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We present numerical investigations into the question of the validity of theBoltzmann prescription in Statistical Mechanics for large systems, addressingthe issue of whether extensivity of energy implies the extensivity of theBoltzmann entropy. The importance of the question stems from the fact that itis currently considered open by some investigators but quite settled by others.We report ab initio results for gas-like Hamiltonian systems with long-range aswell as short-range interactions, based on simulations that explicitly considermore than $2^{30} \approx 10^9$ states of the full Hilbert space. The basis ofthe technique is Monte Carlo algorithms. Despite the largeness of the numbersused, careful inspection shows that the systems studied are still too small tosettle uniquely the issues raised. Therefore, the new approach outlinedrepresents a first step in addressing on first principles the question ofnon-extensive statistical mechanics. General theoretical comments are alsosupplied to supplement the numerical investigations.

Numerical studies for an ab initio investigation into the Boltzmann prescription in statistical mechanics of large systems