Continuum vs thermodynamical limit in Statistical Mechanics

Determining the limiting behavior of discrete systems with a large number of particles in Statistical Mechanics is crucial for developing accurate analytic models, especially when addressing multistability and multiscale effects. Typically, one considers the so called thermodynamical limit or the continuum limit. The guiding principle for selecting the correct limit is to preserve essential properties of the discrete system, including physical attributes such as the interplay between enthalpic and entropic contributions, the influence of boundary conditions, and possible other energetic contributions such as interface effects. In this sense, an important role is played by the fundamental constants. Selecting appropriate rescaling factors for the Planck and Boltzmann constants, according to the specific limit considered, is a key theoretical concern. Despite the importance of this problem, the existing literature often lacks clarity on how different rescalings affect model accuracy. This work aims to clarify these issues by examining classical lattice models - particularly those that exhibit multistable behavior - and by proposing suitable limit rescalings to retain the discrete model's material response when the number of particles increases.