Not exactly. A pressure differential will spin a turbo. However, apply heat to the pressure and the pressure will increase, thereby spinning the turbo up more quickly. In-cylinder this means higher EGTs which in turn increase in-cylinder pressure. If the air inside the cylinder is not heated by combustion (no combustion happens), then on the exhaust stroke it is primarily moving by the piston pushing it out of the cylinder (I am not accounting for the heat created by the compression stroke).
Look at it this way. Take a known volume of air contained in a canister and put a fan on top of it. If the temperature inside the canister is the same as outside, the fan blades will not turn.
However, apply heat to the canister and the air inside is heated. The heated air molecules are excited (hot air balloon anyone) and thereby increase the pressure inside the cansiter. The fan blades will now start to turn because of the pressure differential.
Saying that, heat the air even more and the fan blades will turn faster since the pressure inside the can increased. The increased pressure results in a greater pressure differential across the fan.
PV=nRT
For the same gas, P1V1/T1 = P2V2/T2.
For the same volume, P1/T1=P2/T2. Rewritten, P2 = P1* T2/T1
You can clearly see that an increase in temperature, T2, results in an increase in pressure.
Keep the exhaust gases hot up to the turbine and cool after the turbine (increased pressure differential). Use a large downpipe to minimize backpressure (increase pressure differential).