Which power-law decays make interactions dynamically long-range
In plain words
Interactions that fall off with distance as $1/r^\alpha$ behave as long-range in thermodynamics when $\alpha$ is below the space dimension, but the threshold for long-lived non-equilibrium states may differ. The exact boundary is not established.
Precise statement
For particles in $d$ dimensions with pair potential $V(r) \sim 1/r^{\alpha}$, determine the range of $\alpha$ for which the $N \to \infty$ dynamics follows a Vlasov equation with quasi-stationary states whose lifetime diverges with $N$, as opposed to collisional relaxation on $N$-independent times; the proposed criterion is $\alpha < d - 1$ (force integrable at large distance). An answer is a classification with proof or with simulations controlling $N \to \infty$.
What would settle it
A kinetic-theory derivation of the $N$ dependence of relaxation times for general $\alpha$, confirmed by simulations in $d = 1, 2, 3$.
Status in the literature
The criterion was proposed by Gabrielli, Joyce and Marcos (Phys. Rev. Lett. 105, 210602, 2010); a derivation for general $\alpha$ and $d$ is incomplete.