Recall in Part 1, we saw the density of systems for the (x1, v1) phase space is:
N(x1 ,v1 ,t) = d (x1 - X1 (t)) d (v1 - V1 (t))
Where: x1 = X1 (t)), v1 = V1 (t))
Analogously, in the 12-dimensional phase space (two particles considered) we saw from the Part 1 solutions (#2):
(x1, v1, x2, v2 ) = ( x1 , y1 , z1, vx1, vy1, vz1, x2 , y2 , z2, vx2, vy2, vz2,)
And there is one system occupying the point:
x1 = X1 (t)), v1 = V1 (t), x2 = X2 (t)), v2 = V2 (t) i.e. at time t. The density of systems in this phase space is then:
N( x1, x2 ,v1 , v2 ) =
d (x1 - X1 (t)) d (v1 - V1 (t))d (x2 - X2 (t)) d (v2 - V2 (t))
Basically, then there is one system in 6No -dimensional space, so by analogy with the density of systems expression e.g. for N( x1, x2 ,v1 , v2 ), we can write:
N(x1, x2 ,v1 , v2 .... xNo ,vNo , t) = åNo i=1 d (xi - Xi (t)) d (vi - Vi (t))
As with the Klimontovich equation, the Liouville equation is found by taking the time derivative of the appropriate density. Given the already defined density of systems is the product of 6No terms then the time derivative must involve the product of 6No terms. We can use the expression:
¶ / ¶ t d [ (x1 - X1 (t)] = - ¶ Xi/ ¶ t · Ñ x i d [ (x1 - X1 (t)]
The time density will be:
¶ N / ¶ t + å No i=1 Vi (t) · Ñ x i åNo j=1 d (xj - Xj (t)) d (vj - Vj (t)) +
åNo i=1 V'i åNo j=1 d (xj - Xj (t)) d (vj - Vj (t)) = 0
The next standard step employed by plasma physicists is to use the identity: ad (a - b) = bd (a - b) to replace Vi by vi
Leading to:
V'i (t) = q si / m si [ E (xi,t) + Vi /c x B( xi ,t)]
Given that the products are just the density of systems, N, the equation for the time derivative of N(x1, x2 ,v1 , v2 .... xNo ,vNo , t)
becomes:
¶ N / ¶ t + å No i=1 Vi · Ñ x N +å No i=1 V'i (t) · Ñvi N = 0
which is the Liouville equation, and when combined with Maxwell's equations, viz.
i) Ñ X H = J
+ ¶D / ¶ t
ii)
Ñ
X E = - ¶B / ¶ t
iii) Ñ · B = 0
iv) Ñ · D = r
is an exact description of a plasma.
