  Following Berni Alder [1] and Francis Ree [2], Douglas Henderson was the
third of Bill's California coworkers from the 1960s to die in 2020. Motivated
by Doug's death we undertook better to understand Lyapunov instability and the
breaking of time symmetry in continuum and atomistic simulations. Here we have
chosen to extend our explorations of an interesting pair of nonequilibrium
systems, the steady shockwave and the unsteady rarefaction wave. We eliminate
the need for boundary potentials by simulating the collisions of pairs of
mirror-images projectiles. The resulting shock and rarefaction structures are
respectively the results of the compression and the expansion of simple fluids.
Shockwaves resulting from compression have a steady structure while the
rarefaction fans resulting from free expansions continually broaden. We model
these processes using classical molecular dynamics and Eulerian fluid mechanics
in two dimensions. Although molecular dynamics is time-reversible the reversed
simulation of a steady shockwave compression soon results in an unsteady
rarefaction fan, violating the microscopic time symmetry of the motion
equations but in agreement with the predictions of macroscopic Navier-Stokes
fluid mechanics. The explanations for these results are an interesting
combination of two (irreversible) instabilities, Lyapunov and Navier-Stokes.
