  Traditionally, phase transitions are defined in the thermodynamic limit only.
We discuss how phase transitions of first order (with phase separation and
surface tension), continuous transitions and (multi)-critical points can be
seen and classified for small systems. Boltzmann defines the entropy as the
logarithm of the area W(E,N)=e^S(E,N) of the surface in the mechanical N-body
phase space at total energy E. The topology of the curvature determinant D(E,N)
of S(E,N) allows the classification of phase transitions without taking the
thermodynamic limit. The first calculation of the entire entropy surface S(E,N)
for the diluted Potts model (ordinary (q=3)-Potts model plus vacancies) on a
50*50 square lattice is shown. The regions in {E,N} where D>0 correspond to
pure phases, ordered resp. disordered, and D<0 represent transitions of first
order with phase separation and ``surface tension''. These regions are bordered
by a line with D=0. A line of continuous transitions starts at the critical
point of the ordinary (q=3)-Potts model and runs down to a branching point P_m.
Along this line \nabla D vanishes in the direction of the eigenvector v_1 of D
with the largest eigen-value \lambda_1\approx 0. It characterizes a maximum of
the largest eigenvalue \lambda_1. This corresponds to a critical line where the
transition is continuous and the surface tension disappears. Here the
neighboring phases are indistinguishable. The region where two or more lines
with D=0 cross is the region of the (multi)-critical point. The micro-canonical
ensemble allows to put these phenomena entirely on the level of mechanics.
