Dually Flat Geometries in the State Space of Statistical Models

The study of the geometry of models of statistical physics started with the work of (Weinhold, 1975) and (Ruppeiner, 1979).
A key notion is the thermodynamic length which is a measure for distances in the space of equilibrium states of the model.
Appealing is the relation between model interactions and the curvature of the state space.

On the other hand, (Amari, 1985) showed that the state space of a statistical model can be equipped with a dual set of flat geometries.
The duality between the two geometries can be worked out using Legendre transforms and is the same duality which in termodynamics
relates inverse temperature to energy and Massieu's function to entropy (Naudts, 2004, 2011). One of the two geometries is flat in intensive
coordinates such as inverse temperature and external magnetic field.

The other is flat in the extensive variables energy and total magnetization. Because geodesics are straight lines it is easy to calculate a
thermodynamic length in these flat geometries.

F. Weinhold, J. Chem. Phys 63, 2479 (1975).
G. Ruppeiner, Phys. Rev. A 20, 1608 (1979).
S. Amari, Differential-geometric methods in statistics (Springer, 1985)
J. Naudts, J. Ineq. Pure Appl. Math. 5, 102 (2004).
J. Naudts, Generalized thermostatistics (Springer, 2011)