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A geometric foundation thermo-statistics is presented with the only axiomaticassumption of Boltzmann's principle S(E,N,V)=k\ln W. This relates the entropyto the geometric area e^{S(E,N,V)/k} of the manifold of constant energy in thefinite-N-body phase space. From the principle, all thermodynamics andespecially all phenomena of phase transitions and critical phenomena canunambiguously be identified for even small systems. The topology of thecurvature matrix C(E,N) of S(E,N) determines regions of pure phases, regions ofphase separation, and (multi-)critical points and lines. Within  Boltzmann's principle, Statistical Mechanics becomes a geometric theoryaddressing the whole ensemble or the manifold of all points in phase spacewhich are consistent with the few macroscopic conserved control parameters.This interpretation leads to a straight derivation of irreversibility and theSecond Law of Thermodynamics out of the time-reversible, microscopic,mechanical dynamics. This is all possible without invoking the thermodynamiclimit, extensivity, or concavity of S(E,N,V). The main obstacle against theSecond Law, the conservation of the phase-space volume due to Liouville isovercome by realizing that a macroscopic theory like Thermodynamics cannotdistinguish a fractal distribution in phase space from its closure.Comment: 26 pages, 6 figure

Geometric foundation of thermo-statistics, phase transitions, second law of thermodynamics, but without thermodynamic limit