Hysteresis in First-Order Transitions

First-order transitions have discontinuous order parameters with latent heat. The free energy F(M,H,T) = aM²+bM⁴−HM has two minima when T < T_c — representing coexisting phases. The Maxwell construction (equal-area rule) gives the true transition field H*, but metastable states persist beyond it until the nucleation barrier vanishes at the spinodal. The resulting hysteresis loop reflects path dependence: sweeping H up vs down traces different branches. Wider loops → larger barriers → harder nucleation. At T→T_c, the loop collapses (second-order limit). Real systems: magnets, shape-memory alloys, first-order Mott transitions, cloud nucleation.