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 $\{\hat{\psi}\}$ 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 I\"{t}o 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 $\{\hat{\psi}\}$ (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 $\{\psi\}$ with smooth spatial and temporal dependence. The coarse-grained equations of motion for the collective densities $\{\psi\}$ 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 ${\cal F}[\psi]$ and demonstrate the relation between the ${\cal F}[\psi]$s for different levels of the FNH descriptions with its corresponding set of $\{\psi\}$.