Derivation of the nonlinear fluctuating hydrodynamic equation from underdamped Langevin equation

TL;DR

This paper derives a fluctuating hydrodynamic equation from the underdamped Langevin equation, extending the Kawasaki-Dean formula, and clarifies its relation to steady state distributions and limiting cases.

Contribution

It provides an exact derivation of the fluctuating hydrodynamic equation from the underdamped Langevin dynamics, extending existing formulas.

Findings

The derived equation reduces to the Kawasaki-Dean equation in the massless limit.

The steady state distribution is expressed via kinetic and potential energies.

The equation aligns with the canonical field equation when friction is zero.

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.