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

Statistical Mechanics