Navier-Stokes Equation by Stochastic Variational Method

TL;DR

This paper demonstrates that the stochastic variational method can derive the Navier-Stokes equation from ideal fluid action, linking dissipation to fluctuations via the fluctuation-dissipation theorem.

Contribution

It is the first to show that the stochastic variational method can naturally derive the Navier-Stokes equation from ideal fluid dynamics.

Findings

Dissipation arises as a consequence of stochastic fluctuations.

The stochastic variational method extends to dissipative fluid dynamics.

Potential to describe more general dissipative processes.

Abstract

We show for the first time that the stochastic variational method can naturally derive the Navier-Stokes equation starting from the action of ideal fluid. In the frame work of the stochastic variational method, the dynamical variables are extended to stochastic quantities. Then the effect of dissipation is realized as the direct consequence of the fluctuation-dissipation theorem. The present result reveals the potential availability of this approach to describe more general dissipative processes.