Intuition behind the entropy functional for curvature bounds

Question

What is the intuition behind defining in metric measure spaces a Ricci curvature bound via convexity or concavity of the entropy functional? I know that this is made because in Riemannian manifolds the Ricci curvature bounds can be characterized via these inequalities, but in every paper that result seems to come "from above", without explaining why precisely the entropy encodes that.

Answer 1

Not intending rigour here, but probably it helps to get some intuition. On a Hilbert space or manifold, consider the gradient flow $\dot{\xi}=-\nabla f(\xi)$. Then the strong $k$- convexity of the potential $f$ is equivalent to exponentially fast contraction of flow trajectories, i.e. $d(\xi^{x}(t),\xi^{y}(t))\le e^{-kt}d(x,y)$ with $x,y$ being initial states of two flows.
Now on the other hand, one of the various equivalent ways to characterize lower Ricci curvature bounds is by requiring heat to contract exponentially fast (e.g. by means of Brownian motion trajectories). Physical reasoning behind would be that without global lower bounds, you may find initial starting points, close to singularities in your manifold, where trajectories may not contract at all.
Now where is the entropy? Since the seminal work of Felix Otto (c.f. 'THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION'), we know that the heat equation can itself be interpreted as Wasserstein gradient flow, for which the driving functional is precisely the Boltzmann entropy, i.e. $\dot{\mu}=-\nabla^{W}\operatorname{Ent}(\mu)$. Therefore, (recall first paragraph) strong convexity of the entropy is equivalent to exponential contraction of heat flow is equivalent to lower Ricci curvature bound.
This observation goes back originally to von Renesse & Sturm 'Transport inequalities, Gradient Estimates, Entropy and Ricci Curvature'.