Non-Universal Finite Size Effects with Universal Infinite-Size Free Energy for the $\alpha$-XY model

TL;DR

This paper investigates how finite size effects in the $eta$-XY model depend on interaction range, revealing non-universality in finite systems despite universal behavior in the infinite-size limit, through theoretical and numerical analysis.

Contribution

It demonstrates that finite size effects vary with interaction range in the $eta$-XY model, contrasting with the universal infinite-size free energy.

Findings

Finite size effects depend on interaction range.

Discrepancies between dynamical and canonical averages are predicted and confirmed.

Finite-size effects in canonical ensemble are similar to those in dynamical systems.

Abstract

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.