Characterisation of gradient flows for a given functional

2022/09/22 by Morris Brooks, Jan Maas, Brooks, Morris +1
Engineering · Mathematics · #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2209.11149

openalex publication_date 2022/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a vector field and Y be a co-vector field on a smooth manifold M. Does there exist a smooth Riemannian metric g_αβ on M such that Y_β= g_αβ X^α? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we show that a finite-dimensional ergodic Lindblad equation admits a gradient flow structure for the von Neumann relative entropy if and only if the condition of BKM-detailed balance holds.

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