Moments of Collision Operator

The most important moments of the collision operator (see Section 4.3) are the friction force density,

$${\bf F}_{ss'}({\bf r},t)=\int m_s\,{\bf v}_s\,C_{ss'}\,d^3{\bf v}_s,$$ (3.163)

acting on species-$s$ particles due to collisions with species-$s'$ particles, the friction force density,

$${\bf F}_{ss'}({\bf r},t)=\int m_{s'}\,{\bf v}_{s'}\,C_{s's}\,d^3{\bf v}_{s'},$$ (3.164)

acting on species-$s'$ particles due to collisions with species-$s$ particles, the collisional heating rate density,

$$W_{ss'}({\bf r},t) = \int\frac{1}{2}\,m_s\,v_s^{2}\,C_{ss'}\,d^3{\bf v}_s,$$ (3.165)

experienced by species-$s$ particles due to collisions with species-$s'$ particles, and the collisional heating rate density,

$$W_{ss'}({\bf r},t) = \int\frac{1}{2}\,m_{s'}\,v_{s'}^{2}\,C_{ss'}\,d^3{\bf v}_{s'},$$ (3.166)

experienced by species-$s'$ particles due to collisions with species-$s$ particles. However, as is clear from Equations (3.35) and (3.37),

$${\bf F}_{s's} = - {\bf F}_{ss'},$$ (3.167)

and

$$W_{s's} = - W_{ss'}.$$ (3.168)

Hence, we only need to determine ${\bf F}_{ss'}$ and $W_{ss'}$. Let us calculate these quantities using the Maxwellian collision operator (3.163).

Equations (3.163) and (3.164) imply that

$$F_{ss'\,\alpha} =-\frac{\gamma_{ss'}\,n_s\,n_{s'}}{\pi^{3/2}\,m_{s'}\,v_{t\,s}^{3... ...lpha}{v^{3}}\left(1+\frac{2\,v_\beta\,V_{s\,\beta}}{v_{t\,s}^{2}}\right)\right.$$

$$\phantom{=}\left.-\frac{T_{s'}}{T_s}\,\frac{V_{s\,\beta}}{v^{3}}\... ...pha\beta}-(F_2+2\,F_1)\frac{v_\alpha\,v_\beta}{v^{2}}\right]\right\}d^3{\bf v},$$ (3.169)

where we have integrated by parts. However, it follows from symmetry that

$$\int H(v)\,v_\alpha\,d^3{\bf v} =0,$$ (3.170)

$$\int H(v)\,v_\alpha\,v_\beta\,d^3{\bf v} = \frac{\delta_{\alpha\beta}}{3}\int H(v)\,v^{2}\,d^3{\bf v},$$ (3.171)

$$\int H(v)\,v_\alpha\,v_\beta\,v_\gamma\,d^3{\bf v} =0,$$ (3.172)

where $H(v)$ is a general function. Hence, we obtain

$$F_{ss'\,\alpha} =-\frac{2\,\gamma_{ss'}\,n_s\,n_{s'}\,V_{s\,\alph... ...{v\,v_{t\,s}^{2}}-\frac{T_{s'}}{T_s}\,\frac{F_{2}-F_1}{v^{3}}\right]d^3{\bf v},$$ (3.173)

which yields

$$F_{ss'\,\alpha} =-\frac{16\,\gamma_{ss'}\,n_s\,n_{s'}\,V_{s\,\alp... ...ght)\zeta\,F_1(\zeta)+\frac{T_{s'}}{T_s}\,\zeta\,{\rm erf}(\zeta)\right]d\zeta,$$ (3.174)

where $\zeta=v/v_{t\,{s'}}$ and $\xi = v_{t\,{s'}}^{2}/v_{t\,s}^{2}$. It is easily demonstrated that (Abramowitz and Stegun 1965)

$$\int_0^\infty {\rm e}^{-\xi\,\zeta^{2}}\,\zeta\,F_1(\zeta)\,d\zeta = \frac{1}{2\,\xi\,(1+\xi)^{3/2}},$$ (3.175)

$$\int_0^\infty {\rm e}^{-\xi\,\zeta^{2}}\,\zeta\,{\rm erf}(\zeta)\,d\zeta = \frac{1}{2\,\xi\,(1+\xi)^{1/2}}.$$ (3.176)

Thus, we get

$${\bf F}_{ss'} = - \frac{8\,\gamma_{ss'}\,n_s\,n_{s'}}{3\pi^{1/2}\,\mu_{ss'}\,(v_{t\,s}^2+v_{t\,{s'}}^2)^{3/2}}\,{\bf V}_s.$$ (3.177)

Suppose that we transform to a new frame of reference that moves with velocity ${\bf U}$ with respect to our original frame. In the new reference frame,

$${\bf F}_{ss'}' = \int m_a\,({\bf v}_s-{\bf U})\,C_{ss'}\,d^3{\bf v}_s= {\bf F}_{ss'}- {\bf U}\int\,m_s\,C_{ss'}\,d^3{\bf v}_s = {\bf F}_{ss'},$$ (3.178)

where use has been made of the collisional conservation law (3.31). It follows that ${\bf F}_{ss'}$ is invariant under Galilean transformations. This implies that the quantity ${\bf V}_s$, appearing in Equation (3.178), must be reinterpreted as the relative mean flow velocity, ${\bf V}_{ss'}= {\bf V}_s-{\bf V}_{s'}$, between species-$s$ and species-$s'$ particles. Of course, ${\bf V}_{ss'}= {\bf V}_s$ in our adopted reference frame in which ${\bf V}_{s'}={\bf0}$. It follows that the general expression for the friction force density is

$${\bf F}_{ss'} = - \frac{8\,\gamma_{ss'}\,n_s\,n_{s'}}{3\pi^{1/2}\,\mu_{ss'}\,(v_{t\,s}^{2}+v_{t\,{s'}}^{2})^{3/2}}\,{\bf V}_{ss'}.$$ (3.179)

Note that this expression satisfies the collisional momentum conservation constraint (3.168). Furthermore, it is clear that the collisional friction force acts to reduce the relative mean flow velocity, ${\bf V}_{ss'}$, of species-$s$ and species-$s'$ particles. Note that ${\bf F}_{ss}={\bf0}$. In other words, a plasma species cannot exert a frictional force on itself.

Equations (3.163) and (3.166) imply that

$$W_{ss'} = -\frac{\gamma_{ss'}\,n_s\,n_{s'}}{\pi^{3/2}\,m_{s'}\,v_{t\,s}^{... ...lpha}{v^{3}}\left(1+\frac{2\,v_\beta\,V_{s\,\beta}}{v_{t\,s}^{2}}\right)\right.$$

$$\phantom{=}\left.-\frac{T_{s'}}{T_s}\,\frac{V_{s\,\beta}}{v^{3}}\... ...a\beta}-(F_{2}+2\,F_1)\frac{v_\alpha\,v_\beta}{v^{2}}\right]\right\}d^3{\bf v},$$ (3.180)

where we have integrated by parts. Making use of Equation (3.171)–(3.173), the previous expression reduces to

$$W_{ss'} =-\frac{2\,\gamma_{ss'}\,n_s\,n_{s'}}{\pi^{3/2}\,m_{s'}\,... ...t)\int \exp\left(-\frac{v^{2}}{v_{t\,s}^{2}}\right)\,\frac{F_1}{v}\,d^3{\bf v},$$ (3.181)

which gives

$$W_{ss'} =-\frac{8\,\gamma_{ss'}\,n_s\,n_{s'}\,v_{t\,s'}^{2}}{\pi^... ...}{T_s}\right)\int_0^\infty{\rm e}^{-\xi\,\zeta^{2}}\,\zeta\,F_1(\zeta)\,d\zeta.$$ (3.182)

It follows from Equation (3.176) that

$$W_{ss'} = -\frac{8\,\gamma_{ss'}\,n_s\,n_{s'}}{\pi^{1/2}\,m_s\,m_{s'}\,(v_{t\,s}^{2}+v_{t\,{s'}}^{2})^{3/2}}\,(T_s-T_{s'}).$$ (3.183)

Note that this expression satisfies the collisional energy conservation constraint (3.169). Furthermore, it is clear that the collisional heating acts to reduce the temperature difference of species-$s$ and species-$s'$ particles. Note that $W_{ss}=0$. In other words, a plasma species cannot heat itself by means of collisions.