# Exercises

1. Verify Equations (4.17) and (4.18).

2. Verify Equation (4.30).

3. Derive Equations (4.36)–(4.38) from Equation (4.35).

4. Derive Equations (4.41)–(4.43) from Equations (4.36)–(4.38).

5. Derive Equation (4.53) from Equation (4.49).

6. Consider the Maxwellian distribution

Let

Demonstrate that , , , and .

7. Consider a neutral gas in a force-free steady-state equilibrium. The particle distribution function satisfies the simplified kinetic equation

We can crudely approximate the collision operator as

where is the effective collision frequency, and

Here, . Suppose that the mean-free-path is much less than the typical variation lengthscale of equilibrium quantities (such as , , and ). Demonstrate that it is a good approximation to write

1. Suppose that and are uniform, but that . Demonstrate that the only non-zero components of the viscosity tensor are

where

2. Suppose that is uniform, and , but that . Demonstrate that the only non-zero component of the heat flux density is

where

3. Suppose that , and and , but that is constant. Demonstrate that the only non-zero component of the heat flux density is

where

8. Consider a spatially uniform, unmagnetized plasma in which both species have zero mean flow velocity. Let and be the electron number density and temperature, respectively. Let be the ambient electric field. The electron distribution function satisfies the simplified kinetic equation

We can crudely approximate the electron collision operator as

where is the effective electron-ion collision frequency, and

Here, . Suppose that . Demonstrate that it is a good approximation to write

Hence, show that

where