Stochastic envelope equations for nonequilibrium transitions and application to thermal fluctuations in electroconvection in nematic liquid crystals.

Near the threshold of continuous nonequilibrium transitions in spatially extended pattern-forming systems thermal fluctuations are enhanced in analogy to equilibrium phase transitions. These fluctuations anticipate the deterministic pattern above threshold. A Langevin-equation approach based on Landau's method to determine the stochastic terms in hydrodynamic systems is presented and applied to fluctuations in electrohydrodynamic convection in nematic liquid crystals. The resulting set of equations is then transformed into a universal stochastic envelope (or amplitude) equation of the Ginzburg-Landau type, valid near threshold. The resulting fluctuations are compared with recent experiments and with an estimate based on equilibrium theory. The methods are formulated in a general way so that application to other pattern-forming systems is readily possible.