A free-energy landscape picture and Landau theory for the dynamics of disordered materials

Landau's theory of phase transitions [Nature (London) 138, 840 (1936); Statistical Physics (Pergamon, London, 1959)] is adapted to treat independently relaxing regions in complex systems using nanothermodynamics. The order parameter we use governs the thermal fluctuations, not a specific static structure. We find that the entropy term dominates the thermal behavior, as is reasonable for disordered systems. Consequently, the thermal equilibrium occurs at the internal-energy maximum, so that the potential-energy minima have negligible influence on the dynamics. The dynamics involves normal thermal fluctuations about the free-energy minimum, with a time scale that is governed by the curvature of the internal-energy maximum. The temperature dependence of the fluctuations yields Vogel-Tamman-Fulcher-type [Phys. Z. 22, 645 (1921); J. Am. Ceram. Soc. 8, 339 (1925); Z. Anorg. Allg. Chem. 156, 245 (1926)] relaxation rates and approximate time-temperature superposition, consistent with the Williams-Landell-Ferry [J. Am. Chem. Soc. 77, 3701 (1955)] procedure for analyzing the dynamics of complex fluids, while the size dependence of the fluctuations provides an explanation for the distribution of relaxation times and heterogeneity that are found in glass-forming liquids, thus providing a unified picture for several features in the dynamics of disordered materials.