Geometric thermodynamics, non-equilibrium entropy and revision of B.D. Coleman’s thermomechanics

This paper is a continuation with certain modifications of a previous work by this author involving a new theory for the continuum-mechanical entropy in non-equilibrium states. Treating internal energy as an essential coordinate in the thermomechanical state space, the theory provides a nominal calorimetric theory of materials with memory as an alternative to the widely accepted thermometric theory of Coleman which treats both entropy and temperature as ” primitives” without a compelling connection to experimental measurement. Following a brief review of the previous work, involving a generally defined topology of state space for higher-gradient continua, a revision of Coleman’s theory is proposed that specifies non-equilibrium entropy as the variational extremum of a functional of the forces that define the relevant cotangent bundle of the underlying state space. As a conjecture, the optimal cycle in state space is assumed to be a non-equilibrium generalization of the classical Carnot cycle.