Using contact and symplectic geometry, the authors recover standard thermodynamics by fixing an extended scale variable and conclude that breaking scale symmetry between energy and entropy is required for non-isothermal black hole dynamics.
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Scaling Symmetry in Symplectic Thermodynamics
Using contact and symplectic geometry, the authors recover standard thermodynamics by fixing an extended scale variable and conclude that breaking scale symmetry between energy and entropy is required for non-isothermal black hole dynamics.