** Next:** Plasma Fluid Theory
** Up:** Collisions
** Previous:** Collision Times

- Consider the Maxwellian distribution

Demonstrate that

- The species-
entropy per unit volume is conventionally defined as

The Boltzmann H-theorem thus states that collisions drive the system toward a maximum entropy state
characterized by Maxwellian distribution functions with common mean velocities and common temperatures.
Demonstrate that for a Maxwellian distribution,

the entropy per unit volume takes the form

- The Landau collision operator is written

where

Demonstrate directly that this collision operator satisfies the same conservation laws as the Boltzmann collision operator. Namely,

- The net heating rate per unit volume of type
particles due to Coulomb collisions with type
particles is

where
is the Landau collision operator.
Suppose that both species have Maxwellian distribution functions with zero mean velocities:

Suppose, further, that the kinetic temperatures of the two species are almost the same (i.e.,
).
Demonstrate that

where the collision time,
, is defined in Equation (3.181). In particular, show that in an electron-ion plasma

** Next:** Plasma Fluid Theory
** Up:** Collisions
** Previous:** Collision Times
Richard Fitzpatrick
2016-01-23