Geometric thermodynamics

Speaker:  Arjan van der Schaft (Bernoulli Institute for Mathematics, Computer Science and AI, University of Groningen)
Date:  Friday, 26 November 2021 - 15:30
Place:  Online - us06web.zoom.us/j/7555463367 (ID: 755 546 3367)

Abstract:

Starting from Gibbs' fundamental thermodynamic relation, contact geometry has been recognized as a natural framework for the geometric formulation of classical thermodynamics since the early 1970s. On the other hand, the contact-geometric formulation of thermodynamics makes a distinction between the energy and the entropy representation of the same thermodynamic system. An attractive point of view that is merging the energy and entropy representation is offered by the extension of contact manifolds to symplectic manifolds; in fact cotangent bundles without zero section. This extension is already known from the geometric theory of partial differential equations. From a thermodynamics perspective it amounts to replacing the intensive variables (such as temperature and pressure) by their homogeneous coordinates. This results in a formulation on the cotangent bundle of the manifold of extensive variables, where all geometric objects are homogeneous in the cotangent variables, and all geometric objects are defined with respect to the Liouville form. It makes a clear distinction between the extensive and intensive variables of a thermodynamic system, and enables the definition of port-thermodynamic systems. These are thermodynamic systems that interact with their environment via either power or entropy flow ports. Additional homogeneity with respect to the extensive variables, corresponding to the classical Gibbs-Duhem relation, can be treated within the same geometric framework.

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