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Geometry and Control of Thermodynamic Systems Described by Generalized Exponential Families
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: Received: 8 June 2021 / Approved: 9 June 2021 / Online: 9 June 2021 (10:37:33 CEST)
How to cite: Favretti, M. Geometry and Control of Thermodynamic Systems Described by Generalized Exponential Families. Preprints 2021, 2021060251 Favretti, M. Geometry and Control of Thermodynamic Systems Described by Generalized Exponential Families. Preprints 2021, 2021060251
Abstract
In this paper we investigate the geometric structure and control of exponential families depending on additional parameters, called external parameters. These generalized exponential families emerges 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 fibration 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 parameter space corresponds to an isentropic evolution of the system. We apply the theory to the example of a ideal gas in a rotating rigid container. Most of the results are expressed in local coordinates; in the appendices we hint at possible global extensions of the theory.
Keywords
statistical models; Fisher metric; Ehresmann connection, exponential families
Subject
Physical Sciences, Mathematical Physics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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