Moments of Collision Term

Consider moments of $\Omega$which are related to conserved quantities.

$$\int d^3k_1 C(\vec{r}_1,\vec{k}_1,t)\Omega= -\int d^3k_1d^3r_... ...vec{k}_1} \cdot \frac{\partial V(r_{12})}{\partial \vec{r}_1}$$ (5)

where C is a function of position or momentum. Here we have integrated by parts and assumed that P₂ goes to zero in the limit of large k.

For C equal to a constant the integral in Eq. 5 is equal to zero corresponding to conservation of mass. For the momentum moment ($C=\vec{k}$) we have

$$\int d^3k_1 \vec{k}_1\Omega= -\int d^3k_1d^3r_2d^3k_2 P_2(\vec{r}... ...c{k}_1}{\partial \vec{k}_1} \cdot \frac{\partial V(r_{12})}{\partial \vec{r}_1}$$

$$= -\int d^3r_2 P_2(\vec{r}_1,\vec{r}_2,t)\frac{\partial V(r_{12})}{\partial \vec{r}_1}$$ (6)

where $P_2(\vec{r}_1,\vec{r}_2,t)= \int d^3k_1d^3k_2 P_2(\vec{r}_1,\vec{r}_2,\vec{k}_1, \vec{k}_2,t)$.

The kinetic energy moment ( $C=\frac{k^2}{2m}$) is

$$\int d^3k_1 \frac{{k_1}^2}{2m}\Omega= -\int d^3k_1d^3r_2d^3k_... ...c{k}_1}{m} \cdot \frac{\partial V(r_{12})}{\partial \vec{r}_1}$$ (7)

In the regime of low to moderate densities where multiple collisions between atoms can be ignored the molecular chaos approximation is valid [10,15]. Here

$$\begin{eqnarray*}P_2(\vec{r}_1,\vec{r}_2,\vec{k}_1, \vec{k}_2)= P_1(\vec{r}_1,\vec{k}_1)P_1(\vec{r}_2, \vec{k}_2)g(\vec{r}_1,\vec{r}_2) \end{eqnarray*}$$

and $g(\vec{r}_1,\vec{r}_2)$ is the two point correlation function. Then Eqs. 6 and 7 become

$${\partial}_t \vec{P}=\int d^3k_1\vec{k}_1\Omega= -P_1(\vec{r... ...r}_1,\vec{r}_2) \frac{\partial V(r_{12})}{\partial \vec{r}_1}$$ (8)

and

$${\partial}_t E_{ke} = \int d^3k_1 \frac{{k^2}_1}{2m}\Omega= ... ...\vec{v}_1\cdot \frac{\partial V(r_{12})}{\partial \vec{r}_1}.$$ (9)

For the case of non-local interactions neither the momentum or energy are conserved locally as local regions respond to the forces from neighboring areas. Equations 8 and 9 respectively represent the change of momentum and the rate work is done on the fluid due to an effective averaged force field. Note that the moments depend on the two point correlation function. As mentioned above, this function could in principle be approximated by solving a closed set of the BBGKY hierarchy of equations. For this paper we will assume that $g(\vec{r}_1,\vec{r}_2)$ or, for that matter, $g(\vec{r}_1,\vec{r}_2,\vec{k}_1,\vec{k}_2)$ is known or can be obtained by other simulation techniques like molecular dynamics [16,17] for use in simulations.

To show that total energy is globally conserved first note that the local potential energy is given by

$$\begin{eqnarray*}\phi(r)= \frac{1}{2}\int \!\! d^3r_1 d^3r_2 P(\vec {r}_1, \vec {r}_2 ,t) \delta^{(3)}(\vec {r}_1-\vec{r}) V(r_{12}). \end{eqnarray*}$$

The total energy, E, of this system is then given by the volumetric integral of the kinetic and potential energy,

$$E= \int d^3rd^3k \frac{k^2}{2m} P_1(\vec{r},\vec{k})+ \frac{1... ..._1, \vec {r}_2 ,t) \delta^{(3)}(\vec {r}_1-\vec{r}) V(r_{12})$$ (10)

The time derivative of the local potential energy can be obtained from the BBGKY equations since

$$\begin{eqnarray*}{\partial}_t \phi(r)= \frac{1}{2}\int \!\! d^3r_1 d^3r_2 {\part... ...}_1, \vec {r}_2 ,t) \delta^{(3)}(\vec {r}_1-\vec{r}) V(r_{12}) \end{eqnarray*}$$

so that we can utilize Eq. 2. Terms with $\Theta_{ij}$_ij integrate out to zero. Integration by parts of the remaining terms obtains

It is then easy to see that

$$\begin{eqnarray*}\int d^3r {\partial}_t \phi=- \int d^3r {\partial}_t E_{ke} \end{eqnarray*}$$

thus proving global conservation of energy.

In addition, it can be shown that [9] ${\partial}_t \vec{P}$ and ${\partial}_t E_{ke}+{\partial}_t \phi$ can be written as the divergence of a function so that, in a numerical implementation, total momentum and energy are easily conserved in a closed system.

Here the momentum moment of the collision operator is written as

$$\int d^3k \vec{k}\Omega = -\nabla \cdot \frac{1}{2}\int d^3r_... ...\cdot \nabla_{r_1})^{n-1}]P(\vec{r}_1, \vec{r}_1+\vec{r}_{12})$$ (11)

and the kinetic energy moment is

$$\int d^3k \frac{k^2}{2m}\Omega= -\partial_t \phi - \nabla \cdot (... ...1, \vec{k}_2) \delta^3(\vec{r}_1-\vec{r})V(r_{12})(\frac{\vec{k}_1}{m}-\vec{v})$$

$$+ \frac{1}{2} \nabla \cdot \int d^3r_1d^3k_1d^3r_{12}d^3k_2 \delt... ...t \nabla_{r_1})^{n-1}]P(\vec{r}_1, \vec{r}_1+\vec{r}_{12},\vec{k}_1,\vec{k}_2).$$ (12)

In the limit of small variation of P₂ over a range $L >> \vec{r}_{12}$, where L is the characteristic length of density fluctuation, terms with n > 1 can be ignored. Then

$$\begin{eqnarray*}\int d^3k \vec{k}\Omega \simeq - \nabla \cdot [\frac{1}{2}\int ... ...al V(\vec{r}_{12})}{\partial r_{12}}P(\vec{r}_1, \vec{r}_{2})] \end{eqnarray*}$$

or

$$\int d^3k \vec{k}\Omega \simeq -\nabla \cdot \stackrel {\leftrightarrow}{B}$$ (13)

Similarly, for the kinetic energy moment, including the n=1 term only obtains

$$\int d^3k_1 \frac{{k_1}^2}{2m} \Omega \simeq -\partial_t \phi... ...\stackrel {\leftrightarrow}{B} \cdot {\vec v} + {\vec {q}_v})$$ (14)

where

$$\vec {q}_v(\vec x,t)=\frac{1}{2} \int \!\! d^3r_1 d^3r_2 d^3... ...}{\bf {1}}-V'(r_{12}){r_{12}}^{-1}\vec {r}_{21}\vec {r}_{21}).$$ (15)

Using the molecular chaos approximation

$$B_{ij}(\vec {r},t)=-\frac{1}{2} \int \!\! d^3r_1 d^3r_2 P_... ...vec{r}_1,\vec{r}_2)V'(r_{12}){r_{12}}^{-1}(r_{21})_i(r_{21})_j$$ (16)

and $\vec{q}_v=0$ indicating that without velocity correlations, currents associated with potential energy are zero.

Again, since only the divergence of $\stackrel {\leftrightarrow}{B}$ and are present in Eqs. 13 and 14, it is easy to numerically implement momentum and energy conservation.

For later reference the collision operator moments are summarized below [9,10].

$$\int \!\! d^3 \: k \, \Omega = 0, \!$$ (17a)

$$\int \!\! d^3k \, \vec{k} \Omega = -\nabla \cdot \stackrel {\leftrightarrow}{W},$$ (17b)

$$\int \!\! d^3k \, \ \frac{k^2}{2m} \Omega = - \partial_t \phi- {\nabla {\cdot (\phi {\vec v}) }}- {\nabla \cdot \vec{Z}} \,$$ (17c)

where $-\nabla \cdot {\stackrel {\leftrightarrow}{W}}$ is equal to the righthand side of Eq. 11 and $-\nabla \cdot {\vec{Z}}$ is equal to the last two terms on the righthand side of Eq. 12. This represents a "conservative" form of the moment equations. In the case of n = 1 only, ${\stackrel {\leftrightarrow}{W}}= {\stackrel {\leftrightarrow}{B}}$ and $\vec{Z}=\stackrel {\leftrightarrow}{B} \cdot {\vec v} + {\vec {q}_v}$.