标题： 通过统计力学建模领土动态：对常住外国人口的应用 显示英文标题

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

摘要： 本研究引入了一种统计力学框架，用于分析意大利市镇中常住外国人口的地域分布。 观察到的外国居民百分比被视为一个相互作用的地域单元系统的参考配置。 通过可解释的综合指数总结市镇的社会经济结构，并通过主成分分析进行简化，以构建与建模框架兼容的单变量外部场。 探索了两种互补的方法：一种是伊辛模型的连续变体，另一种是称为朗之万动力学的随机微分替代方法，这两种方法均采用蒙特卡洛方法进行模拟，并结合模拟退火，这是一种常用于探索局部能量极小值的随机优化策略。 这两种方法都能在观察状态附近高效地进行配置空间的局部探索，有助于发现塑造常住外国人口空间分布的社会经济决定因素。 模型的适用性通过能量稳定性、似然性和预测准确性进行评估。 使用与模型无关的共形预测来量化不确定性，该方法产生具有保证边际覆盖的自适应预测区间。 绘制这些区间的幅度可以突出不确定性较高的区域，并支持对地域社会经济动态的细致解释。 通过结合统计力学、多元分析和不确定性量化，本研究提供了一种稳健且可解释的一般方法，用于建模复杂的地域模式，对官方统计具有直接的相关性。 显示英文摘要

摘要： 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.