\sect{Baruch's D04.}

Coarse grained entropy. The usual $\rho(p,q,t)$, i.e. the
normalized state density in the 6N dimensional phase space
$(p,q)$, satisfies Liouville's theorem $d\rho/dt=0$. We wish to
redefine $\rho(p,q,t)$ so that the corresponding entropy increases
with time.

Divide phase space to small sub-volumes $\Omega_{\ell}$ and define
a coarse grained density
\[{\bar\rho}(p,q,t)={\bar\rho}_{\ell}=\frac{1}{\Omega_{\ell}}
\int_{\Omega_{\ell}}\rho(p,q,t)dpdq \qquad  \qquad   (p,q)\in
\Omega_{\ell}\] so that ${\bar\rho}(p,q,t)$ is constant within
each cell $\Omega_{\ell}$. Define the entropy as
\[\eta(t)=-\int {\bar\rho}(p,q,t)\ln {\bar\rho}(p,q,t)dpdq=
-\sum_{\ell}\Omega_{\ell}{\bar \rho}_{\ell}\ln {\bar
\rho}_{\ell}\,.\] Assume that at $t=0$ $\rho(p,q,0)$ is uniform so
that $\rho(p,q,0)={\bar\rho}(p,q,0)$.
\begin{itemize}
\item[(a)] Show that
$\eta(0)=-\int \rho(p,q,t) \ln \rho(p,q,t)dpdq\,.$
\item[(b)] Show that $\eta(t)$ increases with time, i.e.
\[\eta(t)-\eta(0)=-\int \rho \left[\ln \frac{{\bar \rho}}{\rho} +1-
\frac{{\bar \rho}}{\rho}\right]dpdq\geq 0\,.\]
\end{itemize}

Hint: Show that $\ln x +1 -x \leq 0 $ for all $x>0$.\\

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