On the contact Hamiltonian functions of conservative contact systems

Abstract

The dynamics of irreversible thermodynamic systems have been expressed in terms of conservative contact systems where contact vector fields are generated by contact Hamiltonian functions defined on the Thermodynamic Phase Space (TPS). In this paper, we first emphasize the importance of both the Gibbs relation and the Gibbs–Duhem relation of the entropy or energy contact form in developing a first-order invariance constraint that every contact Hamiltonian function must satisfy. This novel insight is then considered together with the zero-order invariance constraint to infer solutions, thereby yielding a generalized family of contact Hamiltonian functions generating non-strict or strict contact vector fields which are equal on the associated Legendre submanifold on which the dynamics of the thermodynamic system is living. Finally, we show sufficient conditions under which the inverse images of zero by the contact Hamiltonian functions or the Legendre submanifold are globally attractive when lifting the system dynamics to the complete TPS. A simulated example is given to support the theoretical developments and to discuss the difference of the dynamic behaviours between the generated strict and non-strict contact vector fields.