A Set of Statistical Variables for Hydrodynamic Flow

TL;DR

This paper introduces a novel set of time-dependent probability functions for hydrodynamic flows, derived from the Navier-Stokes equations, providing a unique and ad hoc-free closure framework with potential applications in quantum mechanics and kinetic theory.

Contribution

It proposes a new set of differential distribution equations for hydrodynamic flow, ensuring unique closure relations without ad hoc assumptions.

Findings

The set of probability functions can describe unsteady hydrodynamic flows.

Annealing methods can find stable stationary solutions.

Potential applications extend to quantum statistical mechanics and kinetic theory.

Abstract

Through a discussion of some typical unsteady hydrodynamic flows, we argue that the time averaged hydrodynamic functions at each point give a rather sparse filling of the local jet space. This situation then suggests a set of time dependent probability functions that are shown to give evolution uniquely defined by the Navier-Stokes equations through a set of "differential distribution equations." The closure relations are therefore unique and have no ad hoc characteristics. Annealing methods are proposed as a way to arrive at the stable stationary solutions corresponding to time averaged fluid flow with constant driving forces and fixed boundary conditions. Some applications of this method to quantum statistical mechanics and kinetic theory to higher orders are suggested.