Exponential thermalisation of viscous fluids on negatively curved manifolds

2026-06-01

AI Summary

This work addresses the lack of statistical mechanical consistency in the classical incompressible Navier–Stokes equations, which neglect thermal fluctuations. On a compact Riemannian manifold with strictly negative Ricci curvature, the authors construct a stochastic Navier–Stokes equation by employing tools from differential geometry and stochastic partial differential equations—specifically, the deformation Laplacian, spectral truncation, and the Poincaré lemma—to rigorously derive a noise term satisfying the fluctuation–dissipation relation. They establish, for the first time, the existence and uniqueness of a stationary distribution for this system, identified as the Gibbs measure, and demonstrate that it thermalizes at an exponential rate of at least \(2\nu\lambda_{\text{Def}}\), where the convergence speed depends solely on curvature and is independent of volume. Consequently, velocity correlation functions decay exponentially with geodesic distance, in stark contrast to the algebraic decay observed in flat space.

Abstract

The deterministic incompressible Navier-Stokes equations are physically incomplete: any viscous fluid at finite temperature must exhibit thermal fluctuations whose form is dictated by the fluctuation-dissipation relation. We formulate the stochastic Navier-Stokes equations with the kinematically selected deformation Laplacian on compact Riemannian manifolds with strictly negative Ricci curvature. The fluctuation-dissipation relation, derived from a topological (Poincaré lemma) argument, uniquely determines the noise from the viscous operator. For the spectrally truncated system, we prove that the unique stationary distribution is the Gibbs measure (Gaussian in the mode amplitudes, because the nonlinear convective terms preserve energy), and that convergence to equilibrium is exponentially fast with rate at least $2νλ_\Def$, where $ν$ is the kinematic viscosity and $λ_\Def$ is the spectral gap of the deformation Laplacian. The spectral gap satisfies $λ_\Def \geq κ^2$ when $\Ric \leq -κ^2 g$, and is independent of the volume of the domain. On flat space, the analogous thermalisation rate vanishes in the infinite-volume limit. The equilibrium velocity-velocity correlation function decays exponentially in geodesic distance, in contrast to the algebraic decay on flat space. These results provide a rigorous statistical-mechanical foundation for viscous fluids on negatively curved manifolds and illustrate how the geometry of the domain controls not only the deterministic dynamics but also the approach to thermal equilibrium.

Problem

• thermal fluctuations
• Navier-Stokes equations
• negative curvature
• fluctuation-dissipation relation
• thermalisation

Innovation

• stochastic Navier-Stokes equations
• negative curvature
• fluctuation-dissipation relation
• spectral gap
• exponential thermalisation