2005 I 4
From QED
The canonical angular momentum is:

The flux may be defined as:
So:

Taking the lagrangian derivative, the left side is zero since it is conserved:

Bounce averaging:

On average over a bounce, Rdvζ / dt is zero, since it returns to the same position with the same velocity after one complete bounce. Therefore:

By Faraday's law:

In terms of :

Interchanging derivatives:

So:

Noting , and over a bounce
average assuming
:
So:

There is also diffusion, since there is a pressure gradient. We can estimate the banana diffusion by using a step size of Λi, and the ion-ion collision frequency, multiplying by fT = ε1 / 2, the fraction of trapped particles:

The radial flux is then:

The spitzer conductivity is modified to:

because the trapped particles do not carry current. But the trapped particles do produce bootstrap current, due to the pressure gradient. The current is:

:

Using p = nT and Δx = Λi = ρiq / ε1 / 2, the banana width:

The current density is then:

Onzager symmetry states that the cross terms in the flux-driving force
matrix will be equal (symmetric about the diagonal). In this problem,
it states that the coefficient of the term in part
a will be equal to the coefficient of the − Tedn / dx term in part
b.
This page was recovered in October 2009 from the Plasmagicians page on Generals_2005_I_4 dated 20:24, 15 May 2007.