Energy Landscapes, Scale-free Networks and Apollonian Packings

The potential energy as a function of the coordinates of all the atoms in a system defines a multi-dimensional surface that is commonly known as an energy landscape.1 Characterizing such energy landscapes has become an increasingly popular approach to study the behaviour of complex systems, such as the folding of a protein2 or the properties of supercooled liquids.3,4 The aim is to answer such questions as, what features of the energy landscape differentiate those polypeptides that are able to fold from those that get stuck in the morass of possible conformations, or those liquids that show super-Arrhenius dynamics (‘fragile’ liquids) from those that are merely Arrhenius (‘strong’ liquids). Such approaches have to be able to cope with the complexity of the potential energy landscape—for example, the number of minima is typically an exponential function of the number of atoms.5 One such approach is the inherent structure mapping pioneered by Stillinger and coworkers.6 In this mapping each point in configuration space is associated with the minimum obtained by following the steepestdescent pathway from that point. Thus, configuration space is partitioned into a set of basins of attraction surrounding the potential energy minima, as illustrated in Fig. 1. One of the original aims of this approach was to remove the vibrational motion from configurations generated in simulations of liquids to give a clearer picture of the underlying ‘inherent structure’, hence the common name for the mapping. Of more interest to us is that it breaks the energy landscape down into more manageable chunks, whose properties can be more easily established and understood. As an example of the utility of this approach, the classical partition function can be expressed as an integral over the whole of configuration space, but performing this integral (except numerically through say Monte Carlo) is nigh impossible, because