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In this survey paper, we give an overview of our recent works on the study ofthe $W$-entropy for the heat equation associated with the Witten Laplacian onsuper-Ricci flows and the Langevin deformation on Wasserstein space overRiemannian manifolds. Inspired by Perelman's seminal work on the entropyformula for the Ricci flow, we prove the $W$-entropy formula for the heatequation associated with the Witten Laplacian on $n$-dimensional completeRiemannian manifolds with the $CD(K, m)$-condition, and the $W$-entropy formulafor the heat equation associated with the time dependent Witten Laplacian on$n$-dimensional compact manifolds equipped with a $(K, m)$-super Ricci flow,where $K\in \mathbb{R}$ and $m\in [n, \infty]$. Furthermore, we prove ananalogue of the $W$-entropy formula for the geodesic flow on the Wassersteinspace over Riemannian manifolds. Our result recaptures an important result dueto Lott and Villani on the displacement convexity of the Boltzmann-Shannonentropy on Riemannian manifolds with non-negative Ricci curvature. To betterunderstand the similarity between above two $W$-entropy formulas, we introducethe Langevin deformation of geometric flows on the cotangent bundle over theWasserstein space and prove an extension of the $W$-entropy formula for theLangevin deformation. Finally, we make a discussion on the $W$-entropy for theRicci flow from the point of view of statistical mechanics and probabilitytheory.

$W$-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds