Non-extensive Hamiltonian systems follow Boltzmann's principle not Tsallis statistics—phase transitions, Second Law of Thermodynamics

Boltzmann's principle S(E,N,V)=klnW(E,N,V) relates the entropy to the geometric area eS(E,N,V) of the manifold of constant energy in the N-body phase space. From the principle all thermodynamics and especially all phenomena of phase transitions and critical phenomena can be deduced. The topology of the curvature matrix C(E,N) (Hessian) of S(E,N) determines regions of pure phases, regions of phase separation, and (multi-)critical points and lines. Thus, C(E,N) describes all kinds of phase transitions with all their flavor. No assumptions of extensivity, concavity of S(E), additivity have to be invoked. Thus Boltzmann's principle and not Tsallis statistics describes the equilibrium properties as well the approach to equilibrium of extensive and non-extensive Hamiltonian systems. No thermodynamic limit must be invoked.