Fourier’s law breakdown for the planar-rotor chain with long-range interactions

Fourier’s law, which linearly relates heat flux to the negative gradient of temperature, is a fundamental principle in thermal physics and widely applied across materials science and engineering. However, its validity in low-dimensional systems with long-range interactions remains only partially understood. We investigate here the thermal transport along a one-dimensional chain of classical planar rotators with algebraically decaying interactions 1 / r α with distance r ( α ≥ 0 ), known as the inertial α -XY model. Using nonequilibrium simulations with thermal reservoirs at the boundaries, we numerically study the thermal conductance σ as a function of system size L , temperature T , and α . We find that the results obey a universal scaling law characterized by a stretched q -exponential function with α -dependent parameters. Notably, a threshold at α c ≈ 2 separates two regimes: for α ≥ α c , Fourier’s law holds with size-independent conductivity κ ≡ L σ , while for α < α c , anomalous transport is observed, corroborating (with higher precision) the results reported in Phys. Rev. E 94, 042117 (2016) . These findings provide a quantitative framework for understanding the breakdown of Fourier’s law in systems with long-range interactions. The simulation is carried out by assuming the equations of motion, which include Langevin heat baths applied to the first and last particles, and are integrated using the Velocity Verlet algorithm. The conductance is calculated from the connection between Lagrangian heat flux and heat equation for typical values of ( α , T , L ) . For large L , the results can be collapsed into an universal q -stretched exponential form, namely L δ α σ ∝ e q α − B α ( L γ α T ) η α , where e q z ≡ 1 + ( 1 − q ) z 1 / ( 1 − q ) . The parameters ( γ α , δ α , B α , η α ) are α -dependent, and q α is the index of the q -stretched exponential. This form is achievable due to the ratio η α / ( q α − 1 ) being almost constant with respect to the lattice size L . These findings provide significant insights into heat conduction mechanisms in systems with long-range interactions. • Thermal conductance in 1D inertial α -XY model follows universal q -stretched law. • We confirm α c ∗ = 2 : α ≥ α c ∗ normal; α < α c ∗ anomalous transport. • We extrapolate to d dimensions: predict α c ∗ = d + 1 for normal vs anomalous transport.