Coexisting Ballistic and Diffusive Heat Transport in Micrometer-Long Molecular Junctions

Boltzmann transport theory, the standard framework for predicting thermal conductivity, assumes that every vibrational mode eventually scatters, acquiring a finite lifetime that yields a convergent, length-independent thermal conductivity: Fourier's law. Here we show that this assumption fails in a real molecular system. Through atomistic simulations of Au-alkane-Au single-molecule junctions spanning five orders of magnitude in length (0.5 nm to 4 $\mu$m), we find that thermal conductivity never converges. Transport is ballistic for up to one hundred nanometers at room temperature, extending nearly two orders of magnitude beyond existing single-molecule measurements. Past this window, conductivity diverges as $L^{1/3}$, the scaling predicted by the Kardar-Parisi-Zhang universality class for momentum-conserving systems. Frequency-resolved decomposition of the heat current reveals the mechanism behind the divergence. Low-frequency acoustic modes never thermalize: protected by momentum conservation, they remain ballistic at every chain length, still carrying 50% of the total heat current at $L = 2 \mu$m. All other modes thermalize collectively as discrete vibrational states merge into scattering-active phonon bands with increasing length. Hence, the diverging conductivity emerges from the boundary between these coexisting transport regimes: as $L$ grows, the onset of scattering shifts progressively toward lower frequencies, suppressing the ballistic channel at a rate that sustains the $L^{1/3}$ divergence, leaving a finite contribution at every length. This coexistence of permanent ballistic and well-behaved diffusive transport, anticipated in abstract one-dimensional lattice models, survives the structural and chemical complexity of real micrometer-sized junctions.