Ricerc@Sapienza

The formulation of transport models in polymeric systems starting from the theory of stochastic processes possessing finite propagation velocity is here presented. Hyperbolic continuous equations are shown to be derived from Poisson-Kac stochastic processes and the extension to higher dimensions is discussed. We analyze the physical implications of this approach, namely: non-Markovian nature, admissible boundary conditions, breaking of concentration-flux paradigm and extension to nonlinear case.

This article analyzes the hydrodynamic (continuous) limits of lattice random walks in one spatial dimension. It is shown that a continuous formulation of the process leads naturally to a hyperbolic transport model, characterized by finite propagation velocity, while the classical parabolic limit corresponds to the Kac limit of the hyperbolic model itself. This apparently elementary problem leads to fundamental issues in the theory of stochastic processes and non-equilibrium phenomena, paving the way to new approaches in the field.

Non-Fickian behaviors in mass transport are very common in polymeric materials. The classical Fickian law, dictating the instantaneous response of the mass flux to changes in the driving force (namely the concentration gradient), seems to be too simplistic for mass transport in viscoelastic materials, generally linked through characteristic times to stress relaxation or structural changes. A constitutive equation in which the evolution of the mass flux is governed by a characteristic relaxation time thus appeared to be the most natural choice in order to catch anomalous behaviors.