Abstract
In this paper, \((M,\varOmega )\) is a connected symplectic manifold on which a Lie group G acts by a Hamitonian action, with a moment map \(J:M\rightarrow {\mathfrak g}^*\). A short reminder of the definitions of statistical states, Gibbs states, entropy, generalized temperatures and associated thermodynamic functions is first given. Then several examples of such Gibbs states are presented, together with the associated thermodynamic functions. Examples are given too of symplectic manifolds on which a Lie group acts by a Hamiltonian action on which no Gibbs state built with a moment map of this action can exist. Most of these examples are obtained with the use of a remarkable isomorphism of a fully oriented three-dimensional Euclidean or pseudo-Euclidean vector space onto the Lie algebra of its Lie group of symmetries.
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C.-M. Marle—Retired from Université Pierre et Marie Curie, today Sorbonne Université.
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Marle, CM. (2021). Gibbs States on Symplectic Manifolds with Symmetries. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_27
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