Next: Collisional Conservation Laws Up: Collisions Previous: Two-Body Elastic Collisions

# Boltzmann Collision Operator

Let be the cross-section for a scattering process by which particles of types and (located at position vector at time ) are incident with velocities and , respectively, and are scattered to velocities and , 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,

 (3.16)

The rate at which particles with the original velocities and are scattered into the range to and to is

 (3.17)

Here, . Moreover, and are the ensemble-averaged distribution functions for particles of types and , respectively. In writing the previous expression, we have assumed that the distribution functions and 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 and are scattered into the range to and to is

 (3.18)

where . Now, it is easily demonstrated from Equations (3.12) and (3.13) that

 (3.19)

The result follows from the fact that the vectors and differ only in direction. Thus, the net rate of change of the distribution function of particles of type with velocities (at position and time ) due to collisions with particles of type [i.e., the collision operator--see Equation (3.9)] is given by

 (3.20)

Here, use has been made of Equation (3.16), as well as . Moreover, , . , and are short-hand for , , , and , 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 with velocities (at position and time ) due to collisions with particles of type is given by

 (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 is a function only of the magnitude of the relative velocity vector, , and its change in direction as a result of the collision. Furthermore, the integral over the final velocities and reduces to an integral over all solid angles for the change in direction of . Thus, we can write

 (3.22)

where . Here, is the angle through which the direction of is deflected as a consequence of the collision (see Figure 3.1), and is an azimuthal angle that determines the orientation of the plane in which the vector is confined during the collision. (See Section 3.7.) Moreover, is a conventional differential scattering cross-section (Reif 1965). Hence, we obtain

 (3.23)

Note, finally, that if we exchange the identities of particles and in Equation (3.22) then , but , , and . Thus, we conclude that

 (3.24)

Next: Collisional Conservation Laws Up: Collisions Previous: Two-Body Elastic Collisions
Richard Fitzpatrick 2016-01-23