  We consider the evolution of a system composed of $N$ non-interacting point
particles of mass $m$ in a cylindrical container divided into two regions by a
movable adiabatic wall (the adiabatic piston).
  We study the thermodynamic limit for the piston where the area $A$ of the
cross-section, the mass $M$ of the piston, and the number
  $N$ of particles go to infinity keeping $A/M$ and $N/M$ fixed. The length of
the container is a fixed parameter which can be either finite or infinite. In
this thermodynamic limit we show that the motion of the piston is deterministic
and the evolution is adiabatic. Moreover if the length of the container is
infinite, we show that the piston evolves toward a stationary state with
velocity approximately proportional to the pressure difference. If the length
of the container is finite, introducing a simplifying assumption we show that
the system evolves with either weak or strong damping toward a well-defined
state of mechanical equilibrium where the pressures are the same, but the
temperatures different. Numerical simulations are presented to illustrate
possible evolutions and to check the validity of the assumption.
