Two Types of Phase Transitions in Natural and Social Systems: First-Order Exponential Rapid Transitions and Second-Order Power-Law Slow Transitions

Traditional equilibrium phase transition theory is insufficient to characterize irreversible, non-equilibrium phase transitions in complex systems. This paper establishes a binary engineering diag-nostic framework: the exponential law is the intrinsic scaling of unconstrained multiplicative systems,dominating first-order fast phase transitions with near-jump global percolation in the supercriticaldomain; the power law is a constraint-derived heavy-tailed morphology, dominating second-orderslow phase transitions via critical gradual evolution. First-order exponential phase transitions areclassified into single-exponential burst type and spatio-temporal bi-exponential diffusion type withdistinct spatio-temporal structures. Power-law distributions are stratified into dynamically expand-ing, steady-state, and frozen types by constraint intensity, which determines phase-transition activity.This framework unifies the exponential origin, constraint-driven power-law formation, and dynamicphase-transition mechanisms, consistent with the unified principle of multiplicative systems. It pro-vides operable diagnostic and intervention tools for epidemics, finance, power grids, technologicaldiffusion, and ecological systems, serving as a systematic engineering complement to classical equi-librium phase transition theory.