Geometric calculations on density manifolds from reciprocal relations in hydrodynamics
Wuchen Li
TL;DR
This work develops a geometric framework for hydrodynamical density manifolds arising from generalized Onsager reciprocal relations, treating non-equilibrium diffusion as gradient flows of free energies in infinite-dimensional density spaces. It derives the Levi-Civita connection, gradient, Hessian, parallel transport, and curvature tensors, thereby extending Wasserstein-2 geometry to nonlinear mobility settings. In one dimension it yields explicit curvature formulas and shows sectional curvature signs are governed by mobility convexity, with concrete results for zero-range models such as independent particles, simple exclusion, and KMP. The findings provide insights into macroscopic fluctuation theory, energy dissipation, and potential geometry-preserving computational approaches for macroscopic diffusion and stochastic Fokker-Planck dynamics, with applications to fluctuation relations and diffusion in complex systems.
Abstract
Hydrodynamics are systems of equations describing the evolution of macroscopic states in non-equilibrium thermodynamics. From generalized Onsager reciprocal relationships, one can formulate a class of hydrodynamics as gradient flows of free energies. In recent years, Onsager gradient flows have been widely investigated in optimal transport-type metric spaces with nonlinear mobilities, namely hydrodynamical density manifolds. This paper studies geometric calculations in these hydrodynamical density manifolds. We first formulate Levi-Civita connections, gradient, Hessian, and parallel transport and then derive Riemannian and sectional curvatures on density manifolds. We last present closed formulas for sectional curvatures of density manifolds in one dimensional spaces, in which the sign of curvatures is characterized by the convexities of mobilities. For example, we present density manifolds and their sectional curvatures in zero-range models, such as independent particles, simple exclusion processes, and Kipnis-Marchioro-Presutti models.