Dynamics of Fluctuations in Unstable Systems. I

Abstract

Dynamics of fluctuations for time-dependent Ginzburg-Landau models are investigated when the system is suddenly brought from a thermodynamically stable state to an unstable state. Because of the growth of fluctuations beyond their thermal level as time goes on, the nonlinear mode coupling term in the stochastic equation begins to play the major role after a certain time t_r when the system crosses over from the initial linear regime into the turbulent regime. When the order parameter is not conserved, we deduce an approximate stochastic equation valid in the turbulent where fluctuations with long wavelengths are separated from those with shorter wavelengths. The latter are then described by the Fokker-Planck equations whose coefficients contain long wavelength fluctuations which are determined separately. These equations then describe the discrete cascade processes where wavelengths of fluctuations are tripled at each step.