Thermostatistics of extensive and non-extensive systems using generalized entropies

We describe in detail two numerical simulation methods valid to study syste ms whose thermostatistics is described by generalized entropies, such as Ts allis. The methods are useful for applications to non-trivial interacting s ystems with a large number of degrees of freedom, and both short- and long- range interactions. The first method is quite general and it is based on th e numerical evaluation of the density of states with a given energy. The se cond method is more specific for Tsallis thermostatistics and it is based o n a standard Monte Carlo Metropolis algorithm along with a numerical integr ation procedure. We show here that both methods are robust and efficient. W e present results of the application of the methods to the one-dimensional Ising model both in a short-range and in a long-range (non-extensive) case. We show that the thermodynamic potentials for different values of the syst em size N and different values of the non-extensivity parameter q can be de scribed by scaling relations which are an extension of the ones holding for the Boltzmann-Gibbs statistics (q = 1). Finally, we discuss the difference s in using standard or non-standard mean value definitions in the Tsallis t hermostatistics formalism and present a microcanonical ensemble calculation approach of the averages. (C) 2001 Elsevier Science B.V. All rights reserv ed.