Generalized Metropolis dynamics with a generalized master equation: An approach for time-independent and time-dependent Monte Carlo simulations of generalized spin systems

TL;DR

This paper introduces a local generalized Metropolis dynamics based on a new master equation for generalized spin systems, enabling efficient Monte Carlo simulations and accurate critical property calculations across different parameters.

Contribution

It proposes a local generalized Metropolis algorithm derived from a new master equation that obeys detailed energy balance, improving simulation efficiency and accuracy.

Findings

The new dynamics obeys detailed energy balance for all q values.

Critical temperatures and exponents vary with q, affecting universality classes.

Short time power laws are effective for parameter estimation even for q≠1.

Abstract

The extension of Boltzmann-Gibbs thermostatistics, proposed by Tsallis, introduces an additional parameter $q$ to the inverse temperature $\beta$. Here, we show that a previously introduced generalized Metropolis dynamics to evolve spin models is not local and does not obey the detailed energy balance. In this dynamics, locality is only retrieved for $q=1$, which corresponds to the standard Metropolis algorithm. Non-locality implies in very time consuming computer calculations, since the energy of the whole system must be reevaluated, when a single spin is flipped. To circumvent this costly calculation, we propose a generalized master equation, which gives rise to a local generalized Metropolis dynamics that obeys the detailed energy balance. To compare the different critical values obtained with other generalized dynamics, we perform Monte Carlo simulations in equilibrium for Ising model. By using the short time non-equilibrium numerical simulations, we also calculate for this model: the critical temperature, the static and dynamical critical exponents as function of $q$. Even for $q\neq 1$, we show that suitable time evolving power laws can be found for each initial condition. Our numerical experiments corroborate the literature results, when we use non-local dynamics, showing that short time parameter determination works also in this case. However, the dynamics governed by the new master equation leads to different results for critical temperatures and also the critical exponents affecting universality classes. We further propose a simple algorithm to optimize modeling the time evolution with a power law considering in a log-log plot two successive refinements.