Hydrodynamic equations for a system with translational and rotational dynamics.

Area of Science:

• Statistical Mechanics
• Fluid Dynamics
• Nonlinear Dynamics

Background:

• Many-particle systems exhibit complex dynamics involving both translation and rotation.
• Existing hydrodynamic models often simplify or neglect rotational degrees of freedom.
• Understanding these coupled dynamics is crucial for various physical phenomena.

Purpose of the Study:

• To derive fluctuating nonlinear hydrodynamics (FNH) equations for systems with coupled translational and rotational motion.
• To investigate the impact of orientational dynamics on collective density evolution.
• To establish a free-energy functional for these complex fluids.

Main Methods:

• Formulating Langevin equations for orientational dynamics using a director variable 'u'.
• Considering microscopic dynamics via Brownian and Fokker-Planck approaches for position and momentum.
• Averaging microscopic equations using a local-equilibrium distribution to obtain coarse-grained stochastic partial differential equations.
• Analyzing the stationary solution of the probability distribution to derive a free-energy functional.

Main Results:

• Exact representations for the time evolution of collective densities {ψ̂} derived from microscopic dynamics.
• Stochastic partial differential equations for coarse-grained densities {ψ} obtained through averaging.
• Identification of different forms for the collective number-density equation based on multiplicative noise interpretation (Itô vs. Stratonovich).
• Derivation of a free-energy functional F[ψ] from the stationary solution of the probability distribution.

Conclusions:

• The developed FNH framework accurately captures the interplay of translational and rotational dynamics in many-particle systems.
• The derived free-energy functional provides insights into the thermodynamic properties of these complex fluids.
• This work offers a foundation for studying phenomena where orientational order influences fluid behavior.