Determine the asymptotic transport regime at σ = 2

Determine whether the asymptotic thermal transport of the one-dimensional long-range Fermi-Pasta-Ulam-Tsingou lattice with quartic interaction decay exponent σ = 2 is ballistic, with conductivity divergence exponent α = 1, or strongly superdiffusive, with α approximately 0.7, and identify the sources of discrepancies among numerical schemes.

Background

The σ = 2 long-range FPUT system has produced conflicting numerical results. Early nonequilibrium simulations reported nearly linear growth of thermal conductivity and nearly flat temperature profiles, suggesting ballistic transport, whereas later studies using periodic driving, correlation functions, and alternative heat-current definitions found strong superdiffusion with an effective exponent near 0.7.

The disagreement is attributed partly to sensitivity to boundary conditions, heat-bath coupling, finite-size effects, and the definition of microscopic heat current. The review therefore identifies the asymptotic transport class and the origin of the numerical discrepancies as unresolved.

References

Conversely, key disagreements persist regarding whether the asymptotic thermal transport at 𝜎 = 2 approaches ballistic ( 𝛼 = 1 ) or strong superdiffusive ( 𝛼 ≈ 0.7 ) regimes, as well as the sources of discrepancies among different numerical schemes.

— Recent progress on thermal transport in one-dimensional long-range interacting Fermi-Pasta-Ulam-Tsingou lattice systems  (2609.18765 - Xiong et al., 16 Sep 2026) in Section 3.1, “Thermal Transport Puzzle at σ = 2: Ballistic vs. Anomalous”

However, current literature on heat conduction lacks systematic studies on finite-size scaling and critical behavior at this boundary. Therefore, it remains unconfirmed whether these boundaries constitute strict dynamic phase transitions.

— Recent progress on thermal transport in one-dimensional long-range interacting Fermi-Pasta-Ulam-Tsingou lattice systems  (2609.18765 - Xiong et al., 16 Sep 2026) in Section 3.3, “Non-monotonic Heat Transport in the Strong Long-Range Region (0 < σ < 1)” and Section 4, Conclusion and Outlook, open question 1