Improving the efficiency of Monte Carlo simulations of systems that undergo temperature-driven phase transitions

Griffith theory of physical fractures, statistical procedures and entropy production: Rosetta stone's legacy

A physical model, based on energy balances, is proposed to describe the fractures in solid structures such as stelae, tiles, glass, and others. We applied the model to investigate the transition of the Rosetta Stone from the original state to the final state with three major fractures. We consider a statistical corner-breaking model with cutting rules. We obtain a probability distribution as a function of the area and the number of vertices. Our generic results are consistent with the current state of the Rosetta Stone and, additionally, predictions related to a fourth fracture are declared. The loss of information on such heritage pieces is considered through entropy production. The explicit quantification of this concept in information theory stays examined.

Editorial of the Special Issue “Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines”

This Special Issue includes a total of 14 articles on skewed probability distributions and applications across disciplines [...]

Directed walk in probability space that locates mean field solutions to spin models

Despite their formal simplicity, most lattice spin models cannot be easily solved, even under the simplifying assumptions of mean field theory. In this paper, we present a method for generating mean field solutions to classical continuous spins. We focus our attention on systems with nonlocal interactions and nonperiodic boundaries, which require careful handling with existing approaches, such as Monte Carlo sampling. Our approach utilizes functional optimization to derive a closed-form optimality condition and arrive at self-consistent mean field equations. We show that this approach significantly outperforms conventional Monte Carlo sampling in convergence speed and accuracy. To convey the general concept behind the approach, we first demonstrate its application to a simple system: a finite one-dimensional dipolar chain in an external electric field. We then describe how the approach naturally extends to more complicated spin systems and to continuum field theories. Furthermore, we numerically illustrate the efficacy of our approach by highlighting its utility on nonperiodic spin models of various dimensionality.

Extended canonical Monte Carlo methods: Improving accuracy of microcanonical calculations using a reweighting technique

Velazquez and Curilef [J. Stat. Mech. (2010); J. Stat. Mech. (2010)] have proposed a methodology to extend Monte Carlo algorithms that are based on canonical ensemble. According to our previous study, their proposal allows us to overcome slow sampling problems in systems that undergo any type of temperature-driven phase transition. After a comprehensive review about ideas and connections of this framework, we discuss the application of a reweighting technique to improve the accuracy of microcanonical calculations, specifically, the well-known multihistograms method of Ferrenberg and Swendsen [Phys. Rev. Lett. 63, 1195 (1989)]. As an example of application, we reconsider the study of the four-state Potts model on the square lattice L×L with periodic boundary conditions. This analysis allows us to detect the existence of a very small latent heat per site qL during the occurrence of temperature-driven phase transition of this model, whose size dependence seems to follow a power law qL(L)∝(1/L)z with exponent z≃0.26±0.02. Discussed is the compatibility of these results with the continuous character of temperature-driven phase transition when L→+∞.