Geometry and control of thermodynamic systems described by generalized exponential families

In this paper we investigate the geometric structure and control of exponential families depending on additional parameters, called external parameters. These generalized expo- nential families emerge naturally when one applies the maximum entropy formalism to derive the equilibrium statistical mechanics framework. We study the associated statistical model, compute the Fisher metric and introduce a natural fibration of the parameter space over the external parameter space. The Fisher Riemannian metric allows to endow this fi- bration with an Ehresmann connection and to study the geometry and control of these statistical models. As an example, we show that horizontal lift of paths in the external pa- rameter space corresponds to an isentropic evolution of the system. We apply the theory to the example of an ideal gas and an ideal gas in a rotating rigid container.