Dynamic Ginzburg-Landau Theory for the Liquid-Solid Phase Transition

The anisotropv of the probability distribution function for the unit vector joining two nearest neighbour atoms is characterized by tensorial order parameters. For cubic symmetry, the most relevant tensor is of rank 4. Starting from an ansatz for the dependence of the (specific) internal energy, volume and entropy; the entropy production is calculated which is caused by a temporial change of the 4-th rank anisotropy tensor. A constitutive law which guarantees that the entropy production is positive leads to a nonlinear relaxation equation. It shows the features typical for a dynamic Ginzburg-Landau theory. The linearized version of the relaxation equation contains an effective relaxation time and a correlation length which exhibit a temperature dependence typical for a mean field theory. For a special case where the anisotropy tensor can be characterized by a scalar order parameter, the nonlinear relaxation equation is studied in some detail. Its stationary and spatially homogeneous solutions are zero and nonzero values for the order parameter depending on whether the temperature T is larger or smaller than the transition temperature. The unordered phase corresponds to a liquid state, the ordered phase to a simple or body centered cubic crystal. The phase transition is of 1st order. There exist also metastable states.