Derivation of Stokes' Law from Kirkwood's Formula and the Green-Kubo Formula via Large Deviation Theory

TL;DR

This paper derives Stokes' law for a sphere's friction in a viscous fluid by connecting Kirkwood's surface stress correlation with the bulk stress correlation via large deviation theory, avoiding direct hydrodynamic equations.

Contribution

It introduces a novel derivation of Stokes' law by linking microscopic stress correlations to macroscopic viscosity through large deviation theory.

Findings

Derivation of Stokes' law from microscopic stress correlations.

Connection between surface and bulk stress correlations.

Application of large deviation theory to fluid dynamics.

Abstract

We study the friction coefficient of a macroscopic sphere in a viscous fluid at low Reynolds number. First, Kirkwood's formula for the friction coefficient is reviewed on the basis of the Hamiltonian description of particle systems. According to this formula, the friction coefficient is expressed in terms of the stress correlation on the surface of the macroscopic sphere. Then, with the aid of large deviation theory, we relate the surface stress correlation to the stress correlation in the bulk of the fluid, where the latter is characterized by the viscosity in the Green-Kubo formula. By combining Kirkwood's formula and the Green-Kubo formula in large deviation theory, we derive Stokes' law without explicitly employing the hydrodynamic equations.