G. de Saxcé - Thermodynamique pentadimensionnelle des milieux continus

Key idea How to geometrise Thermodynamics

• temperature is a vector $W$
• entropy is a vector $S$
• energy becomes a tensor $T$
• gravitation is a covariant derivative $\nabla$ Background
• C. Eckart: first principle is the divergence of the impulse-energy tensor equal to zero.
• Landau and Lifshitz: decomposition of this tensor into a reversible and an irreversible part + link between the entropy vector and the temperature $S=TW$.
• Souriau 1976: the divergence of the entropy vector is greater or equal to zero. Relativity
• an event occurs at a position and a time.
• The Galilean transformations are space-time transformations preserving inertial motions, durations, and distances. Their set is Galileo’s group, a Lie group of dimension 10.
• A former symmetry group would be the group of Lorentz-Poincaré.
• Work with a dimension 5
• Consider a $\mathbb{R}$-principal bundle, and a section. They build.a group of affine transformation, Galilean when acting onto the space-time.

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• Consider Bargmannian transformations, setting the Bargmann’s group, which is very useful in thermodynamics. Temperature, friction, and momentum tensor
• The temperature is a 5-vector, a Bargmannian 5-vector.
• Introduce the friction tensor, the gradient of the temperature vector. This object merges the temperature gradient and the strain velocity.
• Momentum tensor, the energy-momentum-mass tensor. The momentum tensor leads to a reduced form, where we obtain the potentials of thermodynamics. $\Leftarrow$ It is the key First and second principles
• Need to generalise the divergence in a covariant form, divergence of the momentum tensor. This leads to the conservation of mass, linear momentum, and conservation of energy. This geometry setting gives the opportunity to define a reversible medium as a theorem.
• For the second principle, use the additive decomposition of the momentum tensor, into a reversible part, and an irreversible part. After calculations, obtain a local production of entropy, which is a Galilean invariant. It also recovers the classical form of the Clausius-Duhem inequality. Coupling between thermodynamics and galilean gravitation