Boltzmann Collision Operator

Let $\sigma'({\bf v}_s, {\bf v}_{s'}; {\bf v}_s',{\bf v}_{s'}')\,d^3{\bf v}_s'\,d^{3}{\bf v}_{s'}'$ be the number of species-$s$ particles per unit time, per unit flux of species-$s$ particles incident with velocity ${\bf v}_s$ on a species-$s'$ particle of velocity ${\bf v}_{s'}$, that are scattered such that the species-$s$ particles emerge in the velocity range ${\bf v}_s'$ to ${\bf v}_s'+d{\bf v}_{s}'$ and the species-$s'$ particle emerges in the velocity range ${\bf v}_{s'}'$ to ${\bf v}_{s'}'+d{\bf v}_{s'}'$ (Reif 1965). Assuming that the scattering process is reversible in time and space (which is certainly the case for two-body Coulomb collisions), the corresponding quantity for the inverse process must be equal to that for the forward process (Reif 1965). In other words,

$$\sigma' ({\bf v}_s',{\bf v}_{s'}';{\bf v}_s,{\bf v}_{s'})\,d^3{\b... ...}_s,{\bf v}_{s'}; {\bf v}_s',{\bf v}_{s'}')\,d^3{\bf v}_s'\,d^{3}{\bf v}_{s'}'.$$ (3.16)

However, it is easily demonstrated from Equations (3.12) and (3.13) that

$$d^3{\bf v}_s\,d^3{\bf v}_{s'} = d^3{\bf U}_{ss'}\,d^3{\bf u}_{ss'} =d^3 {\bf U}_{ss'}\,d^3{\bf u}_{ss'}'= d^3{\bf v}_s'\,d^3{\bf v}_{s'}'.$$ (3.17)

The result $d^3{\bf u}_{ss'}=d^3{\bf u}_{ss'}'$ follows from the fact that the vectors ${\bf u}_{ss'}$ and ${\bf u}_{ss'}'$ differ only in direction. Hence, we deduce that

$$\sigma' ({\bf v}_s',{\bf v}_{s'}';{\bf v}_s,{\bf v}_{s'})=\sigma' ({\bf v}_s,{\bf v}_{s'}; {\bf v}_s',{\bf v}_{s'}').$$ (3.18)

The rate of decrease in the number of species-$s$ particles located between ${\bf r}$ and ${\bf r}+d{\bf r}$, and having velocities in the range ${\bf v}_s$ to ${\bf v}_s+d{\bf v}_s$, due to scattering of species-$s$ particles by species-$s'$ particles is obtained by multiplying $\sigma'({\bf v}_s, {\bf v}_{s'}; {\bf v}_s',{\bf v}_{s'}')\,d^3{\bf v}_s'\,d^{3}{\bf v}_{s'}'$ by the the relative flux, $u_{ss'}\,f_s({\bf r}, {\bf v}_s,t)\,d^3{\bf v}_s$, of species-$s$ particles incident on a species-$s'$ particle, then multiplying by the number of species-$s'$ particles, $f_{s'}({\bf r},{\bf v}_{s'},t)\,d^3{\bf r}\,d^3{\bf v}_{s'}$, that can do the scattering, and, finally, summing over all possible species-$s$ initial velocities, and all possible species-$s$ and species-$s'$ final velocities. In other words,

$$-\left[\frac{\partial f_s({\bf r},{\bf v}_s,t)}{\partial t}\right]_{ss'} d^3{\bf r}\,d^3{\bf v}_s = \int_{{\bf v}_{s'}}\int_{{\bf v}_s'}\int_{{\bf v}_{s'}'} [u_{ss... ...^3{\bf v}_s]\, [f_{s'}({\bf r}, {\bf v}_{s'},t)\,d^3{\bf r}\,d^3{\bf v}_{s'}]\,$$

$$\times[\sigma' ({\bf v}_s, {\bf v}_{s'};{\bf v}_s',{\bf v}_{s'}')\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}'].$$ (3.19)

Here, $u_{ss'}= \vert{\bf v}_s-{\bf v}_{s'}\vert$. Moreover, $f_s({\bf r}, {\bf v}_s,t)$ and $f_{s'}({\bf r}, {\bf v}_{s'},t)$ are the ensemble-averaged distribution functions for species-$s$ and species-$s'$ particles, respectively.

In writing the previous expression, we have assumed that the distribution functions $f_s$ and $f_{s'}$ are uncorrelated. This assumption is reasonable provided that the mean-free-path is much longer than the effective range of the inter-particle force. (This follows because, before they encounter one another, two colliding particles originate at different points that are typically separated by a mean-free-path. However, the typical correlation length is of similar magnitude to the range of the inter-particle force.) In writing the previous expression, we have also implicitly assumed that the inter-particle force responsible for the collisions is sufficiently short-range that the particle position vectors do not change appreciably (on a macroscopic lengthscale) during a collision. (Both of the previous assumptions are valid in a conventional weakly coupled plasma because the range of the inter-particle force is of order the Debye length, which is assumed to be much smaller than any macroscopic lengthscale. Moreover, the mean-free-path is much longer than the Debye length—see Section 1.7.)

By analogy with Equation (3.19), the rate of increase in the number of species-$s$ particles located between ${\bf r}$ and ${\bf r}+d{\bf r}$, and having velocities in the range ${\bf v}_s$ to ${\bf v}_s+d{\bf v}_s$, due to the recoil of species-$s$ particles that scatter species-$s'$ particles is

$$\left[\frac{\partial f_s({\bf r},{\bf v}_s,t)}{\partial t}\right]_{s's} d^3{\bf r}\,d^3{\bf v}_s = \int_{{\bf v}_{s'}}\int_{{\bf v}_s'}\int_{{\bf v}_{s'}'} [u_{ss... ...\bf v}_s']\, [f_{s'}({\bf r}, {\bf v}_{s'}',t)\,d^3{\bf r}\,d^3{\bf v}_{s'}']\,$$

$$\times[\sigma' ({\bf v}_s', {\bf v}_{s'}';{\bf v}_s,{\bf v}_{s'})\,d^3{\bf v}_s\,d^3{\bf v}_{s'}].$$ (3.20)

where ${\bf u}_{ss'}'={\bf v}_s'-{\bf v}_{s'}'$. Making use of Equations (3.16) and (3.17), as well as $u_{ss'}'=u_{ss'}$, we obtain

$$\left[\frac{\partial f_s({\bf r},{\bf v}_s,t)}{\partial t}\right]_{s's} d^3{\bf r}\,d^3{\bf v}_s = \int_{{\bf v}_{s'}}\int_{{\bf v}_s'}\int_{{\bf v}_{s'}'} [u_{ss... ...3{\bf v}_s]\, [f_{s'}({\bf r}, {\bf v}_{s'}',t)\,d^3{\bf r}\,d^3{\bf v}_{s'}]\,$$

$$\times[\sigma' ({\bf v}_s, {\bf v}_{s'};{\bf v}_s',{\bf v}_{s'}')\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}'].$$ (3.21)

The net rate of change of the distribution function of species-$s$ particles with velocities ${\bf v}_s$ (at position ${\bf r}$ and time $t$) due to collisions with species-$s'$ particles [i.e., the collision operator—see Equation (3.9)] is given by

$$C_{ss'}(f_s,f_{s'}) =\left[\frac{\partial f_s({\bf r},{\bf v}_s,t... ...ss'} + \left[\frac{\partial f_s({\bf r},{\bf v}_s,t)}{\partial t}\right]_{s's}.$$ (3.22)

Hence,

$$C_{ss'}(f_s,f_{s'}) = \int\!\!\int\!\!\int u_{ss'}\,\sigma' ({\bf... ...(f_s'\,f_{s'}'-f_s\,f_{s'})\, d^3{\bf v}_{s'}\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}'.$$ (3.23)

Here, $f_s$, $f_{s'}$, $f_s'$, and $f_{s'}'$ are short-hand for $f_s({\bf r}, {\bf v}_s,t)$, $f_{s'}({\bf r}, {\bf v}_{s'},t)$, $f_s({\bf r}, {\bf v}_s',t)$, and $f_{s'}({\bf r}, {\bf v}_{s'}',t)$, respectively. The previous expression is known as the Boltzmann collision operator (Boltzmann 1995). By an analogous argument, the net rate of change of the distribution function of species-$s'$ particles with velocities ${\bf v}_{s'}$ (at position ${\bf r}$ and time $t$) due to collisions with species-$s$ particles is given by

$$C_{s's}(f_s,f_{s'}) = \int\!\!\int\!\!\int u_{ss'}\,\sigma' ({\bf... ...)\,(f_s'\,f_{s'}'-f_s\,f_{s'})\, d^3{\bf v}_s\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}'.$$ (3.24)

Expression (3.23) for the Boltzmann collision operator can be further simplified for elastic collisions because, in this case, the collision cross-section $\sigma'({\bf v}_s, {\bf v}_{s'};{\bf v}_s',{\bf v}_{s'}')$ is a function only of the magnitude of the relative velocity vector, ${\bf u}_{ss'}$, and its change in direction as a result of the collision. Furthermore, the integral over the final velocities ${\bf v}_s'$ and ${\bf v}_{s'}'$ reduces to an integral over all solid angles for the change in direction of ${\bf u}_{ss'}$. Thus, we can write

$$\sigma' ({\bf v}_s, {\bf v}_{s'};{\bf v}_s',{\bf v}_{s'}')\,d^3{\... ...{\bf v}_{s'}'= \frac{d\sigma(u_{ss'},\chi,\phi)}{d{\mit\Omega}}\,d{\mit\Omega},$$ (3.25)

where ${\mit\Omega}=\sin\chi\,d\chi\,d\phi$. Here, $\chi$ is the angle through which the direction of ${\bf u}_{ss'}$ is deflected as a consequence of the collision (see Figure 3.1), and $\phi$ is an azimuthal angle that determines the orientation of the plane in which the vector ${\bf u}_{ss'}$ is confined during the collision. (See Section 3.6.) Moreover, $d\sigma/d{\mit\Omega}$ is a conventional differential scattering cross-section (Reif 1965). Hence, we obtain

$$C_{ss'}(f_s,f_{s'}) = \int\!\!\int\!\!\int u_{ss'}\,\frac{d\sigma... ...}{d{\mit\Omega}}\,(f_s'\,f_{s'}'-f_s\,f_{s'})\, d^3{\bf v}_{s'}\,d{\mit\Omega}.$$ (3.26)

Note, finally, that if we exchange the identities of species-$s$ and species-$s'$ particles in Equation (3.25) then ${\bf u}_{ss'}\rightarrow-{\bf u}_{ss'}$, but $u_{ss'}\rightarrow u_{ss'}$, $\chi\rightarrow\chi$, and $\phi\rightarrow\phi$. Thus, we conclude that

$$\sigma'({\bf v}_{s'}, {\bf v}_s; {\bf v}_{s'}', {\bf v}_s')= \sigma'({\bf v}_s, {\bf v}_{s'}; {\bf v}_s', {\bf v}_{s'}').$$ (3.27)