Lie and quasi-Lie structures in thermodynamics

The Lie, Jacobi and weaker structures may be introduced into the scheme of classical thermodynamics if one makes use of the so-called thermodynamic phase space (TPS). For a thermodynamic system having n degrees of freedom, TPS is a (2n + 1)-dimensional manifold endowed with a contact structure, The contact structure of TPS allows one to associate two types of flows, and hence two types of vector fields X-f and <(X)over bar (f)>, to any smooth function f on TPS. While the set of all X-f's forms a Lie algebra, this is not the case for <(X)over bar (f)>'s. Moreover, only X-f's (for special choices of f's) generate flows which can be treated as thermodynamic processes; they preserve some Legendre submanifolds of the contact form. In general, both X-f and <(X)over bar (f)> do not correspond to flows having this property. Examples of X-f's are given and briefly discussed.