Universal exponent of anomalous heat conduction in one dimension
In plain words
In one-dimensional chains that conserve momentum, heat conductivity grows with chain length instead of staying fixed. Theory predicts universal growth exponents, but simulations of different chains disagree.
Precise statement
For 1D anharmonic chains conserving energy, momentum and stretch (e.g., FPUT with asymmetric or symmetric potentials), determine the exponent delta in the conductivity $\kappa(L) \sim L^\delta$. Nonlinear fluctuating hydrodynamics predicts $\delta = 1/3$ for generic asymmetric potentials (FPUT-alpha-beta: KPZ sound modes and a Levy 5/3 heat mode); for the symmetric FPUT-beta chain numerics and mode-coupling theory support $\delta = 2/5$. An answer is the exponent for each class, confirmed at sizes where finite-size drift is controlled.
What would settle it
Nonequilibrium and equilibrium simulations at sizes where effective exponents stop drifting, agreeing with the hydrodynamic prediction for each class.
Status in the literature
Unverified note
Simulated exponents range roughly from 0.25 to 0.4 depending on model and size; a 2026 review attributes much of the spread to thermostat-induced finite-size effects (Lepri, Livi, Politi, arXiv:2602.15512).