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Since Boltzmann and Gibbs, canonical distributions have been fundamental in statistical physics. In a modern approach [Jaynes: Phys. Rev. 106, 620, 1957] they appear naturally as a solution to a well set optimization problem: maximizing entropy under a set of expected values constraints. Generally in physics such expected values define the macrostates, and the space in which the canonical distribution is defined are the microstates.

It happens so that the space of possible macrostates can be regarded as a space of parameters for canonical distributions (statistical manifold). In the field of Information Geometry [Amari and Nagaoka: Methods of Information Geometry, American Mathematical Soc. , 2007] these distributions happen to have deeply interesting geometrical properties such as their metric tensor is a covariance matrix and important thermodynamical objects, such as free energy, appear naturally.

The use of information geometry in statistical physics is not new [Ruppeiner: Rev. Mod. Phys. 68, 313, 1996][Brody: Phys. Rev. E 51, 1006, 1995]. But in this work we want to provide a systematic way to create dynamical systems in the space of such probability distributions. This can give new insight on fields such as critical phenomena and renormalization groups as well as deal with statistical problems in which the microstate dynamics is not so well defined (such as economics and ecology).

These dynamics are derived as an application of entropic methods of inference i.e. that is a form of entropic dynamics [Caticha: Entropy 17, 6110, 2015]. The flow of probability distributions is driven by entropy subject to constraints.
As an example we show how entropic dynamics is applied to well-known thermodynamical models with phase transition, such as the Van der Walls gas.