“…However, counterexamples, i.e., momentum conserving systems but yet having a size-independent thermal conductivity, have been found [10][11][12][13]. These counterexamples usually have asymmetric interparticle interactions, including the Toda-like models [12], Lennard-Jones model [10,11,13], the Fermi-Pasta-Ulam-α-β model [14], the diatomic gas model [15,16], the diatomic Toda model [16], and so on. More importantly, so far direct experimental measurements have not provided solid evidence yet to support the predicted abnormality in real low-dimensional materials of finite sizes available.In one dimensional (1D) case, the heat conductivity, denoted by κ, is related to the HCAF, denoted by C(t), by the Green-Kubo formulaHere k B is the Boltzmann constant, L and T are, respectively, the size and the temperature of the system, and C(t) ≡ J(0)J(t) , with J(t) being the total heat current at time t and · representing the equilibrium ensemble average.…”