Differential equation of state of a model system with a singular measure: application to granular materials in steady states

A direct modeling of the radial distribution function g(r) for model amorphous system is carried out in terms of generalized functions, without phenomenological parameters and independent of the type of potential. Within the proposed model, physical parameters and details of the considered system contribute through certain homogenous rheological characteristics like for instance compression (compressibility). The combination of this singular model form for g(r) with familiar integral relations yields an Abelian differential form for the equation of state which for a given sample includes a set of relevant material characteristics. These characteristics could be considered as given in every particular case and could be extracted from alternative sources. An appropriate choice of the parameters of the Abelian differential equation gives the possibility of its explicit solution. Obtained results have been analyzed in the context of a model equation of state for driven granular material near the steady states. The relevant functional equations of state which follow from the respective explicit solutions of the Abelian equations are obtained and discussed.