Topology of event distributions as a generalized definition of phase transitions in finite systems

We propose a definition of first order phase transitions in finite systems based on topology anomalies of the event distribution in the space of observations. This generalizes the definitions based on the curvature anomalies of thermodynamical potentials, provides a natural definition of order parameters, and can be related to the Yang-Lee theorem in the thermodynamical limit. It is directly operational from the experimental point of view. It allows to study phase transitions in Gibbs equilibria as well as in other ensembles such as the Tsallis ensemble.