 First-order collision-flux approximation for a slow piston

For a piston advancing slowly into a gas, the arrival rate of molecules differs slightly from that at a stationary wall. In calculating compression work from molecular impacts to first order in piston speed $u$, however, the stationary molecular arrival flux is sufficient: the energy change per reflected molecule is already proportional to $u$.

A molecule with incoming normal velocity $v_x$ acquires leading energy $2mu{v}_x$. Multiplying by the stationary arrival weighting proportional to $v_x$ and averaging over an isotropic distribution gives the pressure work rate $pSu$, where $S$ is piston area. The first-order correction to the arrival flux multiplies an order-$u$ energy change and contributes only at order $u^2$. This ordering requires piston speed small compared with the typical molecular speed; it does not justify the same approximation for rapid compression or for the force correction itself.