Correlation function director fluctuations

The twist viscosity is consequently inversely proportional to the TCFI of the director angular velocity correlation function. The physical interpretation of this relation is, that the twist viscosity is low when there are large fluctuations in the director orientation. This is usually the case when the order parameter is low. When the order parameter increases it becomes harder for the director to reorient, so that the twist viscosity increases. 

The fluctuation relations for the various viscosity coefficients involve time correlation functions. Some of them are ensemble dependent and others are ensemble independent. Only correlation functions the components of which couple with the angular velocity of the director or its conjugate torque density... 

One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. 

In general, there are three types of terms that contribute to Gmi C ) director fluctuations, molecular reorientation and a cross-term arising from both of these motions. Ukleja et al. [6.8] and Freed [6.9] used different approaches to tackle the existence of a cross-term between reorientation and director fluctuations and produced, in particular, a difference in the sign of the cross-term. At this time, there are still serious questions [6.11] regarding how to properly obtain correlation functions when both collective and individual molecular motions are present. 

When director fluctuations have small angular amplitudes, the correlation functions of Eq. (6.7) give... 

The influence of A 13 on director fluctuations in nematics has been discussed by Sha-laginov [226]. He gives an explicit expression for the correlation function in homeo-tropically aligned cells, where the apparent anchoring energy WoliK -Ki )... 

Up to now, in the frustrated nematic systems the pseudo-Casimir force has only been determined for the simplest structure with a uniform director field d < dc = Xh Xp, where Ah Ap is the extrapolation length of the honieotropic substrate and Ap the extrapolation length of the degenerate in-plane anchoring which preserves the full rotational symmetry about the substrate normal) [12]. Then, within the bare director description the correlation length in the Hamiltonian (8.28) is constant and the partition function of the fluctuation modes can be derived analytically. The derivation of the force in the bent-director and biaxial structures is more complex due to the deformed equilibrium order. 

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