untouched by the piston) are characterized by the original Maxwellian velocity distribution, while the other half have velocities with $v_x \approx -2u$. For such a realization, $$W^R\approx \frac{n_p}{2}\cdot 2mu^2 = n_p m u^2 .$$

![ (a) A typical realization of the reverse process. As the piston moves rapidly into the gas, the particles with which it collides gain large components of velocity ($\sim 2u$) along the direction of motion of the piston. The particles with the attached arrows are meant to represent these fast particles, while the unadorned ones are characterized by thermal velocities. (b) The conjugate twin of the realization depicted in (a). Half the particles are initially moving at great speeds ($\sim 2u$) in the direction of the piston. By the end of the process, after each of these has collided with the piston, the velocity distribution of the entire gas is thermal. []{data-label="fig:piston_Rtyp_Fdom"}](Figure4.eps)

If Fig. \[fig:piston\_Rtyp\_Fdom\]a illustrates the trajectory just described, then Fig.\[fig:piston\_Rtyp\_Fdom\]b illustrates its conjugate twin. Here the piston begins at $A$, with half the particles streaming rightward at $v_x \approx +2u$. As the piston moves from $A$ to $B$ at speed $u$, each of these very fast particles collides once with the piston, losing most 