  In this note we determine the first two derivatives of the classical
Boltzmann-Shannon entropy of the conjugate heat equation on general evolving
manifolds. Based on the second derivative of the Boltzmann-Shannon entropy, we
construct Perelman's F and W entropy in abstract geometric flows. Monotonicity
of the entropies holds when a technical condition is satisfied. This condition
is satisfied on static Riemannian manifolds with nonnegative Ricci curvature,
for Hamilton's Ricci flow, List's extended Ricci flow, M\"uller's Ricci flow
coupled with harmonic map flow and Lorentzian mean curvature flow when the
ambient space has nonnegative sectional curvature. Under the extra assumption
that the lowest eigenvalue is differentiable along time, we derive an explicit
formula for the evolution of the lowest eigenvalue of the Laplace-Beltrami
operator with potential in the abstract setting.
