodynamic limit and the quasistatic limit. In this paper, we provide detailed analytical proofs and numerical evidence supporting these results. Moreover, to demonstrate the coercivity landscape in concrete physical systems, we study the magnetic hysteresis in the Curie-Weiss model and analyze its finite-size effects. We reveal that finite-time coercivity scaling shows model-specific behavior only in the fast-driving regime, while exhibiting universal characteristics elsewhere.