Mean-Field Approaches

Here it is useful to discuss mean field models of non-ideal systems. First, returning to the BBGKY collision operator (Eq. 4) and making the molecular chaos approximation we have

$$\Omega=-\frac{\partial P_1(\vec{r}_1,\vec{k}_1,t)}{m\partial ... ..._1,\vec{r}_2,t) \frac{\partial V(r_{12})}{\partial \vec{r}_1}.$$ (35)

This approximation of the collision operator is of the form of the body force term, $F \cdot \nabla_k P_1$in Eq. 1. Absorbing the collision term into the body force is called the Vlasov approximation. For $\vert\vec{r}_1-\vec{r}_2\vert > d$ where d is of order a few "effective" hard sphere diameters, $g(\vec{r}_1,\vec{r}_2) \approx 1$. After expanding $\rho$ about r₁, the contribution to the collision operator associated with the attractive intermolecular interaction, $\Omega_m$ can be approximated by, $\Omega_m=\nabla V_m \cdot \nabla_k P_1$, where $V_m=2a\rho (\vec{r}_1)+\kappa \nabla^2 \rho (\vec{r}_1)$ with $a=\frac{1}{2}\int_d d^3rV(r)$ and $\kappa=\frac{1}{6}\int_d d^3rr^2V(r)$. V_m can be thought of as a mean field potential produced by neighboring particles and $-\nabla V_m$ is the associated mean-field force. Note, this approximation of V_m ignores corrections due to spatial gradients in $g(\vec{r}_1,\vec{r}_2)$, which implicitly depends on r through the interaction potential and temperature field. The contribution of this attractive intermolecular interaction to the pressure tensor can be written as

$$\begin{eqnarray*}\stackrel {\leftrightarrow}{B}_m=[-a\rho^2+\kappa (\frac{1}{2}\... ...^2 + \rho \nabla^2 \rho)]{\bf I}-\kappa\nabla \rho \nabla \rho. \end{eqnarray*}$$

This form of pressure tensor has been explored [24] for the case of isothermal systems. A second contribution to the forcing, $\Omega_r$, is due to the repulsion of molecules and corresponds to evaluation of the collision operator for For $\vert\vec{r}_1-\vec{r}_2\vert < d$.

$$\Omega_r=-\frac{\partial P_1(\vec{r}_1)}{\partial \vec{k}_1} ... ..._1,\vec{r}_2,t) \frac{\partial V(r_{12})}{\partial \vec{r}_1}.$$ (36)

Again, expanding $\rho(\vec{r}_2)$ about $\vec{r}_1$, it is easy to see that $\rho^2$ corrections to the pressure tensor will be obtained. A similar result is obtained in the Enskog hard sphere model [15]. Addition corrections will result from consideration of gradient expansions of $g(\vec{r}_1,\vec{r}_2,t)\frac{\partial V(r_{12})}{\partial \vec{r}_1}$. Finally, long and short range contributions from the potential energy may be determined in a like fashion.