Hydrodynamics and correlation functions in ordered systems: Nematic liquid crystals

Abstract

A rigorous hydrodynamic theory is presented which is formulated and derived, from the start, in terms of measurable correlation functions. The theory is explained on the example of nematic liquid crystals; it applies easily also to other ordered systems like glasses, isotropic ferro- and antiferromagnets, superfluids, and, of course, normal fluids. The formal developments are based upon the projector technique of Zwanzig and Mori. Dynamical information is inserted in terms of the conservation laws (of mass, momentum, and energy), and static information about the order of the phase is contained in certain susceptibilities which diverge as k → 0. For nematics, we obtain the usual seven hydrodynamic modes, whose dispersion relations involve eight dissipative and essentially two reactive coefficients. Our method produces, in one unified procedure, also the hydrodynamic correlation functions, and Kubo relations for the transport coefficients. Several surprising results emerge, concerning, e.g., cross correlations with hydrodynamically undetermined strength or the fact that the reactive parameter λ is not determined by a sum rule. Application to super-fluids is sketched briefly, and we have a short note on incompressible nematics.