Derivation of the nonlinear fluctuating hydrodynamic equation from underdamped Langevin equation

Abstract

We derive the fluctuating hydrodynamic equation for the number and momentum densities exactly from the underdamped Langevin equation. This derivation is an extension of the Kawasaki-Dean formula in underdamped case. The steady state probability distribution of the number and momentum densities field can be expressed by the kinetic and potential energies. In the massless limit, the obtained fluctuating hydrodynamic equation reduces to the Kawasaki-Dean equation. Moreover, the derived equation corresponds to the field equation derived from the canonical equation when the friction coefficient is zero.

16 pages

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• Stochastic dynamics of collective modes for Brownian dipoles
• Mean-field microrheology of a very soft colloidal suspension: inertia induces shear-thickening
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• Singular density correlations in chiral active fluids in three dimensions
• Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility