A Micro-Thermodynamic Formalism

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We consider the micro-canonical ensemble of a classical Hamiltonian dynamical system, the Hamiltonian being parameter dependent and in the possible presence of other first integrals. We describe a thermodynamic formalism in which a 1st law of thermodynamics, or fundamental relation, is based upon the bulk-entropy, S. Under an ergodic hypothesis, S is shown to be an adiabatic invariant. Expressions for derivatives and thermodynamic relations are derived within the micro-canonical ensemble itself.