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.

翻译：不可压缩Navier-Stokes方程在物理上是不完备的：任何有限温度下的粘性流体必然存在热涨落，其形式由涨落-耗散关系决定。我们在紧致黎曼流形上，具有严格负Ricci曲率的条件下，使用运动学选择的变形拉普拉斯算子构造了随机Navier-Stokes方程。通过拓扑论证（庞加莱引理）推导出的涨落-耗散关系，从粘性算子中唯一地确定了噪声项。对于谱截断系统，我们证明了唯一平稳分布是吉布斯测度（模态振幅高斯分布，因为非线性对流项保持能量），并且向平衡态的收敛是指数快的，速率至少为$2νλ_\Def$，其中$ν$是运动粘度，$λ_\Def$是变形拉普拉斯算子的谱间隙。当$\Ric \leq -κ^2 g$时，谱间隙满足$λ_\Def \geq κ^2$，且不依赖于区域体积。在平坦空间中，类似的热化速率在无穷体积极限下趋于零。平衡态速度-速度相关函数沿测地线距离呈指数衰减，这与平坦空间中的代数衰减形成对比。这些结果提供了负曲率流形上粘性流体严谨的统计力学基础，并阐明了区域几何不仅控制确定性动力学，还控制热平衡的趋近过程。