An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes

It is shown that the intrinsic geometry associated with equilibrium thermodynamics, namely the contact geometry, provides also a suitable framework in order to deal with irreversible thermodynamical processes. Therefore we introduce a class of dynamical systems on contact manifolds, called conservative contact systems, defined as contact vector fields generated by some contact Hamiltonian function satisfying a compatibility condition with some Legendre submanifold of the contact manifold. Considering physical systemsʹ modeling, the Legendre submanifold corresponds to the definition of the thermodynamical properties of the system and the contact Hamiltonian function corresponds to the definition of some irreversible processes taking place in the system. Open thermodynamical systems may also be modeled by augmenting the conservative contact systems with some input and output variables (in the sense of automatic control) and so-called input vector fields and lead to the definition of port contact systems. Finally complex systems consisting of coupled simple thermodynamical or mechanical systems may be represented by the composition of such port contact systems through algebraic relations called interconnection structure. Two examples illustrate this composition of contact systems: a gas under a piston submitted to some external force and the conduction of heat between two media with external thermostat.