Limiting theorems in statistical mechanics. The mean-field case.

The Curie-Weiss model is defined by ah Hamiltonian according to spins interact. For some particular values of the parameters, the sum of the spins normalized with square-root normalization converges or not toward Gaussian distribution. In the thesis we investigate some correlations between the behaviour of the sum and the central limit for interacting random variables.

Statistical mechanics of polymer models: rigorous results and applications

Monomer-dimer models are amongst the models in statistical mechanics which found application in many areas of science, ranging from biology to social sciences. This model describes a many-body system in which monoatomic and diatomic particles subject to hard-core interactions get deposited on a graph. In our work we provide an extension of this model to higher-order particles. The aim of our work is threefold: first we study the thermodynamic properties of the newly introduced model. We solve analytically some regular cases and find that, differently from the original, our extension admits phase transitions. Then we tackle the inverse problem, both from an analytical and numerical perspective. Finally we propose an application to aggregation phenomena in virtual messaging services.

Tolerance optimization with statistical approach in the CAT environment

Nowadays, product development in all its phases plays a fundamental role in the industrial chain. The need for a company to compete at high levels, the need to be quick in responding to market demands and therefore to be able to engineer the product quickly and with a high level of quality, has led to the need to get involved in new more advanced methods/ processes. In recent years, we are moving away from the concept of 2D-based design and production and approaching the concept of Model Based Definition. By using this approach, increasingly complex systems turn out to be easier to deal with but above all cheaper in obtaining them. Thanks to the Model Based Definition it is possible to share data in a lean and simple way to the entire engineering and production chain of the product. The great advantage of this approach is precisely the uniqueness of the information. In this specific thesis work, this approach has been exploited in the context of tolerances with the aid of CAD / CAT software. Tolerance analysis or dimensional variation analysis is a way to understand how sources of variation in part size and assembly constraints propagate between parts and assemblies and how that range affects the ability of a project to meet its requirements. It is critically important to note how tolerance directly affects the cost and performance of products. Worst Case Analysis (WCA) and Statistical analysis (RSS) are the two principal methods in DVA. The thesis aims to show the advantages of using statistical dimensional analysis by creating and examining various case studies, using PTC CREO software for CAD modeling and CETOL 6σ for tolerance analysis. Moreover, it will be provided a comparison between manual and 3D analysis, focusing the attention to the information lost in the 1D case. The results obtained allow us to highlight the need to use this approach from the early stages of the product design cycle.