Boltzmann Collision Operator

Let $\sigma({\bf v}_1, {\bf v}_2; {\bf v}_1',{\bf v}_2')$ be the cross-section for a scattering process by which particles of types $1$ and $2$ (located at position vector ${\bf r}$ at time $t$ ) are incident with velocities ${\bf v}_1$ and ${\bf v}_2$ , respectively, and are scattered to velocities ${\bf v}_1'$ and ${\bf v}_2'$ , respectively (Reif 1965). Assuming that the scattering process is reversible in time and space (which is certainly the case for two-body Coulomb collisions), the cross-section for the inverse process must be the same as that for the forward process (Reif 1965). In other words,

$$\sigma ({\bf v}_1',{\bf v}_2';{\bf v}_1,{\bf v}_2)=\sigma ({\bf v}_1,{\bf v}_2; {\bf v}_1',{\bf v}_2').$$ (3.16)

The rate at which particles with the original velocities ${\bf v}_1$ and ${\bf v}_2$ are scattered into the range ${\bf v}_1'$ to ${\bf v}_1'+d{\bf v}_1'$ and ${\bf v_2}'$ to ${\bf v_2}'+d{\bf v}_2'$ is

$$u\,f_1({\bf r}, {\bf v}_1,t)\,f_2({\bf r}, {\bf v}_2,t)\,\sigma ({\bf v}_1, {\bf v}_2;{\bf v}_1',{\bf v}_2')\,d^3{\bf v}_1'\,d^3{\bf v}_2'.$$ (3.17)

Here, $u= \vert{\bf v}_1-{\bf v}_2\vert$ . Moreover, $f_1({\bf r}, {\bf v}_1,t)$ and $f_2({\bf r}, {\bf v}_2,t)$ are the ensemble-averaged distribution functions for particles of types $1$ and $2$ , respectively. In writing the previous expression, we have assumed that the distribution functions $f_1$ and $f_2$ 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 (3.17), the rate at which particles with the original velocities ${\bf v}_1'$ and ${\bf v}_2'$ are scattered into the range ${\bf v}_1$ to ${\bf v}_1+d{\bf v}_1$ and ${\bf v_2}$ to ${\bf v_2}+d{\bf v}_2$ is

$$u'\,f_1({\bf r}, {\bf v}_1',t)\,f_2({\bf r}, {\bf v}_2',t)\,\sigma ({\bf v}_1', {\bf v}_2';{\bf v}_1,{\bf v}_2)\,d^3{\bf v}_1\,d^3{\bf v}_2,$$ (3.18)

where ${\bf u}'={\bf v}_1'-{\bf v}_2'$ . Now, it is easily demonstrated from Equations (3.12) and (3.13) that

$$d^3{\bf v}_1\,d^3{\bf v}_2 = d^3{\bf U}\,d^3{\bf u} =d^3 {\bf U}\,d^3{\bf u}'= d^3{\bf v}_1'\,d^3{\bf v}_2'.$$ (3.19)

The result $d^3{\bf u}=d^3{\bf u}'$ follows from the fact that the vectors ${\bf u}$ and ${\bf u}'$ differ only in direction. Thus, the net rate of change of the distribution function of particles of type $1$ with velocities ${\bf v}_1$ (at position ${\bf r}$ and time $t$ ) due to collisions with particles of type $2$ [i.e., the collision operator--see Equation (3.9)] is given by

$$\left(\frac{\partial f_1}{\partial t}\right)_2\equiv C_{12}(f_1,f... ...\bf v}_2')\,(f_1'\,f_2'-f_1\,f_2)\, d^3{\bf v}_2\,d^3{\bf v}_1'\,d^3{\bf v}_2'.$$ (3.20)

Here, use has been made of Equation (3.16), as well as $u'=u$ . Moreover, $f_1$ , $f_2$ . $f_1'$ , and $f_2'$ are short-hand for $f_1({\bf r}, {\bf v}_1,t)$ , $f_2({\bf r}, {\bf v}_2,t)$ , $f_1({\bf r}, {\bf v}_1',t)$ , and $f_2({\bf r}, {\bf v}_2',t)$ , respectively. The previous expression is known as the Boltzmann collision operator. By an analogous argument, the net rate of change of the distribution function of particles of type $2$ with velocities ${\bf v}_2$ (at position ${\bf r}$ and time $t$ ) due to collisions with particles of type $1$ is given by

$$\left(\frac{\partial f_2}{\partial t}\right)_1\equiv C_{21}(f_1,f... ...\bf v}_2')\,(f_1'\,f_2'-f_1\,f_2)\, d^3{\bf v}_1\,d^3{\bf v}_1'\,d^3{\bf v}_2'.$$ (3.21)

Expression (3.20) for the Boltzmann collision operator can be further simplified for elastic collisions because, in this case, the collision cross-section $\sigma({\bf v}_1, {\bf v}_2; {\bf v}_1',{\bf v}_2')$ is a function only of the magnitude of the relative velocity vector, ${\bf u}$ , and its change in direction as a result of the collision. Furthermore, the integral over the final velocities ${\bf v}_1'$ and ${\bf v}_2'$ reduces to an integral over all solid angles for the change in direction of ${\bf u}$ . Thus, we can write

$$\sigma ({\bf v}_1, {\bf v}_2;{\bf v}_1',{\bf v}_2')\,d^3{\bf v}_1'\,d^3{\bf v}_2'= \frac{d\sigma(u,\chi,\phi)}{d{\mit\Omega}}\,d{\mit\Omega},$$ (3.22)

where ${\mit\Omega}=\sin\chi\,d\chi\,d\phi$ . Here, $\chi$ is the angle through which the direction of ${\bf u}$ 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 ${\bf u}$ is confined during the collision. (See Section 3.7.) Moreover, $d\sigma/d{\mit\Omega}$ is a conventional differential scattering cross-section (Reif 1965). Hence, we obtain

$$C_{12}(f_1,f_2) = \int\!\!\int\!\!\int u\,\frac{d\sigma (u,\chi,\phi)}{d{\mit\Omega}}\,(f_1'\,f_2'-f_1\,f_2)\, d^3{\bf v}_2\,d{\mit\Omega}.$$ (3.23)

Note, finally, that if we exchange the identities of particles $1$ and $2$ in Equation (3.22) then ${\bf u}\rightarrow-{\bf u}$ , but $u\rightarrow u$ , $\chi\rightarrow\chi$ , and $\phi\rightarrow\phi$ . Thus, we conclude that

$$\sigma({\bf v}_2, {\bf v}_1; {\bf v}_2', {\bf v}_1')= \sigma({\bf v}_1, {\bf v}_2; {\bf v}_1', {\bf v}_2').$$ (3.24)