Thermostatistics of extensive and non–extensive systems using generalized entropies

We describe in detail two numerical simulation methods valid to study systems whose thermostatistics is described by generalized entropies, such as Tsallis. The methods are useful for applications to non-trivial interacting systems with a large number of degrees of freedom, and both short–range and long–range interactions. The first method is quite general and it is based on the numerical evaluation of the density of states with a given energy. The second method is more specific for Tsallis thermostatistics and it is based on a standard Monte Carlo Metropolis algorithm along with a numerical integration procedure. We show here that both methods are robust and efficient. We present results of the application of the methods to the one–dimensional Ising model both in a short–range case and in a long–range (non–extensive) case. We show that the thermodynamic potentials for different values of the system size N and different values of the non–extensivity parameter q can be described by scaling relations which are an extension of the ones holding for the Boltzmann–Gibbs statistics (q = 1). Finally, we discuss the differences in using standard or non–standard mean value definitions in the Tsallis thermostatistics formalism and present a microcanonical ensemble calculation approach of the averages.