C. Cold plus Thermal Gr¨ uneisen Models For gases, where the pressure and temperature tensors are proportional to one another, a systematic expansion of the Boltzmann equation can be, and has been, tried $^{10,11,14,17}$. Xu and Josyula 11 as well as Holian and Mareschal 14 developed solutions of generalized relaxation-time Boltzmann equations for the shockwave problem. For dense fluids only Enskog’s hard-sphere-based theory is available. More flexible empirical models need to be developed for dense-fluid shockwaves. A trial set of two-temperature evolution equations, the simplest plausible set generalizing the van der Waals model above, makes use of Gr¨ uneisen’s “cold curve” representation of the energy and pressure to define “thermal” contributions. These thermal parts include both the effects of thermal agitation (heat and temperature) and of mechanical distortion (work, through compression with viscous deformation): E = Φ cold + E thermal ; P$_{(}$$_{xx}$ and yy ) = P cold + P thermal + P viscous . e thermal = c ( K$_{x}$ + K$_{y}$ ) /N ; P thermal = γρe thermal , For the molecular dynamics simulations discussed in Section II the cold parts of the pressure and energy, as well as their time dependence, are naturally defined by imagining a perfect static triangular lattice of particles: E $^{cold}$/N = e cold = (30 /π )(1 − r ) 3 ; P $^{cold}$V/N = − ( dE $^{cold}$/dV ) = (45 /π ) r (1 − r ) 2 . ρ ˙ e cold = −∇ u : P cold . Here r is the separation of the six nearest neighbors in a cold triangular lattice, so that ρ = √ 4 / 3 /r $^{2}$. Just as in the equilibrium Gr¨ uneisen model the thermal energy and the nonviscous part of the thermal pressure are taken to be proportional to temperature: where γ is Gr¨ uneisen’s constant and ck is a heat capacity. The Krook-Boltzmann relaxation terms, with relaxation time τ , are the simplest means for guaranteeing thermal equilibrium, with the two temperatures approaching one another far from the shockfront. Because molecular dynamics simulations indicate that temperature becomes a tensor in strong shockwaves, a tentative two-temperature formulation can be based on separating