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In this paper, we develop a new approach to prove the $W$-entropy formula forthe Witten Laplacian via warped product on Riemannian manifolds and give anatural geometric interpretation of a quantity appeared in the $W$-entropyformula. Then we prove the $W$-entropy formula for the Witten Laplacian oncompact Riemannian manifolds with time dependent metrics and potentials, andderive the $W$-entropy formula for the backward heat equation associated withthe Witten Laplacian on compact Riemannian manifolds equipped with Lott'smodified Ricci flow. We also extend our results to complete Riemannianmanifolds with negative $m$-dimensional Bakry-Emery Ricci curvature, and tocompact Riemannian manifolds with $K$-super $m$-dimensional Bakry-Emery Ricciflow. As application, we prove that the optimal logarithmic Sobolev constant oncompact manifolds equipped with the $K$-super $m$-dimensional Bakry-Emery Ricciflow is decreasing in time.

$W$-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials