Liquid Instability and Freezing. II. Multi-Mode One Dimensional Analysis of a Nonlinear Diffusion Equation

In connection with freezing, dynamical behaviour of (metastable) liquids is studied based on the nonlinear diffusion equation (ND eq.) derived in a previous paper. From the ND eq., an H -theorem for the free-energy functional \(F(N({r},t))\) of Brout is derived in the form \(dF(N({r},t))/dt{}0\). This shows that the ND eq. gives a dynamic extention of the mean field equation of freezing studied by Kirkwood-Monroe, Brout and others. While two-mode analysis of the ND eq. essentially reproduces the previous results obtained with a reductive perturbation method, N -mode analysis ( N >2) shows that a crystalline state bifurcates from a uniform one before the system reaches at the metastability limit. Thermodynamic arguments follow the bifurcation analysis to determine a transition (freezing) point.