Thermodynamics of algebraically decaying pair-inertia kernels

Determine the thermodynamics of the algebraically decaying kernel family K_\alpha(r)=(1+\eta r/\alpha)^{-\alpha}, including the required system-size normalization, equilibrium correlations, relaxation behavior across decay exponents, and whether the limits \alpha\to0^+ and the thermodynamic limit commute.

Background

The paper rigorously bounds the free energy for exponentially decaying kernels and identifies a Kac rescaling for uniformly non-decaying kernels, but those results do not cover algebraic tails. The authors propose the family K_\alpha(r)=(1+\eta r/\alpha){-\alpha}, which interpolates between exponential behavior at large \alpha and a constant-kernel limit as \alpha\to0+.

Studying the Laplacian determinant for this family is intended to establish the appropriate thermodynamic normalization and enable comparisons of equilibrium correlations and relaxation. The order of the algebraic-decay and thermodynamic limits is also unresolved.

References

The dependence on the decay of the interaction kernel remains another open question. A useful family is \begin{equation} K_{\alpha}(r)= \left(1+\frac{\eta r}{\alpha}\right){-\alpha}. \end{equation} For every fixed $\alpha>0$, this kernel decays algebraically at large $r$, while at fixed separation it approaches $e{-\eta r}$ as $\alpha\rightarrow\infty$ and the constant kernel as $\alpha\rightarrow0+$. The present exponential bounds do not determine the thermodynamics of this family. Studying its Laplacian determinant would establish the required size normalization and allow a comparison of equilibrium correlations and relaxation across different decay exponents. It would also clarify whether the $\alpha\rightarrow0+$ and thermodynamic limits commute.

— Statistical mechanics of classical fractons on a line  (2609.25999 - Sadki et al., 22 Sep 2026) in Section Summary and future directions