TOPICAL REVIEW: Information geometry in vapour-liquid equilibrium

Using the square-root map p → √p a probability density function p can be represented as a point of the unit sphere {\cal 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 {\mathfrak M} \subset {\cal S} . The metric on {\mathfrak M} induced by the ambient spherical geometry of {\cal 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 characterization 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 {\mathfrak 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 {\mathfrak 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 {\mathfrak M} is highly nontrivial. The scalar curvature of {\mathfrak 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.