Discussion and Conclusion

To properly construct a collision operator, based on the BBGKY equations, the pair potential and two point correlation function are needed as input. Any model, whether discrete Boltzmann or CFD/free energy approaches, must at minimum make an implicit assumption about pair correlations since they are related to the equation of state, chemical potential and the transport properties of the fluid modeled.

The two point correlation function, which in turn depend on the pair potential, can be approximated by a variety of methods based on either closure of the BBGKY hierarchy or by molecular dynamics simulations. For the case of BBGKY closure, methods [14] of approximating the pair correlation function such as Percus Yevick (PY) theory and the Hypernetted chain (HNC) theory work well for many low to moderate density fluids. At the lowest order approximation the equilibrium two point correlation function coupled with the molecular chaos assumption could be utilized. To go to a higher order approximation or model dense fluids a better understanding of the nonequilibrium form of pair correlations and velocity correlations is needed. In this case, molecular dynamics may prove a valuable tool for probing the functional form of velocity correlations or other interesting behavior like the vortex motion found in hard sphere simulations [16]. In addition, experimental data from neutron scattering, which can also provide information about both pair and velocity correlations, could help guide construction of the collision operator.

The methods described in this paper should be readily adaptable to existing thermal discrete Boltzmann models [12]. Rescaling the temperature via Eq. 5 without explicit reference to the potential energy $\phi$may be subject to large numerical error. It may be preferable to use Eq. 8 and estimate ${\partial}_t \phi$ by a numerical algorithm such as predictor-corrector since $\phi$ can be determined at every time step.

While a few numerical tests of the above scheme have been performed in the regime of low density, demonstrating that energy was clearly conserved to the numerical accuracy of the computer, further tests are needed to examine the stability of such numerical algorithms especially near a critical temperature or for the case of a deep quench where density variations can be large. Indeed, thermal lattice Boltzmann methods are notoriously unstable [22]. To better model more realistic systems with greater density or density gradients other implementations of the discrete Boltzmann method such as finite difference or volumetric methods should be considered.

It would also be interesting to examine the effect of resolution on simulations. At a coarse resolution the above system of equations should describe the time evolution of a macroscopic fluid element. At finer resolution, density variations at the atomic level could be probed although computational requirements for such studies may be prohibitive.

In summary, starting with the BBGKY formalism, the moments to be satisfied by the collision operator so that energy conservation in a discrete Boltzmann equation is rigorously maintained were presented. Several methods for constructing the collision operator based on these moment conditions were described. To illustrate the method, a simple numerical simulation was described. It was also shown that the isothermal non-ideal gas model of Shan and Chen was recovered by this more general approach.