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We study a heavy piston of mass $M$ that moves in one dimension. The pistonseparates two gas chambers, each of which contains finitely many ideal, unitmass gas particles moving in $d$ dimensions, where $ d\geq 1$. Using averagingtechniques, we prove that the actual motions of the piston converge inprobability to the predicted averaged behavior on the time scale $M^ {1/2} $when $M$ tends to infinity while the total energy of the system is bounded andthe number of gas particles is fixed. Neishtadt and Sinai previously pointedout that an averaging theorem due to Anosov should extend to this situation.  When $ d=1$, the gas particles move in just one dimension, and we prove thatthe rate of convergence of the actual motions of the piston to its averagedbehavior is $\mathcal{O} (M^ {-1/2}) $ on the time scale $M^ {1/2} $. Theconvergence is uniform over all initial conditions in a compact set. We alsoinvestigate the piston system when the particle interactions have beensmoothed. The convergence to the averaged behavior again takes place uniformly,both over initial conditions and over the amount of smoothing.  In addition, we prove generalizations of our results to $N$ pistonsseparating $N+1$ gas chambers. We also provide a general discussion ofaveraging theory and the proofs of a number of previously known averagingresults. In particular, we include a new proof of Anosov's averaging theoremfor smooth systems that is primarily due to Dolgopyat.

Rigorous Results for the Periodic Oscillation of an Adiabatic Piston