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## TEMPERATURE PROFILE 

Fig. S3 presents the temperature profile of the outer tube for the system (10,10)@(15,15) for different times after a thermal gradient of 7 K/nm is imposed between the two extremities of the outer tube. As we can see an approximately linear variation of the temperature with x is attained after about 50 ps, the period P of p ( t ) should be larger than 100 ps in order to keep the CNT oscillator operating in the constant thermal gradient regime. 

<img src='content_image/369061.jpg'>

## DEPENDENCE UPON TUBE LENGTHS AND THERMAL GRADIENTS 

As mentioned in the main text the behavior of the thermophoretically driven CNT oscilla- tor depends upon both the inner tube length and the thermal gradient. In order to evaluate this dependence, we identified the cases where there is an oscillation (regular or chaotic) of the inner tube. In such cases the motion is limited within the outer tube. On the other hand, for some set of parameters, the inner tube was strongly accelerated and ejected off the outer tube. The evolution of the inner tube position was obtained by numerical integration 

