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 the rotational Brownian motion of a corresponding fixed-length director ${\bf u}$. The time evolution of a set of collective densities $\{\hatψ\}$ is obtained as an exact representation of the corresponding microscopic dynamics. For the Smoluchowski dynamics, noise in the Langevin equation for the director ${\bf u}$ is multiplicative. We obtain that the equation of motion for the collective number-density has two different forms, respectively, for the Ito and Stratonvich interpretation of the multiplicative noise in the ${\bf u}$-equation. Without the ${\bf u}$ variable, both reduce to the Standard Dean-Kawasaki form. Next, we average the microscopic equations for the collective densities $\{\hatψ\}$ (which are, at this stage, a collection of Dirac delta functions) over 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. The coarse-grained equations of motion for the collective densities $\{ψ\}$ constitute the fluctuating non-linear hydrodynamics for the fluid with both rotational and translational dynamics. From the stationary solution of the corresponding Fokker-Planck equation, we obtain a free energy functional ${\cal F}[ψ]$ and demonstrate the relation between the ${\cal F}[ψ]$s for different levels of the FNH descriptions with its corresponding set of $\{ψ\}$.