Fine-grained Gibbs entropy ($S_{\mathrm{fg}}=-k_B\int \rho \log (\rho /{\rho }_{\mathrm{ref}})d\Gamma$)

For a normalized probability density $\rho$ on classical phase space with canonical volume element $d\Gamma$, the fine-grained Gibbs entropy is $-k_B\int \rho \log (\rho /{\rho }_{\mathrm{ref}})d\Gamma$, whenever the integral exists. Here ${\rho }_{\mathrm{ref}}>0$ is a constant reference density fixing the units inside the logarithm, and $k_B$ is the Boltzmann constant. No averaging into finite cells is performed.

For a regular Hamiltonian flow ${\Phi }_t$, Liouville theorem in Hamiltonian mechanics gives $d\Gamma ({\Phi }_{t}z)=d\Gamma (z)$ and ${\rho }_t({\Phi }_{t}z)={\rho }_0(z)$. Substitution directly provesAn irreversible macroscopic entropy increase therefore requires a different description or additional assumptions; it is not a failure of this volume-preservation argument.