Navier-Stokes, Gross-Pitaevskii and Generalized Diffusion Equations using Stochastic Variational Method

TL;DR

This paper applies the stochastic variational method to derive fundamental equations like Navier-Stokes, Gross-Pitaevskii, and generalized diffusion equations, providing new insights and correction terms within a unified framework.

Contribution

It extends the stochastic variational method to continuum mediums, deriving key physical equations and identifying correction terms for the Navier-Stokes equation.

Findings

Derived Navier-Stokes, Gross-Pitaevskii, and generalized diffusion equations using stochastic variational method.

Obtained a correction term for the Navier-Stokes equation.

Compared the correction term with the diffusion equation to interpret its meaning.

Abstract

The stochastic variational method is applied to particle systems and continuum mediums. As the brief review of this method, we first discuss the application to particle Lagrangians and derive a diffusion-type equation and the Schr\"{o}dinger equation with the minimum gauge coupling. We further extend the application of the stochastic variational method to Lagrangians of continuum mediums and show that the Navier-Stokes, Gross-Pitaevskii and generalized diffusion equations are derived. The correction term for the Navier-Stokes equation is also obtained in this method. We discuss the meaning of this correction by comparing with the diffusion equation.