Geometry of Thermodynamic Processes

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.