PRIN 2022 CALL. DECREE NO. 104 OF 2 FEBRUARY 2022
PROJECT TITLE: Mathematical Modelling of Heterogeneous Systems
CUP CODE D53D23005900006
Budget: € 85.441 (of which MUR funding € 55478)
Person in Charge U.R. Giuseppe Florio
Other Research Units or any Sub-Units
R.U. University of Rome “Sapienza” (PI: E.N.M. Cirillo)
R.U. University of Florence (RU: L. Fusi)
Brief project description
The project “Mathematical Modelling of Heterogeneous Systems” (MMHS) is aimed at developing a rigorous and predictive mathematical description of complex systems characterized by structural and multiscale heterogeneity. Such systems, which include composite materials, biological tissues, complex fluids, and granular systems, exhibit emergent macroscopic properties that cannot be inferred from classical homogeneous models. The project seeks to bridge this gap through an integrated approach combining homogenization theory, partial differential equations, statistical mechanics, and asymptotic analysis.
The scientific activity was organized from a multiscale perspective, with the aim of connecting microscopic or mesoscopic models with effective macroscopic laws. In this context, particular attention was devoted to the study of transport and diffusion mechanisms in heterogeneous media, phase transitions in multistable systems, and the modeling of complex materials with nonlinear behavior. The project also developed through collaboration among research units with complementary expertise, ensuring integration between theoretical and numerical approaches supported by experimental activities.
Purpose
The main objective of the project is the construction of mathematical models capable of describing emergent phenomena in heterogeneous systems, with particular reference to the relationship between microscopic structure and macroscopic behavior. In particular, the project aims to derive effective transport and diffusion laws, to understand the mechanisms of metastability and nucleation, and to develop predictive models for complex materials and non-Newtonian fluids. A further goal is the integration of approaches from statistical mechanics, mathematical analysis, and continuum mechanics, in order to build a coherent theoretical framework applicable to different scientific contexts.
Expected results
The expected results included the development of multiscale mathematical models, the rigorous derivation of macroscopic laws from microscopic models, and the characterization of complex phenomena such as anomalous diffusion, phase transitions, and instabilities in heterogeneous systems. The production of high-level scientific results was also envisaged, documented through publications in international journals and participation in conferences, as well as the strengthening of collaboration among the research units involved.
Achieved results
The activities carried out led to the achievement of scientifically significant results consistent with the project’s initial objectives. In particular, a rigorous theory of transport and diffusion processes in heterogeneous media was developed, making it possible to derive effective macroscopic equations from complex microscopic structures. It was shown how different mechanisms of microscopic disorder can generate different macroscopic behaviors, also clarifying the role of interface conditions in composite materials.
In the field of multi-stable systems and phase transitions, significant results have been obtained in the study of metastability and nucleation mechanisms in stochastic lattice models. The analysis highlighted the link between microscopic dynamics and macroscopic behavior, providing a detailed description of transitions between metastable and stable states. These results contribute to a deeper understanding of transition phenomena in heterogeneous systems.
Further advances have been achieved in the modeling and analysis of materials with complex internal structure. In parallel, thermal and entropic effects in heterogeneous materials have been studied, showing how microscopic fluctuations and disorder can significantly alter macroscopic properties, for example in the case of smearing in phase transitions.
In the field of fluid mechanics, non-Newtonian and viscoplastic flows in complex geometries have been analyzed through analytical and numerical approaches. Results have been obtained on stability and the onset of instabilities in stratified suspensions and advanced rheological models, improving the predictive capability of models for complex fluids.
Overall, the project has produced an organic body of results that integrates rigorous mathematical analysis, stochastic modeling, and continuum mechanics, establishing a solid theoretical basis for the description of heterogeneous systems across different scales. These results have been validated and disseminated through a large number of scientific publications in high-impact international journals.
These results were also disseminated through a website available at the page https://www.sbai.uniroma1.it/progetti/mmhs/index.php
