Information geometry in vapour-liquid equilibrium
  Using the square-root map p-->\sqrt{p} a probability density function p can be represented as a point of the unit sphere S in the Hilbert space of square-integrable functions. If the density function depends smoothly on a set of parameters, the image of the map forms a Riemannian submanifold M in S. The metric on M induced by the ambient spherical geometry of S is the Fisher information matrix. Statistical properties of the system modelled by a parametric density function p can then be expressed in terms of information geometry. An elementary introduction to information geometry is presented, followed by a precise geometric characterisation of the family of Gaussian density functions. When the parametric density function describes the equilibrium state of a physical system, certain physical characteristics can be identified with geometric features of the associated information manifold M. Applying this idea, the properties of vapour-liquid phase transitions are elucidated in geometrical terms. For an ideal gas, phase transitions are absent and the geometry of M is flat. In this case, the solutions to the geodesic equations yield the adiabatic equations of state. For a van der Waals gas, the associated geometry of M is highly nontrivial. The scalar curvature of M diverges along the spinodal boundary which envelopes the unphysical region in the phase diagram. The curvature is thus closely related to the stability of the system. 
Dorje C. Brody and Daniel W. Hook