  We study finite size effects in a family of systems in which a parameter
controls interaction-range. In the long-range regime where the infinite-size
free energy is universal, we show that the finite size effects are not
universal but depend on the interaction-range. The finite size effects are
observed through discrepancies between time-averages of macroscopic variables
in Hamiltonian dynamics and canonical averages of ones with infinite degrees of
freedom. For a high energy regime, the relation to a pair of the discrepancies
is theoretically predicted and numerically confirmed. We also numerically show
that the finite-size effects of macroscopic variables in the canonical ensemble
are close to ones in the dynamical systems.
