Heat Conduction in Low Dimensions: From Fermi-Pasta-Ulam Chains to Single-Walled Nanotubes

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Heat conduction in 1-dimensional anharmonic systems is anomalous in the sense that the conductivity κscales with a positive power of the system size, κ Lα. In two dimensions, previous simulations and theoretical arguments gave a logarithmic divergence. For rectangular systems of size L x L⊥ there should be a cross-over from the 2-d to the 1-d behaviour as the aspect ratio r = L / L⊥ increases from r=1 to r >> 1. When taking periodic boundary conditions in the transverse direction, this should be of direct relevance for the heat conduction in single-walled carbon nanotubes. In particular, one expects that k nanotubes of diameter R should conduct heat better than a single nanotube of the same length and of radius kR. We study this cross-over numerically by simulating the Fermi-Pasta-Ulam model. Apart from giving a precise estimate of the exponent α, our most intriguing results are that the divergence does not seem to be logarithmic in d=2 but also power-like, and that the cross-over does not happen at a fixed aspect ratio. Instead, it happens at r=r* with r* -> ∞ for L -> ∞.