Rosenbluth Potentials

It is convenient to define

$$G_2({\bf v}_1) = \int u\,f_2\,d^3{\bf v}_2,$$ (3.125)

$$H_2({\bf v}_1) = \int u^{\,-1}\,f_2\,d^3{\bf v}_2.$$ (3.126)

Now, from Equation (3.106),

$$w_{ij} = \frac{\delta_{ij}}{u} - \frac{u_i\,u_j}{u^3}.$$ (3.127)

Moreover,

$$\frac{\partial u}{\partial u_i} = \frac{u_i}{u},$$ (3.128)

$$\frac{\partial u_i}{\partial u_j} = \delta_{ij}.$$ (3.129)

Hence, it is easily demonstrated that

$$w_{ij} = \frac{\partial^2 u}{\partial u_i\,\partial u_j},$$ (3.130)

$$\frac{\partial w_{ij}}{\partial u_j} = \frac{\partial w_{jj}}{\partial u_i} = 2\,\frac{\partial}{\partial u_i}\left(\frac{1}{u}\right).$$ (3.131)

According to Equations (3.115) and (3.116),

$${\bf B}_{12} =\frac{\gamma_{12}}{m_2}\int {\bf w}\cdot\frac{\partial f_2}{\par... ...2}}{m_2}\int \frac{\partial}{\partial {\bf u}}\cdot {\bf w}\,f_2\,d^3{\bf v}_2,$$ (3.132)

$${\bf D}_{12} =\frac{\gamma_{12}}{m_1}\int {\bf w}\,f_2\,d^3{\bf v}_2,$$ (3.133)

where we have integrated the first equation by parts, making use of Equation (3.110). Thus, we deduce from Equations (3.130) and (3.131) that

$${\bf B}_{12} =\frac{2\,\gamma_{12}}{m_2}\,\frac{\partial H_2}{\partial {\bf v}_1},$$ (3.134)

$${\bf D}_{12} = \frac{\gamma_{12}}{m_1}\,\frac{\partial^2 G_2}{\partial {\bf v}_1\partial{\bf v}_1}.$$ (3.135)

The quantities $H_2({\bf v})$ and $G_2({\bf v})$ are known as Rosenbluth potentials (Rosenbluth, MacDonald, and Judd 1957), and can easily be seen to satisfy

$$\nabla^2_v H_2 = -4\pi\,f_2({\bf v}),$$ (3.136)

$$\nabla^2_v G_2 = 2\,H_2({\bf v}),$$ (3.137)

where $\nabla^2_v$ denotes a velocity-space Laplacian operator. The former result follows because $\nabla^2_v(1/v)=-4\pi\,\delta ({\bf v})$ , and the latter because $\nabla^2_v(v)=2/v$ . In particular, if $f_2({\bf v})$ is isotropic in velocity space then we obtain

$$\frac{d}{dv}\left(v^2\,\frac{dH_2}{dv}\right) =-4\pi\,v^2\,f_2(v),$$ (3.138)

$$\frac{d}{dv}\left(v^2\,\frac{dG_2}{dv}\right) =2\,v^2\,H_2(v).$$ (3.139)

Suppose that $f_2({\bf v})$ is a Maxwellian distribution of characteristic number density $n_2$ , mean flow velocity zero, and temperature $T_2$ . In other words,

$$f_2({\bf v})=n_2\left(\frac{m_2}{2\pi\,T_2}\right)^{3/2}\exp\left(-\frac{m_2\,v^2}{2\,T_2}\right).$$ (3.140)

In this case, Equation (3.138) reduces to

$$\frac{d^2}{d\zeta^{\,2}}\!\left(\zeta\,H_2\right) = - \frac{4}{\s... ...2}{\sqrt{\pi}}\,\frac{n_2}{v_{t\,2}}\,\frac{d}{d\zeta}\,{\rm e}^{-\zeta^{\,2}},$$ (3.141)

where $\zeta=v/v_{t\,2}$ , and $v_{t\,2}=\sqrt{2\,T_2/m_2}$ . Hence, requiring $H_2(\zeta)$ to be finite at $\zeta=0$ , we can integrate the previous expression to give

$$H_2(\zeta) = \frac{n_2}{v_{t\,2}}\,\frac{{\rm erf}(\zeta)}{\zeta},$$ (3.142)

where

$${\rm erf}(\zeta) = \frac{2}{\sqrt{\pi}}\int_0^\zeta {\rm e}^{-t^2}\,dt$$ (3.143)

is a so-called error function (Abramowitz and Stegun 1965b). This function is such that

$${\rm erf}(\zeta)= \frac{2}{\sqrt{\pi}}\left[\zeta- \frac{\zeta^{\,3}}{3}+{\cal O}\left(\zeta^{\,5}\right)\right]$$ (3.144)

when $0< \zeta\ll 1$ , and

$${\rm erf}(\zeta)= 1- \frac{{\rm e}^{-\zeta^{\,2}}}{\sqrt{\pi}\,\zeta}\left[1+{\cal O}\left(\frac{1}{\zeta^{\,2}}\right)\right]$$ (3.145)

when $\zeta\gg 1$ . Equation (3.139) yields

$$\frac{d^2}{d\zeta^{\,2}}\!\left(\zeta\,G_2\right) = 2\,n_2\,v_{t\,2}\,{\rm erf}(\zeta),$$ (3.146)

which can be integrated, subject to the constraint that $G_2$ be finite at $\zeta=0$ , to give

$$G_2(\zeta) = \frac{n_2\,v_{t\,2}}{2\,\zeta}\left[\zeta\,\frac{d\,{\rm erf}}{d\zeta}+\left(1+2\,\zeta^{\,2}\right){\rm erf}(\zeta)\right].$$ (3.147)

According to Equations (3.128), (3.129), (3.142), and (3.147),

$$\frac{\partial H_2}{\partial{\bf v}} = - n_2\,F_1(\zeta)\,\frac{{\bf v}}{v^3},$$ (3.148)

$$\frac{\partial^2 G_2}{\partial{\bf v}\,\partial{\bf v}} = \frac{n_2\,v_{t\,2}^{\,2}}{2\,v^3}\left[-F_2(\zeta)\,{\bf I} + 3\,F_3(\zeta)\,\frac{{\bf v}{\bf v}}{v^2}\right],$$ (3.149)

where

$$F_1(\zeta) = {\rm erf}(\zeta)-\zeta\,\frac{d\,{\rm erf}}{d\zeta},$$ (3.150)

$$F_2(\zeta) = \left(1-2\,\zeta^{\,2}\right){\rm erf}(\zeta)-\zeta\,\frac{d\,{\rm erf}}{d\zeta},$$ (3.151)

$$F_3(\zeta) = \left(1-\frac{2}{3}\,\zeta^{\,2}\right){\rm erf}(\zeta) - \zeta\,\frac{d\,{\rm erf}}{d\zeta}.$$ (3.152)

Finally, it follows from Equations (3.114), (3.134), (3.135), (3.148), and (3.149) that

$${\bf A}_{12}({\bf v}) =-\frac{\gamma_{12}\,n_2}{m_2}\left\{2\,F_1(\zeta)\,\frac{\bf v}{... ...c{{\bf v}{\bf v}}{v^2}\right]\cdot\frac{\partial f_1}{\partial{\bf v}}\right\}.$$ (3.153)

Suppose that $f_1({\bf v})$ is a Maxwellian distribution of characteristic number density $n_1$ , mean flow velocity zero, and temperature $T_1$ . In other words,

$$f_1({\bf v})=n_1\left(\frac{m_1}{2\pi\,T_1}\right)^{3/2}\exp\left(-\frac{m_1\,v^2}{2\,T_1}\right).$$ (3.154)

It follows that

$$\frac{\partial f_1}{\partial{\bf v}}=-\frac{m_1}{T_1}\,{\bf v}\,f_1.$$ (3.155)

Hence, Equations (3.113) and (3.153) yield

$$C_{12}({\bf v}) = -\frac{1}{m_1}\,\frac{\partial}{\partial{\bf v}}\cdot\left({\bf R}_{12}\,f_1\right),$$ (3.156)

where

$${\bf R}_{12}= \frac{2\,\gamma_{12}\,n_2}{m_2\,v_{t\,2}^{\,3}}\left(\frac{T_2-T_1}{T_1}\right)\frac{F_1(\zeta)}{\zeta^{\,3}}\,{\bf v},$$ (3.157)

and use has been made of the fact that $3\,F_3-F_2=2\,F_1$ . The ensemble-averaged kinetic equation, Equation (3.9), can thus be written in the form

$$\frac{\partial f_1}{\partial t} + \frac{\partial}{\partial {\bf r... ...artial}{\partial{\bf v}}\cdot\left[({\bf F}_1+{\bf R}_{12})\,f_1\right]={\bf0},$$ (3.158)

where

$${\bf F}_1 =e_1\,({\bf E}+{\bf v}\times {\bf B})$$ (3.159)

is the ensemble-averaged Lorentz force. In deriving Equation (3.158), we have made use of the easily proved result

$$\frac{\partial}{\partial {\bf v}}\cdot {\bf F}_1 = {\bf0}.$$ (3.160)

According to Equation (3.158), collisions with particles of type $2$ give rise to a velocity dependent effective force, ${\bf R}_{12}$ , acting on individual particles of type $1$ . As expected, this force is zero if the temperatures of the two species are equal. On the other hand, if particles of type $2$ have a higher kinetic temperature than particles of type $1$ (i.e., if $T_2>T_1$ ) then the collisional force acts to speed up the latter particles--in other words, the force always acts in the same direction as the particle's instantaneous velocity. [This follows because $F_1(\zeta)\geq 0$ .] Conversely, if particles of type $2$ have a lower kinetic temperature than particles of type $1$ , then the collisional force acts to slow down the latter particles--in other words, the force always acts in the opposite direction to the particle's instantaneous velocity. In both cases, the collisional force is clearly acting to equalize the kinetic temperatures.

Suppose that $f_1({\bf v})$ is a Maxwellian distribution of characteristic number density $n_1$ , mean flow velocity ${\bf V}$ , and temperature $T_2$ . In other words,

$$f_1({\bf v})=n_1\left(\frac{m_1}{2\pi\,T_2}\right)^{3/2}\exp\left[-\frac{m_1\,({\bf v}-{\bf V})^2}{2\,T_2}\right].$$ (3.161)

It follows that

$$\frac{\partial f_1}{\partial{\bf v}}=-\frac{m_1}{T_2}\,({\bf v}-{\bf V})\,f_1.$$ (3.162)

Hence, Equation (3.153) yields

$${\bf A}_{12} =- \frac{\gamma_{12}\,n_2}{m_2}\left[-\frac{F_2(\zet... ...,{\bf V} + \frac{3\,F_3(\zeta)}{v^5}\,({\bf v}\cdot{\bf V})\,{\bf v}\right]f_1,$$ (3.163)

which implies that

$$C_{12} = -\frac{1}{m_1}\,\frac{\partial}{\partial {\bf v}}\cdot ({\bf R}_{12}\,f_1),$$ (3.164)

where

$${\bf R}_{12} =- \frac{\gamma_{12}\,n_2}{m_2}\left[-\frac{F_2(\zet... ...3}\,{\bf V} + \frac{3\,F_3(\zeta)}{v^5}\,({\bf v}\cdot{\bf V})\,{\bf v}\right].$$ (3.165)

As before, collisions with particles of type $2$ give rise to a velocity dependent effective force, ${\bf R}_{12}$ , acting on individual particles of type $1$ . In particular, if ${\bf v}$ is parallel to ${\bf V}$ , then

$${\bf R}_{12} =- \frac{2\,\gamma_{12}\,n_2}{m_2\,v_{t\,2}^{\,3}}\,\frac{F_1(\zeta)}{\zeta^{\,3}}\,{\bf V}.$$ (3.166)

We conclude that particles of type $1$ moving parallel to the mean drift velocity ${\bf V}$ (of particles of type $1$ relative to particles of type $2$ ) experience a velocity dependent force due to collisions with particles of type $2$ , which acts to reduce their speed. Of course, this has the effect of reducing the drift velocity.

It is easily demonstrated that $F_1(\zeta)\rightarrow 1$ as $\zeta\rightarrow \infty$ . Hence, Equations (3.157) and (3.166) yield $R_{12}\propto v^{\,-2}$ and $R_{12}\propto v^{\,-3}$ , respectively, in the limit $v\gg v_{t\,2}$ , implying that collisions only have a relatively weak effect on high speed particles. In fact, collisions are unable to prevent an imposed electric field from accelerating super-thermal particles (whose number is generally only a very small fraction of the total number of particles) to relativistic speeds (Rose and Clark 1961). Such particles are known as runaway particles.