Phase transitions in finite systems = topological peculiarities of the microcanonical entropy surface

It is discussed how phase transitions of first order (with phase separation and surface tension), continuous transitions and (multi)-critical points can be defined and classified for finite systems from the topology of the energy surface $e^{S(E,N)}$ of the mechanical N-body phase space or more precisely of the curvature determinant $D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial N^2-(\partial^2S/\partial E\partial N)^2$ without taking the thermodynamic limit. The first calculation of the entire entropy surface $S(E,N)$ for a q=3-states Potts lattice gas on a 50*50 square lattice is shown. There are two lines, where $S(E,N)$ has a maximum curvature $\sim 0$. One is the border between the regions in {$E,N$} with $D(E,N)>0$ and with $D(E,N)<0$, the other line is critical starting as a valley in $D(E,N)$ running from the continuous transition in the ordinary q=3-Potts model, converting at $P_m$ into a flat ridge/plateau (maximum) deep inside the convex intruder of $S(E,N)$ which characterizes the first order liquid-gas transition. The multi-critical point $P_m$ is their crossing.