From clusters of moving molecules to continua: Material elements as open systems

We discuss the discrete-to-continuum transition in the description of matter, starting from a cluster of flowing molecules with equal mass and ending up with a non-simple fluid. We account for the fluctuations beyond a local affine approximation of the velocity distribution of molecules within a space window adopted to compute some prominent statistics. The resulting continuum picture accounts for local mass variation in each space window corresponding to a point in the continuum scale. From a statistical viewpoint, every material element is thus considered as a grand-canonical ensemble. So-called C-derivatives account for macroscopic-to-mesoscopic relative motion. When considered for second-rank tensors, they extend Truesdell’s derivative and reduce to Oldroyd’s one when the macroscopic-to-mesoscopic relative motion vanishes. Fluctuations beyond an affine component best approximating the kinetic energy are summarized into a second-rank symmetric tensor whose time variation enters a balance equation governing transfer from velocity fluctuations to heat. Eventually, we discuss essential elements of thermodynamics in the present setting. What emerges is the possibility of a non-Fourier type heat transfer. The results address computational schemes for field representations of sparse phase dynamics, such as granular materials, and the one of bodies with transport of scattered molecules, such as pollutants in fluids or proteins in biological tissues.