Non-universal finite size effects with universal infinite-size free energy

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 and indicates a second-order phase transition, 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 the subcritical regime, it is numerically shown that convergences towards the canonical averages become slower as the interaction-range becomes shorter. For the supercritical regime, the relation to a pair of the discrepancies is theoretically predicted and numerically confirmed.