Finite-size effects in molecular simulations: a physico-mathematical view

Molecular simulation of condensed matter systems has always been characterized by the aim for an optimal balance between a precise physical description of the simulated substance, and the efficient use of computational resources. A major challenge for the accurate representation of a physical system in a simulation, therefore, consists in determining the appropriate size of the simulated sample. The latter must be sufficiently large in order to represent the bulk of the substance, and thus to reproduce its characteristic thermodynamic features. This problem is known under the name of ‘finite-size effects’, and several criteria have been adopted in order to determine these effects, thereby inferring about the validity of a simulation study. In this article, we discuss the application of a rigorous mathematical theorem, the so-called ‘two-sided Bogoliubov inequality’, to estimate the finite-size effects. The theorem provides upper and lower bounds for the free energy cost of partitioning a system into equivalent, non-interacting subsystems, and it can be used to obtain a rigorous definition of the minimal size of a system with its full thermodynamic features. The corresponding criterion based on this theorem is complementary to those existing in the literature, and it can be applied to both classical and quantum systems.The need for accurate and physically consistent results of current simulations is enormously increased by the use of simulation data in machine learning procedures. Physically inconsistent data, produced by simulations of insufficient size, results in a substantial error in the modeling procedure that propagates further into the study of several other systems or larger scales beyond the molecular one. Furthermore, the statistical nature of machine learning implies questions about the number of parameters and the size of the training set. Such problems are the equivalent of the size effects discussed in the first part of the review. Here this feature is treated employing the same statistical mechanics framework developed for the first problem.