384 Chapter 13. Fokker-Planck theory of collisions
13.2 Statistical argument for the development of the Fokker-Planck equation
The Fokker-Planck theory (Rosenbluth et al. 1957) is based on a logical argument on how a
distribution function attains its present form due to collisions at someearlier time. Consider
a particle with velocityvat timetthat is subject to random collisions which change its
velocity. We defineF(v,∆v)as the conditional probability that if a particle has a velocity
vat timet,then at some later timet+∆t, collisions will have caused the particle to have
velocityv+∆v.
Clearly at timet+∆t the particle must have some velocity, so the sum of all the
conditional probabilities must be unity, i.e.,Fmust be normalized such that
∫
F(v,∆v)d∆v=1. (13.13)This definition of conditional probability can be used to show how a present distribution
functionf(v,t)comes to be the way it is because of the way it was at timet−∆t.This
can be expressed as
f(v,t)=∫
f(v−∆v,t−∆t)F(v−∆v,∆v)d∆v (13.14)which sums up all the possible ways for obtaining a given present velocity weighted by
the probability of each of these ways occurring. This analysis presumes that the present
status depends only on what happened during the previous collision and is independent
of all events before that previous collision. A partial differential equation forfcan be
constructed by Taylor expanding the integrand as follows:
f(v−∆v,t−∆t)F(v−∆v,∆v)= f(v,t)F(v,∆v)−∆t∂f
∂tF(v,∆v)−∆v·∂
∂v
(f(v,t)F(v,∆v))+1
2
∆v∆v:∂
∂v∂
∂v(f(v,t)F(v,∆v)).
(13.15)
Substitution of Eq.(13.15) into Eq.(13.14) gives
f(v,t)= f(v,t)∫
F(v,∆v)d∆v−∆t
∂f
∂t∫
F(v,∆v)d∆v−∫
∆v·∂
∂v(f(v,t)F(v,∆v))d∆v+1
2
∫
∆v∆v:∂
∂v∂
∂v(f(v,t)F(v,∆v))d∆v
(13.16)where in the top line advantage has been taken off(v,t)not depending on∆v. Upon