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

Abstract

This study proposes an energy-based framework for Official Statistics aimed at investigating the local stability of territorial socio-demographic configurations. The observed distribution of a continuous variable is interpreted as the reference configuration of an interacting territorial system, where endogenous interactions among local units and exogenous socio-economic drivers jointly determine the underlying energy landscape. Socio-economic characteristics are summarized through interpretable composite indices and synthesized by Principal Component Analysis to construct a univariate external field acting on the territorial system. Two complementary formulations are investigated: a Continuous Ising model and a stochastic alternative based on Langevin dynamics. Both models are simulated through Monte Carlo procedures with Simulated Annealing to explore energetically favourable configurations within the local neighbourhood of the observed state. Rather than searching for a global optimum, the proposed framework investigates local perturbations of the reference configuration in order to analyse the influence of socio-economic drivers acting as external forces on territorial dynamics. The stochastic exploration is analysed through the stability of the energy trajectories, the descriptive agreement between observed and simulated territorial configurations, and the statistical properties of the stationary distributions. Uncertainty is quantified through a model-agnostic Conformal Prediction framework employed as a diagnostic tool for evaluating the variability of stationary simulated configurations. The methodology is illustrated through the analysis of the distribution of the resident foreign population across Italian municipalities, but it is formulated in general terms and is applicable to a broad class of territorial systems in Official Statistics.