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This study develops fluctuating nonlinear hydrodynamics for systems with both translational and rotational motion. It derives new equations for collective densities, incorporating rotational dynamics and yielding a free-energy functional.

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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.