  Soft-mode turbulence (SMT) is the spatiotemporal chaos observed in
homeotropically aligned nematic liquid crystals, where non-thermal fluctuations
are induced by nonlinear coupling between the Nambu-Goldstone and convective
modes. The net and modal relaxations of the disorder pattern dynamics in SMT
have been studied to construct the statistical physics of nonlinear
nonequilibrium systems. The net relaxation dynamics is well-described by a
compressed exponential function and the modal one satisfies a dual structure,
dynamic crossover accompanied by a breaking of time-reversal invariance.
Because the net relaxation is described by a weighted mean of the modal ones
with respect to the wave number, the compressed-exponential behavior emerges as
a superposition of the dual structure. Here, we present experimental results of
the power spectra to discuss the compressed-exponential behavior and the dual
structure from a viewpoint of the harmonic analysis. We also derive a
relationship of the power spectra from the evolution equation of the modal
autocorrelation function. The formula will be helpful to study non-thermal
fluctuations in experiments such as the scattering methods.
