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.