A Stochastic Interacting Particle Model for the Marble Sulphation Process

The marble sulphation process, the phenomenon of the formation of a patina of gypsum on the surface of calcium carbonate monuments, is characterized by different scales: from the nanoscale where molecules interact and react, to the macroscale where averaged quantities are measured. We propose a first stochastic model at the nanoscale. It is given by a first order dynamics for the molecules given by a system of stochastic differential equations driven by Wiener processes, coupled with point processes properly describing the chemical reactions. In particular we suggest a new model for the activating energy related to the reaction. By an appropriate rescaling procedure, we heuristically derive both a measure-valued equation for the empirical measure and a natural mean-field approximation, and compare it with the closest deterministic model in literature. Possible developments of the proposed stochastic model are discussed.