Modelling Territorial Dynamics through Statistical Mechanics: An Application to Resident Foreign Population

This study introduces a statistical mechanics framework to analyze the territorial distribution of the Resident Foreign Population across Italian municipalities. The observed percentages of foreign residents are treated as the reference configuration of an interacting system of territorial units. The socio-economic structure of municipalities is summarized through interpretable composite indices and reduced via Principal Components Analysis to construct a univariate external field compatible with the modelling framework. Two complementary approaches are explored: a continuous variant of the Ising model and a stochastic-differential alternative known as Langevin dynamics which are both simulated by adopting Monte Carlo approaches with Simulated Annealing, a stochastic optimization strategy commonly used for exploring local energy minima. Both methods enable efficient local exploration of the configuration space in the neighborhood of the observed state, facilitating the discovery of socio-economic determinants that shape the spatial distribution of resident foreign population. Model adequacy is assessed in terms of energy stability, likelihood, and predictive accuracy. Uncertainty is quantified using model-agnostic Conformal Prediction, which yields adaptive prediction intervals with guaranteed marginal coverage. Mapping the amplitude of these intervals highlights areas of higher uncertainty and supports a nuanced interpretation of territorial socio-economic dynamics. By combining statistical mechanics, multivariate analysis, and uncertainty quantification, the study provides a robust and interpretable general method for modelling complex territorial patterns, with direct relevance for Official Statistics.