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In this paper, we prove the characterization of the $(K, \infty)$-superPerelman Ricci flows by various functional inequalities and gradient estimatefor the heat semigroup generated by the Witten Laplacian on manifolds equippedwith time dependent metrics and potentials. As a byproduct, we derive theHamilton type dimension free Harnack inequality on manifolds with $(K,\infty)$-super Perelman Ricci flows. Based on a new second order differentialinequality on the Boltzmann-Shannon entropy for the heat equation of the WittenLaplacian, we introduce a new $W$-entropy quantity and prove its monotonicityfor the heat equation of the Witten Laplacian on complete Riemannian manifoldswith the $CD(K, \infty)$-condition and on compact manifolds with $(K,\infty)$-super Perelman Ricci flows. Our results characterize the $(K,\infty)$-Ricci solitons and the $(K, \infty)$-Perelman Ricci flows. We alsoprove a second order differential entropy inequality on $(K, m)$-super Ricciflows, which can be used to characterize the $(K, m)$-Ricci solitons and the$(K, m)$-Ricci flows. Finally, we give a probabilistic interpretation of the$W$-entropy for the heat equation of the Witten Laplacian on manifolds with the$CD(K, m)$-condition.

$W$-entropy, super Perelman Ricci flows and $(K, m)$-Ricci solitons