Brenner’s potential [24] (explicitly taking into account rotations and vibrations) to validate Eq. (1) and to extract the relevant parameters ( γ , α , and | F | ). Non-bonded interactions were described by a Lennard-Jones 6-12 potential with σ = 3 . 37 ˚ A and ϵ = 4 . 2038 meV [25, 26]. The equations of motion were integrated with a third-order Nordisieck predictor-corrector algorithm [27] using a time step of 0.5 fs. We considered the (10,10)@(15,15) double-walled CNT as a model for the oscillator, with lengths of 100 nm and 5 nm for the outer and inner tubes, respectively [29]. The atoms within 0.5 nm of the outer tube extremities were kept fixed during the simulations. To create the thermal gradient on the outer tube, the Berend- sen’s thermostat [28] was applied to about 1100 atoms that were within 5 nm of each fixed extremity. We considered thermal gradients of 7 and 17 K/nm that are of the same order of magnitude that can be experimentally obtained [30]. Such large thermal gradient values also provided high signal-to-noise ratios in our MD simulations. Initially, we established a constant thermal gradient on the outer tube while the inner tube was separately equilibrated at 300 K. Then, we inserted the inner tube in the middle the outer tube and subtracted inner tube center of mass velocity from the velocity of its individual atoms. Fig. 2 (a) displays the time evolution of the center of mass of the inner tube and the resulting fitting curves, obtained with the following parameters: | F | = 1 . 92 nN, γ = 1 . 48 (2 . 24) pN ps $^{2}$/ ˚ A $^{2}$, and α = 8 . 08 (6 . 26) pN nm/K for 7 (17) K/nm. Despite the simplicity of the model the fitting is quite good. Discrepancies are present and are due to other effects not included in the model such as the tube rotations and vibrations. The imposed thermal gradient causes an acceleration of the inner tube initially at rest. The acceleration is larger for larger thermal gradients. When reaching the outer tube extremity, the inner tube experiences a restoring force which causes its retraction. Due to the presence of a constant thermal gradient, the inner tube is accelerated towards the extremity of the outer tube. 

In order to simulate a temporal dependence of the driving force we used a periodic square wave-like pulse function p ( t ) (0 ≤ p ( t ) ≤ 1), which will be responsible to “turn on” (current turned on) and “turn off” (current turned off) the thermal gradient. The dT/dx behavior as a function of x is determined by the device geometry, electrical current pulse through p ( t ), and by the thermal CNT conductivity values. Our MD simulations showed that a constant dT/dx is obtained (after a transient period) for a large range of different temperatures [29]. These results also showed that thermal gradients can be a very effective way to initialize the oscillatory regime of the inner tube. 

