Consider the following expressions that define the Fokker-Planck coefficients of dynamic friction and of diffusion in velocity
Question:
Consider the following expressions that define the Fokker-Planck coefficients of dynamic friction and of diffusion in velocity space:
(a) With reference to Fig. 6, verify that
For a general inverse-power interparticle force of the form
where K is a constant and p is a positive integer number, show that [see (5.18)]
where μ is the reduced mass and σm is the momentum transfer cross section given by
where
Verify also that [see (5.19)]
(b) For the case of Maxwell molecules (p = 5), where the results are independent of fβ1, show that the Fokker-Planck coefficients are given by
where
(c) Calculate the Fokker-Planck coefficients for the case of the coulomb interaction (p = 2) using the results of part (a) and of problem 20.6, in terms of integrals over fβ1.
(d) Calculate the Fokker-Planck coefficients for electron-electron interactions, when fβ1 is the Maxwellian distribution function. Refer to (5.49), (5.50), and (5.51).
Figure 6.
Equations
Data from problem 20.6.
Show that for the case of coulomb interactions (p = 2) we have
where b0 = qq1/(4π∈0μg2), Aℓ(p) is as defined, and A = λD/b0. For ∧ ≫ 1 verify that
Note that, since K = qq1/(4π∈0) for the coulomb interaction potential, we have (μg2/K)2 b02 = 1.
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