Phase transitions in "small" systems

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. "Small" systems are systems where the linear dimension is of the characteristic range of the interaction between the particles; i.e. also astrophysical systems are "small" in this sense. 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 S(E, N) or more precisely, of the curvature determinant D(E, N) = partial derivative(2)S/partial derivative E-2 x partial derivative(2)S/partial derivative N-2 - (partial derivative(2)S/partial derivative E partial derivative N)(2) allows the classification of phase transitions without taking the thermodynamic limit. Micro-canonical thermostatistics and phase transitions will be discussed here for a system coupled by short range forces in another situation where entropy is not extensive. 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 x 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 del D vanishes in the direction of the eigenvector upsilon(1) of D with the largest eigen-value lambda(1) approximate to 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.