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Schmeltz-Fisher Theory

This theory combines the statistical mechanics of Fisher with the melting behavior of solids to model the transition between liquid crystal and isotropic liquid phases.
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The statement of the theorem

Consider the free energy density ff of the system, which depends on the local density ρ\rho and the order parameter SS. The transition is modeled by minimizing the free energy functional F\mathcal{F}: \nF[ρ,S]=∫[fliquid(ρ,S)+fsolid(ρ,S)]d3r\mathcal{F}[\rho, S] = \int \left[ f_{liquid}(\rho, S) + f_{solid}(\rho, S) \right] d^3r \nThe melting transition is characterized by the condition that the free energy difference between the liquid and solid phases vanishes at the transition temperature TmT_m: \nΔf=fsolid(ρm,Sm,Tm)−fliquid(ρm,Sm,Tm)=0\Delta f = f_{solid}(\rho_m, S_m, T_m) - f_{liquid}(\rho_m, S_m, T_m) = 0 \nThis requires the system to satisfy the equilibrium condition δFδρ=0\frac{\delta \mathcal{F}}{\delta \rho} = 0 and δFδS=0\frac{\delta \mathcal{F}}{\delta S} = 0 simultaneously.