Physical completion of the Navier-Stokes equations

TL;DR

This paper rigorously derives the fluctuation-dissipation relation for the nonlinear Navier-Stokes equations using topological arguments, confirming the GENERIC framework without linearization.

Contribution

It provides a topological proof of the fluctuation-dissipation relation for nonlinear Navier-Stokes equations, validating the reversible/irreversible decomposition in a physically realistic setting.

Findings

Derivation of fluctuation-dissipation relation without linearization

The nonlinear convective term is Hamiltonian and drops out of equilibrium conditions

The resulting stochastic system is globally well-posed with a unique Gibbs equilibrium

Abstract

The incompressible Navier-Stokes equations contain viscous dissipation but no thermal noise. I show, using a topological argument based on Poincar\'e's lemma, that the fluctuation-dissipation relation for the full nonlinear dynamics can be derived without the linearisation or structural assumptions that all previous derivations require. The nonlinear convective term is Hamiltonian (energy-preserving and phase-space-volume-preserving) and drops out of the Fokker-Planck equilibrium condition exactly, so the noise derived from linearised fluctuations near equilibrium is in fact exact for the full nonlinear system. This result proves, rather than assumes, the reversible/irreversible decomposition that the GENERIC framework postulates, provided Poincar\'e's lemma holds on the phase space. The resulting stochastic system, with a physical molecular-scale spectral cutoff, is trivially globally well-posed: a finite-dimensional stochastic differential equation with non-degenerate noise and a confining Lyapunov function. It has a unique Gibbs equilibrium and converges to it exponentially. The difficulty of the Clay Millennium Prize Problem arises entirely from two idealisations, zero temperature and infinite spectral resolution, neither of which is satisfied by any physical fluid.