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Geometry of Thermodynamic Processes.

Arjan Van der Schaft1, Bernhard Maschke2

  • 1Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, Jan C. Willems Center for Systems and Control, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary
This summary is machine-generated.

Contact geometry offers a new framework for thermodynamic systems, unifying energy and entropy representations. This research develops a global geometric metric for state properties and formulates non-equilibrium processes using Hamiltonian dynamics.

Keywords:
interconnectionmetricsnon-equilibrium processessymplectizationthermodynamics

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Area of Science:

  • Thermodynamics
  • Differential Geometry
  • Mathematical Physics

Background:

  • Contact geometry has been a recognized framework for geometric formulation of thermodynamic systems and their state properties since the 1970s.
  • Symplectization of contact manifolds offers a new perspective, facilitating easy switching between energy and entropy representations of thermodynamic systems.

Purpose of the Study:

  • To extend the geometric formulation of thermodynamic systems using contact and symplectic geometry.
  • To define a global degenerate Riemannian metric for thermodynamic state properties.
  • To develop a geometric formulation for non-equilibrium thermodynamic processes.

Main Methods:

  • Utilizing the symplectization of contact manifolds.
  • Defining a global degenerate Riemannian metric on homogeneous Lagrangian submanifolds.
  • Formulating non-equilibrium processes via Hamiltonian dynamics with specific Hamiltonian functions.
  • Extending geometric correspondences to port-thermodynamic systems and interconnection ports.

Main Results:

  • A global geometric definition of a degenerate Riemannian metric for thermodynamic state properties, unifying local metrics.
  • A geometric formulation of non-equilibrium thermodynamic processes using Hamiltonian dynamics.
  • Extension of geometric framework to port-thermodynamic systems and interconnection ports.

Conclusions:

  • The developed geometric framework provides a unified approach to thermodynamic systems, including non-equilibrium processes.
  • The framework, illustrated with examples, shows potential for analysis and control of thermodynamic systems.
  • This work bridges contact and symplectic geometry for a comprehensive understanding of thermodynamics.