  Modal relaxation dynamics has been observed experimentally to clarify
statistical-physical properties of soft-mode turbulence, the spatiotemporal
chaos observed in homeotropically aligned nematic liquid crystals. We found a
dual structure, dynamical crossover associated with violation of time-reversal
invariance, the corresponding time scales satisfying a dynamical scaling law.
To specify the origin of the dual structure, the memory function due to
non-thermal fluctuations has been defined by a projection-operator method and
obtained numerically using experimental results. The results of the memory
function suggest that the non-thermal fluctuations can be divided into Markov
and non-Markov contributions, the latter is called the turbulent fluctuation
(TF). Consequently, the relaxation dynamics is separated into three
characteristic stages: bare-friction, early, and late stages. If the
dissipation due to TFs dominates over that of the Markov contribution, the
bare-friction stage contracts; the early and late stages then configure the
dual structure. The memory effect due to TFs results in the time-reversible
relaxation at the early stage, and the disappearance of the memory by turbulent
mixing leads to a simple exponential relaxation at the late stage. Furthermore,
the memory effect due to TFs is shown to originate from characteristic spatial
coherency called the patch structure.
