Figure 1: Partitioning of phase space into neighbourhoods.
The potential energy hypersurface V(x) (as a function of the state x of the system) defines a natural partitioning of phase space into neighbourhoods ησ, based on the sign of c(x), the local minimum curvature of V(x) (blue: negative and red: positive). Each neighbourhood (stable or not) can be assigned a well-defined free energy by integration of the partition function over that neighbourhood. In this example, the point 
corresponds to fcc W while the point 
corresponds to bcc W (there are three symmetrically equivalent basins corresponding to bcc W) and the path joining them is the well-known Bain path. Also shown are paths of steepest descent from given ‘unrelaxed’ positions (
, 
) towards corresponding minima (
, 
) within the corresponding neighbourhoods (ησ, 
). For the mechanically stable phase, 
lies at a local minimum while for the mechanically unstable phase, 
lies at an inflection point along a path of steepest descent.
