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Hyperbolic heat/mass transport and stochastic modelling - Three simple problems

Dipartimento di Ingegneria Chimica DICMA Facoltà di Ingegneria, La Sapienza Università di Roma via Eudossiana 18, 00184, Roma, Italy

Starting from the correspondence between the Cattaneo hyperbolic heat equation and the stochastic formulation based on Poisson-Kac processes, that holds solely for one-dimensional spatial models, this article analyzes three paradigmatic problems in the hyperbolic theory of heat and mass transport. The problems considered involve unbounded, semi-bounded and bounded domains, and are aimed at : (i) highlighting analogies and differences between the two approaches (Cattaneo vs Poisson-Kac), (ii) addressing the role of a bounded propagation velocity in order to regularize the properties of the solutions of heat/mass transport problems. A typical example of the latter phenomenology is expressed by boundary-layer regulatization of interfacial fluxes. The case of transport in bounded domains permits to pinpoint unambiguously the need of a stochastic interpretation of the transport equation in order to unveil the occurrence of physical inconsistencies that may occur in the linear Cattaneo hyperbolic model in some range of parameter values.
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