  Boltzmann's principle S=k ln W allows to extend equilibrium thermo-statistics
to ``Small'' systems without invoking the thermodynamic limit. The clue is to
base statistical probability on ensemble averaging and not on time averaging.
It is argued that due to the incomplete information obtained by macroscopic
measurements thermodynamics handles ensembles or finite-sized sub-manifolds in
phase space and not single time-dependent trajectories. Therefore, ensemble
averages are the natural objects of statistical probabilities. This is the
physical origin of coarse-graining which is not anymore a mathematical ad hoc
assumption. From this concept all equilibrium thermodynamics can be deduced
quite naturally including the most sophisticated phenomena of phase transitions
for ``Small'' systems. Boltzmann's principle is generalized to non-equilibrium
Hamiltonian systems with possibly fractal distributions ${\cal{M}}$ in 6N-dim.
phase space by replacing the conventional Riemann integral for the volume in
phase space by its corresponding box-counting volume. This is equal to the
volume of the closure $\bar{\cal{M}}$. With this extension the Second Law is
derived without invoking the thermodynamic limit. The irreversibility in this
approach is due to the replacement of the phase-space volume by the volume of
its closure $\bar{\cal{M}}$. The physical reason for this replacement is that
macroscopic measurements cannot distinguish ${\cal{M}}$ from $\bar{\cal{M}}$.
Whereas the former is not changing in time due to Liouville's theorem, the
volume of the closure can be larger. In contrast to conventional coarse
graining the box-counting volume is defined in the limit of infinite
resolution. I.e. there is no artificial loss of information.
