Calculation of correlation function of the director fluctuations in cholesteric liquid crystals by WKB method. (English) Zbl 1071.82055

Summary: The spatial correlation functions of the thermal fluctuations in systems with smoothly varying structure are calculated by means of the WKB method. As a particular physical problem we consider the behavior of director fluctuations in cholesteric liquid crystals possessing one-dimensional spatial periodicity. The problem leads to the solution of set of two second order differential equations with periodic coefficients. It is shown that in this physical system there exist regions where the WKB approximation is not valid. The analysis of these regions is similar to that of the turning points in quantum mechanics. Contrary to standard approach in our problem the turning point has fourth-order singularity and only decaying solutions have physical sense. We find WKB solutions for normal modes of director fluctuations in cholesteric liquid crystals far from the turning point as well as in its vicinity. We obtain that two fluctuating modes interact in the vicinity of the turning point, but any of these modes does not produce another. The amplitudes of modes change in such a way the product of amplitudes is constant. As a result we obtain explicit expressions for spatial correlation function in cholesteric liquid crystals with the large pitch which are valid in the entire domain. Finally we discuss the use of the correlation function in light scattering experiments.


82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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