  In 1870s, L. Boltzmann proved the famous $H$-theorem for the Boltzmann
equation in the kinetic theory of gas and gave the statistical interpretation
of the thermodynamic entropy. In 2002, G. Perelman introduced the notion of
$W$-entropy and proved the $W$-entropy formula for the Ricci flow. This plays a
crucial role in the proof of the no local collapsing theorem and in the final
resolution of the Poincar\'e conjecture and Thurston's geometrization
conjecture. In our previous paper \cite{Li11a}, the author gave a probabilistic
interpretation of the $W$-entropy using the Boltzmann-Shannon-Nash entropy. In
this paper, we make some further efforts for a better understanding of the
mysterious $W$-entropy by comparing the $H$-theorem for the Boltzmann equation
and the Perelman $W$-entropy formula for the Ricci flow. We also suggest a way
to construct the "density of states" measure for which the Boltzmann
$H$-entropy is exactly the $W$-entropy for the Ricci flow.
