论文标题
给定功能的梯度流的表征
Characterisation of gradient flows for a given functional
论文作者
论文摘要
令$ x $为矢量字段，$ y $是平滑歧管$ m $的共同矢量场。是否存在$ m $上的平滑Riemannian度量$ g_ {αβ} $，以便$y_β= g_ {αβ} x^α$？本说明的主要结果给出了必要和充分的条件，以使这是正确的。作为此结果的应用，我们表明有限的千古lindblad方程在且仅当BKM-detailed Balance的状态保持时，就可以接受Von Neumann相对熵的梯度流量结构。
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.
