Alternate Methods of Representing Collision Term

The collision term $\Omega$ may be represented as a Hermite expansion [12,15,19] $\xi =\frac{k}{m}$:

$$\Omega = \omega(\xi) \sum_{n = 0}^{\infty} \frac{1}{n!} {\mathsf p}^{(n)}(x,\, t) {\cal H}^{(n)}(\xi),$$ (26)

where the weight function

$$\omega(\xi) = ( 1/ 2\pi )^{D/2} \exp ( - \xi^2 / 2 ) \, .$$ (27)

Here, the microscopic velocity, $\vec{\xi}$ is in units of $c_o=\sqrt{KT_o/m_o}$ and T_o and m_o are units of the temperature and mass respectively. A dimensionless temperature may be represented by $\theta=Tm_o/T_om$. Readers are referred to Refs. [7] [12] for more background and details. The generalized Hermite polynomials $\{ {\cal H}^{(n)} \}$ form a complete ortho-normal basis in $\xi$-space with respect to the weight function $\omega(\xi)$:

$$\int \!\! d\xi^3 \omega(\xi) {\cal H}^{(m)}_{i} {\cal H}^{(n)}_{j} = \delta_{m n} \delta_{i j} \, ,$$ (28)

where $i \equiv (i_1,\, i_2,\, \ldots,\, i_m)$, $j \equiv (j_1,\, j_2,\, \ldots,\, j_n)$, and $\delta_{i j} = 1$ when j is a permutation of i, and $\delta_{i j} = 0$ otherwise. Thus the coefficients ${\mathsf p}^{(n)}$ can be determined from the above orthogonal relation (Eq. 28):

$${\mathsf p}^{(n)} = \int \!\! d\xi \, {\cal H}^{(n)} \Omega \, .$$ (29)

The generalized Hermite polynomial ${\cal H}^{(n)}$ and coefficient ${\mathsf p}^{(n)}$ in the expansion are symmetric tensors of rank n. An element of these tensors is denoted by the same symbol, ${\cal H}^{(n)}$ or ${\mathsf p}^{(n)}$, with n subscripts for $n \geq 1$. And in the case of n = 0, the subscript is omitted. The product ${\mathsf p}^{(n)} {\cal H}^{(n)}$ denote contraction on all the n subscripts. The first three generalized Hermite polynomials are

$${\cal H}^{(0)} 1 \, ,$$ = (30a)

$${\cal H}_i^{(1)} \xi_i \, ,$$ = (30b)

$${\cal H}_{ij}^{(2)} \xi_i \xi_j - \delta_{ij} \, ,$$ = (30c)

The p⁽ⁿ⁾ may be directly obtained, to any order, from the moment equations where p⁰ =0, $p^1= \vec {\nabla} \cdot \stackrel { \leftrightarrow}{W}$ and so forth. When expanding P and P^eq' in Hermite polynomials the modified BGK form is found to be equivalent, to second order, to Hermite representation of the collision operator (Eq. 26) when derived from the moments associated with conserved quantities because the first two moments of P are the same as the unscaled equilibrium distribution. Because the Hermite expansion is orthonormal higher order terms will not directly affect the conserved quantities associated with the first and second moments.

Since the BGK formalism lends itself to analysis of the hydrodynamic limit another useful form of the collision operator is

$$\Omega = -\frac{1}{\lambda}(P-P^{eq})+ \omega(\xi) \sum_{n ... ...ac{1}{n!} {\mathsf \psi}^{(n)}(x,\, t) {\cal H}^{(n)}(\xi).$$ (31)

In this case, the mean velocity and temperature in the equilibrium distribution are not rescaled. The $\psi^{(n)}$ would be determined by the moment equations. A similar form has been suggested in regards to the revised Enskog kinetic theory (RET) [20]. However, in RET all the energy is kinetic and there is no non-local interaction potential.

It is easy to see the relation between the BGK form with correction terms (Eq. 31) and the modified BGK (Eq. 18) form by expanding P^eq' about the microscopic velocity $\xi$ and temperature. First write

$$P^{eq'}= \frac {n}{(2 \pi R(T+\Delta T)^{D/2}}\exp(-\frac{(\vec{\xi}+\vec{u})^2}{2R(T + \Delta T)}).$$ (32)

Here $\vec{u}= -\vec{v}'= -(\vec{v}+\Delta \vec{u})$ with $\Delta \vec{u}=-\frac{\lambda}{\rho} \nabla \cdot \stackrel {\leftrightarrow}{W},$

and

$$\Delta T= T'-T= \frac{ 2 \lambda}{3 \rho}(- \partial_t \phi -... ...c {1}{3R}(2\Delta \vec{u} \cdot \vec{v}+ {\Delta \vec {u}}^2)$$ (33)

Note, in this example, $\vec{\xi}$ is not rescaled in units of $c_o=\sqrt{KT_o/m_o}$. Next, expand P^eq' to second order in velocity and first order in temperature,

$$\begin{eqnarray*}P^{eq'} \approx \omega + \vec{u} \cdot \frac{\partial \omega}{\... ...}u_i u_j \frac{\partial^2 \omega}{\partial \xi_i \xi_j }+\cdots \end{eqnarray*}$$

where $\omega( \xi ,T) = \frac{1}{(2 \pi RT)^{3/2}}exp(-\frac{\xi^2}{2RT})$. Separating terms with $\Delta T$ and $\Delta \vec{u}$, we can then write the collision operator $\Omega$ (Eq. 18) as

$$\Omega -\frac{1}{\lambda}(P-P^{eq'}) \approx - \frac{1}{\lambda}(P- n \o... ...\xi}{RT}+ \frac{1}{2}\frac{v_iv_j}{RT}(-\delta_{ij} + \frac{\xi_i \xi_j}{RT})))$$ =

$$n\frac{\omega}{\lambda}\frac{\Delta \vec{u} \cdot \xi}{RT}+ n\fra... ...Delta u_j + \Delta u_i \Delta u_j}{RT} (-\delta_{ij} + \frac{ \xi_i \xi_j}{RT})$$ +

$$n\frac{\omega}{2\lambda}\frac{\Delta T}{T} (-\delta_{ii} + \frac{\xi_i \xi_i}{RT})$$ + (34)

Equation 34 can be written as

$$\begin{eqnarray*}\Omega \approx -\frac{1}{\lambda}(P-P^{eq}) + \Omega' \end{eqnarray*}$$

where

$$\begin{eqnarray*}P^{eq} \approx n \omega(1+ \frac{\vec{v} \cdot \xi}{RT}+ \frac{1}{2}\frac{v_iv_j}{RT}(-\delta_{ij} + \frac{\xi_i \xi_j}{RT}) \end{eqnarray*}$$

and $\Omega'$ can be identified with the Hermite expansion in Eq. 31. Indeed, this expansion still satisfies the moment Eqs. 17a-17c so that mass, momentum and energy are conserved.

Similar results were obtained by considering a Hermite expansion expressed as a function of the peculiar velocity, $(\vec{c}=\vec{\xi}-\vec{v})$, instead of the microscopic velocity $\vec{\xi}$. Here P^eq' was written as

$$\begin{eqnarray*}P^{eq'}= \frac {n}{(2 \pi R(T+\Delta T)^{D/2}}\exp(-\frac{(\vec{c}+\Delta\vec{u})^2}{2R(T + \Delta T)}). \end{eqnarray*}$$

Again the moment Eqs. 17a-17c were satisfied when expanding to order $\Delta \vec{u}^2$ and $\Delta T$.

A minimal version of $\Omega'$ that satisfies the moment Eqs. 17a-17c is

$$\begin{eqnarray*}\Omega'= - \frac{\omega \nabla \cdot \stackrel {\leftrightarrow... ...{\nabla} \cdot \vec{Z}) (-\delta_{ii} + \frac{\xi_i \xi_i}{RT}). \end{eqnarray*}$$

However to be consistent with the moments of the collision operator described in Eq. 25, the collision operator should also include the terms with $\Delta u_i v_j$ as found in the expansion of the modified BGK form.