Hydrodynamic equations for a system with translational and rotational dynamics

We obtain the equations of fluctuating hydrodynamics for many-particle systems whose microscopic units have both translational and rotational motion. The orientational dynamics of each element are studied in terms of Langevin equations for the rotational motion of a corresponding fixed-length director u. We consider the microscopic dynamics for two separate choices of basic variables: Brownian dynamics for position, Fokker-Planck dynamics for position, and momentum. In each case of the microscopic dynamics, the time evolution of a corresponding set of collective densities {ψ[over ̂]} has been obtained as an exact representation. For the Brownian dynamics, noise in the Langevin equation for the director u is multiplicative. The corresponding equation of motion for the collective number-density ρ[over ̂] has two different forms, respectively, for the It[over ̂]o and Stratonvich interpretation of the multiplicative noise in the u equation. Without the u variable, both forms reduce to the standard Dean-Kawasaki form. We average the microscopic equations for the collective densities {ψ[over ̂]} (which are, at this stage, a collection of Dirac δ functions) over the phase space variables and obtain a corresponding set of stochastic partial differential equations for the coarse-grained densities {ψ} with smooth spatial and temporal dependence. For averaging, we use a general local-equilibrium distribution involving an extended set of dynamical variables for the rotational motion. The coarse-grained equations of motion for the collective densities {ψ} constitute the fluctuating nonlinear hydrodynamics (FNH) for the fluid with both rotational and translational dynamics. From the stationary solution of the (deterministic) equation for the probability distribution P[ψ], we obtain a free-energy functional F[ψ]. The F[ψ]s for the different FNH descriptions with their corresponding set of {ψ} are also worked out.