Growth of fluctuations in quenched time-dependent Ginzburg-Landau model systems

A dynamical theory is presented for the enhanced fluctuations that occur in a time-dependent Ginzburg-Landau model system with the order parameter not conserved which is quenched from a thermodynamically stable to an unstable state. In a certain weak-coupling, long-time, and long-distance limit, diffusion and saturation effects can be treated separately. As a result explicit expressions are found for the probability distribution functional, the two-point reduced distribution function, and the pair correlation function of the fluctuations, which evolve from an arbitrary initial probability distribution functional. The behavior of the latter two functions is also displayed graphically. A central role is played by the time-independent nonlinear transformation of the order parameter which takes care of the saturation effects. The nature of such a transformation is discussed in a general context. If the problem is viewed as a nonequilibrium critical phenomenon, the theory corresponds to the Landau mean-field theory. An expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ is suggested to improve our treatment, where $d$ is the dimensionality of space.