Villani's Lazy Gas experiment: Meaning of intermediate distribution having maximum entropy?

Question

Taken from Villani's 2006 book on optimal transport to describe the figure that follows:

The Lazy Gas experiment: Take a perfect gas in which particles do not interact, and ask him to move from a certain prescribed density field at time t = 0, to another prescribed density field at time t = 1. Since the gas is lazy, he will find a way to do so by spending a minimal amount of work (least action path). Measure the entropy of the gas at each time, and check that it always lie above the line joining the final and initial entropies. If such is the case, then we know that we live in a nonnegatively curved space (see Figure 16.2).

The "gas" described is essentially a variable whose sample observations form, or can be represented as, a probability distribution or density so that Boltzmann's entropy translates to Shannon entropy since they are twin concepts designed for their respective fields.

Question

Is there some significance or meaning in, and reason why, the intermediate state $\left(t=\frac{1}{2}\right)$ of a gas or distribution has higher entropy than its initial (source) and final (target) states (densities)? The chapters ends there, but where is this experiment even going?

Answer 1

The claim follows immediately from a deep result of von Renesse and Sturm (https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.20060) which tells that for any smooth and connected Riemannian manifold and $K>0$ the following statements are equivalent:

1. Ricci curvature is bounded from below by $K$, i.e. $\operatorname{Ric}(M)\ge K$
2. the entropy $H:=-S$ is displacement $K$-convex, i.e. \begin{equation} H(\mu_{t})\le (1-t)H(\mu_{0})+tH(\mu_{1})-\frac{K}{2}t(1-t)d_{W}^{2}(\mu_{0},\mu_{1}), \end{equation} with $d_{W}$ being the Wasserstein-2-distance.