  In order to better understand the occurrence of phase transitions, we adopt
an approach based on the study of energy landscapes: The relation between
stationary points of the potential energy landscape of a classical
many-particle system and the analyticity properties of its thermodynamic
functions is studied for finite as well as infinite systems. For finite
systems, each stationary point is found to cause a nonanalyticity in the
microcanonical entropy, and the functional form of this nonanalytic term can be
derived explicitly. With increasing system size, the order of the nonanalytic
term grows unboundedly, leading to an increasing differentiability of the
entropy. Therefore, in the thermodynamic limit, only asymptotically flat
stationary points may cause a phase transition to take place. For several spin
models, these results are illustrated by predicting the absence or presence of
a phase transition from stationary points and their local curvatures in
microscopic configuration space. These results establish a relationship between
properties of energy landscapes and the occurrence of phase transitions. Such
an approach appears particularly promising for the simultaneous study of
dynamical and thermodynamical properties, as is of interest for example for
protein folding or the glass transition.

Statistical Mechanics