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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 toevolve 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 tothe standard Metropolis algorithm. Non-locality implies in very time consumingcomputer calculations, since the energy of the whole system must bereevaluated, when a single spin is flipped. To circumvent this costlycalculation, we propose a generalized master equation, which gives rise to alocal generalized Metropolis dynamics that obeys the detailed energy balance.To compare the different critical values obtained with other generalizeddynamics, we perform Monte Carlo simulations in equilibrium for Ising model. Byusing the short time non-equilibrium numerical simulations, we also calculatefor this model: the critical temperature, the static and dynamical criticalexponents as function of $q$. Even for $q\neq 1$, we show that suitable timeevolving power laws can be found for each initial condition. Our numericalexperiments 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 differentresults for critical temperatures and also the critical exponents affectinguniversality classes. We further propose a simple algorithm to optimizemodeling the time evolution with a power law considering in a log-log plot twosuccessive refinements.

physical review. e, statistical, nonlinear and soft matter physics

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