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OK -- a few more thought experiments.
Consider an infinitely long insulated cylinder. There are two insulated pistons placed far apart with some gas in between (say 1 km apart, with 1 atm of N2 @ 300 K). This makes any processes adiabatic within the tube adiabatic.
If I accelerate one piston inward (say at 9.8 m/s^2), there will be an adiabatic compression at that end (and that compression will be occurring faster and faster). The gas at that end will warm. In the quasistatic limit, the gas throughout would be the same temperature, but does a quasi-static approximation apply in this continuously changing situation? I conclude there will a definite (and continuously changing) temperature gradient -- hottest near the moving "back piston", and coolest near the stationary "front piston".
I could also pull out on the far side with the same sort of acceleration. Same questions (but with cooling rather than warming, of course). I conclude there will a definite (and continuously changing) temperature gradient -- hottest near the "back piston", and coolest near the moving "front piston".
I could also move BOTH sides with the same acceleration (maintaining a constant volume) , so that there would be a continued adiabatic compression at one end and a continued adiabatic expansion at the other. Could this be considered quasi-static, so that we can assume the gas will relax to a uniform temperature, or does the fact that the ends are continuously changing (accelerating) mean we might never reach a quasi-static situation and the compression& expansion would maintain a temperature gradient across the tube?
Of course, I could ALSO do this with a 1 km long tube mounted in a spaceship accelerating at 9.8 m/s^2. Or do it with a 1 km long tube standing on the earth. John Denker's previous analysis concludes that we would indeed achieve a uniform temperature in any of these cases.
FOLLOW-UP # 1: How would the analysis change if either or both of the pistons was a thermal reservoir held at the original temperature (eg 300 K), rather than an insulated piston?
FOLLOW-UP #2: For the one piston moving outward, at first the motion is slow and we could treat this as a typical adiabatic expansion, which cools the gas. But by the time the piston is moving very fast, there will be essentially no molecules hitting the piston, and we have approximately an adiabatic free expansion, which would NOT cool the gas. Is the amount of cooling a function of the speed that the piston is expanding? Presumably it must be. If could be interesting to see what that function is.
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