Thermal energy is related to the speed at which atoms vibrate, therefor thermal energy is a form of motion. In a stationary object, all of these vibrating atoms 'impact' each other in such a way that the net sum of their motion is zero. if all (or a significant number) of the atoms were aligned so that their thermal energy 'pointed' in the same direction, the net sum of thermal movement would no longer be zero, and the object should jerk. Does this ever happen? If so, wouldn't cooling one side of an object cause a difference in thermal motion vectors which would cause the object to move in the direction that the cooler side was facing? Why or why not?
This is done, and is called the Thermoelectric Effect. The "object" does not jerk, but it does induce an electric current. The issue with having, say, atoms in a lattice move like this is that the atomic lattice elements are so many thousands of times more massive than the electrons that the electrons go flying while the nuclei just don't move. Think same $F$, but about $100,000 \times $ more mass.