Liouville geometry of classical thermodynamics

Starting from Gibbs’ fundamental thermodynamic relation, contact geometry has been recognized as a natural framework for the geometric formulation of classical thermodynamics since the early 1970s [21]. This spurred a series of papers; see e.g. [29, 30, 31, 32, 33, 34, 4, 19, 11, 13, 16, 8, 26, 6, 17, 35, 23, 39, 10, 12], and [7] for a recent introduction and survey. Other geometric work emphasizing the variational formulation of thermodynamics includes [28, 15]. On the other hand, as discussed in [5], the contact-geometric formulation of thermodynamics makes a distinction between the energy and the entropy representation of the same thermodynamic system. By itself this need not be considered as a major flaw since the two representations are conformally equivalent. Nevertheless, it was shown in [5], and later in [36, 27, 37], that an attractive point of view that is merging the energy and entropy representation is offered by the extension of contact manifolds to symplectic manifolds. Compared with the odd-dimensional contact manifold this even-dimensional symplectic manifold has one more degree of freedom, called a gauge variable in [5]. From a thermodynamics perspective it amounts to replacing the intensive variables by their homogeneous coordinates. In fact, this symplectization of contact manifolds is rather well-known in differential geometry [2, 25]; dating back to [20]. As argued in [37], the extension of contact manifolds to symplectic manifolds, in fact to cotangent bundles without zero section, has additional advantages for the geometric formulation of thermodynamics as well. First, it yields a clear distinction between the extensive and intensive variables of the thermodynamic system. Secondly, it enables the definition of port-thermodynamic systems, which are thermodynamic systems that interact with their environment via either power or entropy flow ports. Finally, symplectization has computational benefits; as was already argued within differential geometry by Arnold [2, 3].