Collision Operator for Maxwellian Distributions

Suppose that the species-$s$ and species-$s'$ distribution functions are Maxwellian, but are characterized by different number densities, mean flow velocities, and kinetic temperatures. Let us calculate the collision operator. Without loss of generality, we can choose to work in a frame of reference in which the species-$s'$ mean flow velocity is zero. It follows that

$$f_s({\bf v}) = \frac{n_s}{\pi^{3/2}\,v_{t\,s}^{3}}\exp\left(-\frac{\vert{\bf v} - {\bf V}_s\vert^{2}}{v_{t\,s}^{2}}\right),$$ (3.145)

$$f_{s'}({\bf v}) = \frac{n_{s'}}{\pi^{3/2}\,v_{t\,{s'}}^{3}}\exp\left(-\frac{v^2}{v_{t\,{s'}}^{2}}\right),$$ (3.146)

where $v_{t\,s}=\sqrt{2\,T_s/m_s}$ and $v_{t\,{s'}}=\sqrt{2\,T_{s'}/m_{s'}}$ are the species-$s$ and species-$s'$ thermal velocities, respectively. Moreover, $n_s$, ${\bf V}_s$, and $T_s$ are the number density, mean flow velocity, and temperature of species $s$, whereas $n_{s'}$, ${\bf0}$, and $T_{s'}$ are the corresponding quantities for species $s'$

Given that $f_{s'}({\bf v})$ is isotropic in velocity space, Equations (3.110) and (3.111) yield

$$\frac{d}{dv}\!\left(v^{2}\,\frac{dH_{s'}}{dv}\right) =-4\pi\,v^{2}\,f_{s'}(v),$$ (3.147)

$$\frac{d}{dv}\!\left(v^{2}\,\frac{dG_{s'}}{dv}\right) =2\,v^{2}\,H_{s'}(v).$$ (3.148)

Making use of Equation (3.146), we obtain

$$\frac{d^2(\zeta\,H_{s'})}{d\zeta^2} =-4\pi\,v_{t\,{s'}}^{2}\,\zeta\,f_{s'}(\zeta) = -\frac{4}{\sqrt{\... ...\sqrt{\pi}} \,\frac{n_{s'}}{v_{t\,{s'}}}\frac{d}{d\zeta}\,{\rm e}^{-\zeta^{2}},$$ (3.149)

$$\frac{d^2(\zeta\,G_{s'})}{d\zeta^{2}} =2\,v_{t\,{s'}}^{2}\,\zeta\,H_{s'}(\zeta),$$ (3.150)

where $\zeta=v/v_{t\,{s'}}$. Equation (3.149) can be integrated, subject to the boundary condition that $H_{s'}(\zeta)$ remain finite at $\zeta=0$, to give

$$H_{s'}(\zeta) = \frac{n_{s'}}{v_{t\,{s'}}}\,\frac{{\rm erf}(\zeta)}{\zeta},$$ (3.151)

where

$${\rm erf}(\zeta) = \frac{2}{\sqrt{\pi}}\int_0^\zeta {\rm e}^{-t^{2}}\,dt$$ (3.152)

is an error function (Abramowitz and Stegun 1965). Hence, Equation (3.150) yields

$$\frac{d^2}{d\zeta^{2}}\!\left(\zeta\,G_{s'}\right) = 2\,n_{s'}\,v_{t\,{s'}}\,{\rm erf}(\zeta),$$ (3.153)

which can be integrated, subject to the constraint that $G_{s'}$ be finite at $\zeta=0$, to give

$$G_{s'}(\zeta) = \frac{n_{s'}\,v_{t\,{s'}}}{2\,\zeta}\left[\zeta\,\frac{d\,{\rm erf}}{d\zeta}+\left(1+2\,\zeta^{2}\right){\rm erf}(\zeta)\right].$$ (3.154)

It follows that

$$\frac{\partial H_{s'}}{\partial v_\alpha} = - n_{s'}\,F_1(\zeta)\,\frac{v_\alpha}{v^{3}},$$ (3.155)

$$\frac{\partial^2 G_{s'}}{\partial v_\alpha\,v_\beta} = \frac{n_{s'}\,v_{t\,{s'}}^{2}}{2\,v^{3}}\left\{-F_2(\zeta)\,\de... ...pha\beta}+ [F_2(\zeta)+2\,F_1(\zeta)]\,\frac{v_\alpha\,v_\beta}{v^{2}}\right\},$$ (3.156)

where

$$F_1(\zeta) = {\rm erf}(\zeta)-\zeta\,\frac{d\,{\rm erf}}{d\zeta},$$ (3.157)

$$F_2(\zeta) = \left(1-2\,\zeta^{2}\right){\rm erf}(\zeta)-\zeta\,\frac{d\,{\rm erf}}{d\zeta}.$$ (3.158)

Thus, Equation (3.112) yields

$$C_{ss'} = \frac{\gamma_{ss'}\,n_{s'}}{m_s\,m_{s'}}\,\frac{\partial}{\part... ...{v_\alpha\,v_\beta}{v^{2}}\right\}\frac{\partial f_s}{\partial v_\beta}\right].$$ (3.159)

Now, it is clear from Equation (3.145) that

$$\frac{\partial f_s}{\partial v_\alpha} = -\frac{2\,(v_\alpha-V_{s\,\alpha})}{v_{t\,s}^{2}}\,f_s.$$ (3.160)

The previous two equations imply that

$$C_{ss'} = \frac{\gamma_{ss'}\,n_{s'}}{m_s\,m_{s'}}\,\frac{\partial}{\part... ...zeta)+2\,F_1(\zeta)]\,\frac{v_\alpha\,v_\beta}{v^{2}}\right\}\right]f_s\right).$$

Suppose that the drift velocity, ${\bf V}_s$, is much smaller than the thermal velocity, $v_{t\,s}$, of species-$s$ particles. In this case, we can expand the distribution function (3.145) such that

$$f_s({\bf v}) \simeq \frac{n_s}{\pi^{3/2}\,v_{t\,s}^{3}}\exp\left(... ...t\,s}^{2}}\right) \left(1+\frac{2\,{\bf v}\cdot{\bf V}_s}{v_{t\,s}^{2}}\right).$$ (3.161)

Neglecting terms that are second order in $V_s/v_{t\,s}$, the previous two equations lead to the following final expression for the collision operator for species with Maxwellian distribution functions:

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

$$\left.\left.\phantom{===}-\frac{T_{s'}}{T_s}\,\frac{V_{s\,\beta}}... ...(\zeta)+2\,F_1(\zeta)]\,\frac{v_\alpha\,v_\beta}{v^{2}}\right\}\right]\right\},$$ (3.162)

where $\zeta=v/v_{t\,s'}$.