Write an MD code for \(N\) Lennard-Jones particles in a two-dimensional \(L \times L\) square box, and include a one-body gravity term in the energy: \(\sum_{i=1}^{N} m g y_{i}\). Apply periodic boundary conditions in the \(x\)-direction but a repulsive Weeks-Chandler-Andersen (WCA) (Weeks et al., 1971) potential on the top and bottom walls. The WCA potential is the repulsive part of a LennardJones potential for \(r / D<(2)^{1 / 6}\), with the potential shifted up by \(\varepsilon\). Show that the average kinetic energy per particle is independent of the height \(y\) in the box but the average scaled density \(n D^{2}\) depends on the vertical position in the box.