Nonequilibrium potential for the Ginzburg-Landau equation in the phase-turbulent regime

The steady state distribution functional of the supercritical complex Ginzburg-Landau equation with weak noise is determined asymptotically for long-wave-length fluctuations including the phaseturbulent regime. This is done by constructuring a non-equilibrium potential solving the Hamilton-Jacobi equation associated with the Fokker-Planck equation. The non-equilibrium potential serves as a Lyapunov functional. In parameter space it consists of two branches which are joined at the Benjamin-Feir instability. In the Benjamins-Feir stable regime the non-equilibrium potential has minima in the plane-wave attractors and our result generalizes to arbitrary dimension an earlier result for one dimension. Beyond the Benjamin-Feir instability the potential in the function space has a minimum which is degererate with respects to arbirary long-wavelength phase variations. The dynamics on the minimum set obey the generalized Kuramoto-Sivashinsky equation.