Geometry of the Entropy

In this work we investigate the material point model (MP-model) and exploit the geometrical meaning of the " entropy form " introduced by B.Coleman and R.Owen ([7]). We analyze full and partial integrability (close-ness) condition of the entropy form for the model of thermoelastic point and for the the deformable ferroelectric crystal media point. We show that the extended thermodynamical space introduced by R.Hermann and widely exploited by R. Mrugala with his collaborators and other researchers, extended possibly by time, with its canonical contact structure is an appropriate setting for the development of material point models in different physical situations. This allows us to formulate the model of a material point and the corresponding entropy form in terms similar to those of the homogeneous thermodynamics, [26]. Closeness condition of the entropy form is reformulated as the requirement that the admissible processes curves belongs to the constitutive surface Σ of the model. Our principal result is the description of the constitutive surfaces of the material point model as the Legendre submanifolds Σ S (equilibrium submanifolds of homogeneous thermodynamics) of the space P shifted by the flow of Reeb vector field. Contents 1. Introduction. 2 Part I. Coleman-Owen model, entropy form and integrability. 2. Entropy form of a thermoelastic system. 3 2.1. State space and thermodynamical phase space. 6 3. Integrability (closeness) conditions. 6 4. Partial integrability, dynamical equations and constitutive relations. 7 5. Entropy form in deformable ferroelectric crystal media 9 Part II. Contact geometry of entropy form. 6. Contact structure of homogeneous thermodynamics. 12 7. Gibbs space. Legendre surfaces of equilibrium. 13 8. Extended thermodynamical phase space, its contact structure. 14 9. Entropy form as a flat connection in the Gibbs bundle.