Memory function of turbulent fluctuations in soft-mode turbulence
  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. 
Takayuki Narumi, Junichi Yoshitani, Masaru Suzuki, Yoshiki Hidaka,   Fahrudin Nugroho, Tomoyuki Nagaya, and Shoichi Kai
cond-mat.soft cond-mat.stat-mech