Elementary preamble to a theory of granular gases

Granular materials partake almost dramatically at times of the properties of solids and, under different circumstances, of some properties of gases. Some scientists have suggested that they represent a new state of matter, to be treated per se; because of their ambiguity the search of an overall continuum model for these media is quite intriguing. Even if one confines inquiry to fast flows of sparse granules (as in granular gases, see e.g. [2]), the range of phenomena that may occur and that should be described by an adequate theory is vast. Within this area one is lured into adopting such appealing terms as gross granular heat and gross granular temperature, though the appropriateness of the borrowing can be challenged as the definitions of those gross quantities involve kinetic entities, whose detailed evolution can be in part ascertained rather than being totally unpredictable on principle and, at most, appraised through statistics. In any case, when exploiting vague analogies with the thermodynamics of perfect gases, one must make allowance for the fact that, in generic granular flows, speeds of agitation need not be, as they are in gases, many orders of magnitude greater than the average velocity or the velocity of any reasonably behaved observer. Thus one must be wary of a naive transfer to each moving element in a continuum of properties inferred from experiments within containers fixed to the walls of a laboratory; questions of objectivity become important if not paramount. One way to avoid difficulties would be to evaluate the relevant variables from some intrinsic local frame. The traditional way in continuum mechanics is to appeal to gross position and velocity gradients, a way that cannot be satisfactorily pursued in our context. Notice that akin problems