Interactions in a multi-scale representation of sparse media: from mechanics to thermodynamics

We develop further a discrete-to-continuum approach to sparse media for which we do not grant the common axiom of permanent identification of the material elements. After refining our previous work on this topic, we focus our attention on the identification of stresses and self-actions at continuum scale in terms of the molecular interactions. Moreover, we introduce and discuss a notion of tensorial temperature by considering a grand canonical statistical ensemble constituted by molecules freely flowing in and out a control volume