0 300 600 900 1200 Time (ps) -800 -400 0 400 800 Velocity (m/s) 100 nm 500 nm 1000 nm FIG. S5: Time evolution of the velocity of the inner tube center of mass obtained from integration of Eq. (1) for different outer tube lengths ( p ( t ) = 1). The data correspond to the following parameters: F$_{vdw}$ = 1.92 nN, α = 8.08 pN nm/K, γ = 1 . 48 pN ps $^{2}$/ ˚ A $^{2}$, thermal gradient of 7 K/nm, and inner tube length of 5 nm. motions covering all long outer tubes. For low thermal gradients (below ∼ 1 K/nm), steady velocity state is expected to be achieved for even longer outer tubes. By reducing the temperature, a decreased in the friction is expected since nanotube vibrations (one of the causes of energy dissipation) are reduced. This is indeed observed in the γ values of the fitted parameters for the cases of 7 and 17 K/nm. Thus, we expect a decreasing of the γ value for lower thermal gradients than 7 K/nm. Since we have not performed molecular dynamics simulations for low thermal gradients (below 1 K/nm), which provide the necessary parameters for Eq. (1), we can only estimate the behavior for such cases. The estimation is done by numerically integration Eq. (1) for different parameters of γ and α . The parameter F$_{vdw}$ will not change since we are not 16