See the animation below. Consider that as the air in the hot end heats up, the high pressure air forces in on the sealed displacement piston, and the flexible diaphragm will be forced to collapse, bringing the piston to the hot end of the cylinder, forcing the air to the cold end. As the air cools and contracts, the pressure drops, and the lower pressure causes the diaphragm on the sealed displacement piston to distend, forcing the cold air to the hot end. The linear oscillation would of course be used to drive a linear alternator, simplified in this diagram as magnets moving outside of a fixed coil.
There is no question that this design can forcefully shove the piston one way or the other as the temperature changes, the question is of oscillation. It seems to me that the mass of the piston would have the same effect as a flywheel, and that a forceful oscillation could be achievable. The specifics of the oscillation would certainly be determined by factors that include the mass of the piston and the density and composition of the working fluid.
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If you commented on this for the last post, I appreciate hearing from you again. Thanks!