Fluctuation and distribution of macroscopic order in formation processes of a dissipative structure

We report the temporal developments of the variance and distribution of fluctuations of macroscopic order in a structure-formation process using an electrohydrodynamic instability of a nematic liquid crystal at f>$f_c$, where f and $f_c$ are the applied and the critical field frequencies separating two different instabilities: a convective flow instability and a forced oscillatory instability of the director. At f>$f_c$, the dissipative structure appears essentially as the forced oscillatory instability of the director in a three-dimensional pattern in space. The temporal development of the variance is qualitatively in good agreement with the result expected from the noisy Landau equation for the early stage, which resembles the result in a transient process of laser radiation, theoretically and experimentally studied by Arecchi and Degiorgio, but not for the late stage. This comes from the spatial inhomogeneity of the macroscopic order. The distributions of the macroscopic order fluctuation have profiles similar to the Gaussian type (or lognormal type in the presence of externally applied noise) for the early stage, and have an asymmetric profile with negative skewness for the late stage. This would suggest that the growth process of the dissipative structure for the late stage is dominated by totally different kinetics from that for the initial stage.