For a one-component fluid on a solid substrate, a thermal singularity may occur at the contact line where the liquid-vapor interface intersects the solid surface. Physically, the liquid-vapor interface is almost isothermal at the liquid-vapor coexistence temperature in one-component fluids while the solid surface is almost isothermal for solids of high thermal conductivity. Therefore, a temperature discontinuity is formed if the two isothermal interfaces are of different temperatures and intersect at the contact line. This leads to the so-called thermal singularity. The localized hydrodynamics involving evaporation/condensation near the contact line leads to a contact angle depending on the underlying substrate temperature. This dependence has been shown to lead to the motion of liquid droplets on solid substrates with thermal gradients (Xu and Qian 2012 Phys. Rev. E 85 061603). In the present work, we carry out molecular dynamics (MD) simulations as numerical experiments to further confirm the predictions made from our previous continuum hydrodynamic modeling and simulations, which are actually semi-quantitatively accurate down to the small length scales in the problem. Using MD simulations, we investigate the motion of evaporative droplets in one-component Lennard-Jones fluids confined in nanochannels with thermal gradients. The droplet is found to migrate in the direction of decreasing temperature of solid walls, with a migration velocity linearly proportional to the temperature gradient. This agrees with the prediction of our continuum model. We then measure the effect of droplet size on the droplet motion. It is found that the droplet mobility is inversely proportional to a dimensionless coefficient associated with the total rate of dissipation due to droplet movement. Our results show that this coefficient is of order unity and increases with the droplet size for the small droplets (∼10 nm) simulated in the present work. These findings are in semi-quantitative agreement with the predictions of our continuum model. Finally, we measure the effect of liquid-vapor coexistence temperature on the droplet motion. Through a theoretical analysis on the size of the thermal singularity, it can be shown that the droplet mobility decreases with decreasing coexistence temperature. This is observed in our MD simulations.