Microthermodynamic formalism

We consider the microcanonical 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 first law of thermodynamics, or fundamental relation, is based upon the bulk-entropy, $S_{\Omega }.$ Under an ergodic hypothesis, $S_{\Omega }$ is shown to be an adiabatic invariant. Expressions for derivatives and thermodynamic relations are derived within the microcanonical ensemble itself.