from $L$ to $2L$ (Fig.\[fig:piston\_Ftyp\_Rdom\]a). Since the volume of the box is doubled, the free energy difference between the equilibrium states corresponding to the piston locations $A$ and $B$ is $$\Delta F = -n_p \beta^{-1} \ln 2.$$ During a realization of this process, whenever a particle collides with the moving piston, the particle suffers a change of kinetic energy, $\delta K = -2m u ( v_x - u ) < 0$, where $v_x$ is the component of the particle’s velocity parallel to the motion of the piston, prior to the collision. The total work $W$ is the sum of such contributions.

If the process described above is the [*forward*]{} process, then the [*reverse*]{} process involves pushing the piston into the gas at speed $u$, from $B$ to $A$ (Fig.\[fig:piston\_Rtyp\_Fdom\]a), starting from an initial state of thermal equilibrium. Each particle-piston collision now produces a change $\delta K = 2m u ( v_x + u ) > 0$ in the particle’s kinetic energy. In the context of the piston and gas example, I will use the terms [*expansion*]{} and [*compression*]{} to denote the forward and reverse processes, respectively.

Now suppose the piston speed is much greater than the thermal particle speed: $$\label{eq:fastPiston}
