u \gg v_{\rm th}=\sqrt{3/m\beta}.$$ In this case, the particle density profile typically changes very little during expansion (Fig.\[fig:piston\_Ftyp\_Rdom\]a): most particles remain in the left half of the box, and few if any collide with the piston, consequently $$W^F \approx 0.$$ (The superscript indicates the forward process, i.e. expansion.)

![ (a) A typical realization of the forward process. Due to the great speed of the piston (Eq. \[eq:fastPiston\]), most particles remain in the left half the box for the duration of the process. (b) The conjugate twin of the realization depicted in (a). Almost all particles begin in the left half of the box, and the piston then moves rapidly through a largely empty region. (The vertical gray arrows specify the direction of increasing time.) []{data-label="fig:piston_Ftyp_Rdom"}](Figure3.eps)

During the compression process (Fig.\[fig:piston\_Rtyp\_Fdom\]a), the piston typically collides with about half of the gas particles – roughly speaking, those initially located in right half of the box. This generates a shock wave of particles streaming leftward at approximately twice the piston speed. At time $\tau$ the front of this wave reaches the wall at $x=0$, just as the piston arrives at $A$. Thus at the end of such a realization, half the particles (those 