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 $\mathbf{u}$. 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 $\left\{\widehat{\psi }\right\}$ has been obtained as an exact representation. For the Brownian dynamics, noise in the Langevin equation for the director $\mathbf{u}$ is multiplicative. The corresponding equation of motion for the collective number-density $\widehat{\rho }$ has two different forms, respectively, for the $\mathrm{I}\widehat{\mathrm{t}}\mathrm{o}$ and Stratonvich interpretation of the multiplicative noise in the $\mathbf{u}$ equation. Without the $\mathbf{u}$ variable, both forms reduce to the standard Dean-Kawasaki form. We average the microscopic equations for the collective densities $\left\{\widehat{\psi }\right\}$ (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 $\left\{\psi \right\}$ 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 $\left\{\psi \right\}$ 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 $\mathcal{P}\left[\psi \right]$, we obtain a free-energy functional $\mathcal{F}\left[\psi \right]$. The $\mathcal{F}\left[\psi \right]\mathrm{s}$ for the different FNH descriptions with their corresponding set of $\left\{\psi \right\}$ are also worked out.

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• Continuum mechanics
• Nonlinear dynamics in fluids
• Statistical hydrodynamics
• Stochastic differential equations