Probability Distribution Function of the Order Parameter: Mixing Fields and Universality

We briefly review the use of the order parameter probability distribution function as a useful tool to obtain the critical properties of statistical mechanical models using computer Monte Carlo simulations. Some simple discrete spin magnetic systems on a lattice, such as Ising, general spin-$S$ Blume-Capel and Baxter-Wu, $Q$-state Potts, among other models, will be considered as examples. The importance and the necessity of the role of mixing fields in asymmetric magnetic models will be discussed in more detail, as well as the corresponding distributions of the extensive conjugate variables.

💡 Research Summary

The paper provides a comprehensive review of how the probability distribution function (PDF) of the order parameter can be employed as a powerful tool for extracting critical properties of statistical‑mechanical models from Monte Carlo simulations. Starting from the simplest case, the Ising model, the authors demonstrate that the full magnetization distribution, when properly rescaled by the standard finite‑size scaling variables (L^{\beta/\nu}m) and ((T-T_c)L^{1/\nu}), collapses onto a universal curve that is independent of lattice size. This establishes the PDF as a more informative alternative to traditional observables such as mean magnetization or susceptibility, because the entire shape of the distribution encodes information about symmetry, scaling functions, and corrections to scaling.

The discussion then moves to asymmetric magnetic systems, exemplified by the general‑spin (S) Blume‑Capel model. In such models the presence of a non‑magnetic state (spin 0) breaks the up‑down symmetry, so that temperature and magnetic field alone no longer provide a complete scaling basis. The authors introduce the concept of “mixing fields”: linear combinations of temperature and field, \