Non-thermal Brownian motion of a particle in soft-mode turbulence (SMT) in the electroconvection of a nematic liquid crystal has been experimentally investigated to clarify the statistical and thermodynamical aspects of SMT using the Lagrangian picture in hydrodynamics. The effective temperature for SMT is obtained in two different ways based on the definition of the diffusion coefficients due to non-thermal particle fluctuations: the Einstein relation and the fluctuation theorem. The temperatures from both methods agree well and exhibit a high value of 10 6 K. They depend on the coarsegraining time, which reflects the anomalous properties of the macroscopic fluctuations in the SMT.
For soft-mode turbulence, which is essentially the spatiotemporal chaos caused by the nonlinear interaction between convective modes and Goldstone modes in electroconvection of homeotropic nematics, a type of order-disorder phase transition was revealed, in which a new order parameter was introduced as pattern ordering. We calculated the spatial correlation function and the anisotropy of the convective patterns as a 2D XY system because the convective wave vector could freely rotate in the homeotropic system. We found the hidden order in the chaotic patterns observed beyond the Lifshitz frequency f(L), and a transition from a disordered to a hidden ordered state occurred at the f(L) with the increase of the frequency of the applied voltages.
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