Extending canonical Monte Carlo methods

TL;DR

This paper extends canonical Monte Carlo methods to better handle systems with negative heat capacities by incorporating a fluctuation-dissipation relation, enabling more efficient simulations of anomalous thermodynamic behaviors.

Contribution

It introduces a generalized framework based on the dynamical ensemble to extend Monte Carlo algorithms for systems with negative heat capacities, demonstrated on Potts models.

Findings

Successfully extended Monte Carlo methods to systems with negative heat capacities.

Reduced decorrelation time divergence from exponential to power-law with system size.

Applied methods to 2D 10-state Potts model, showing improved efficiency.

Abstract

In this work, we discuss the implications of a recently obtained equilibrium fluctuation-dissipation relation on the extension of the available Monte Carlo methods based on the consideration of the Gibbs canonical ensemble to account for the existence of an anomalous regime with negative heat capacities $C<0$. The resulting framework appears as a suitable generalization of the methodology associated with the so-called \textit{dynamical ensemble}, which is applied to the extension of two well-known Monte Carlo methods: the Metropolis importance sample and the Swendsen-Wang clusters algorithm. These Monte Carlo algorithms are employed to study the anomalous thermodynamic behavior of the Potts models with many spin states $q$ defined on a $d$-dimensional hypercubic lattice with periodic boundary conditions, which successfully reduce the exponential divergence of decorrelation time $\tau$ with the increase of the system size $N$ to a weak power-law divergence $\tau\propto N^{\alpha}$ with $\alpha\approx0.2$ for the particular case of the 2D 10-state Potts model.