  A geometric foundation thermo-statistics is presented with the only axiomatic
assumption of Boltzmann's principle S(E,N,V)=k\ln W. This relates the entropy
to the geometric area e^{S(E,N,V)/k} of the manifold of constant energy in the
finite-N-body phase space. From the principle, all thermodynamics and
especially all phenomena of phase transitions and critical phenomena can
unambiguously be identified for even small systems. The topology of the
curvature matrix C(E,N) of S(E,N) determines regions of pure phases, regions of
phase separation, and (multi-)critical points and lines. Within
  Boltzmann's principle, Statistical Mechanics becomes a geometric theory
addressing the whole ensemble or the manifold of all points in phase space
which are consistent with the few macroscopic conserved control parameters.
This interpretation leads to a straight derivation of irreversibility and the
Second Law of Thermodynamics out of the time-reversible, microscopic,
mechanical dynamics. This is all possible without invoking the thermodynamic
limit, extensivity, or concavity of S(E,N,V). The main obstacle against the
Second Law, the conservation of the phase-space volume due to Liouville is
overcome by realizing that a macroscopic theory like Thermodynamics cannot
distinguish a fractal distribution in phase space from its closure.
