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
   $$f({\bf v}) = \frac{n}{\pi^{3/2}\,v_t^{\,3}}\,\exp\left(-\frac{v^2}{v_t^{\,2}}\right).$$
   Let
   $$I_n = \int \frac{f}{n}\left(\frac{v}{v_t}\right)^n\,d^3{\bf v}.$$
   Demonstrate that $I_{-2}=2$ , $I_0=1$ , $I_2=3/2$ , and $I_4=15/4$ .

7. Consider a neutral gas in a force-free steady-state equilibrium. The particle distribution function $f$ satisfies the simplified kinetic equation
   $${\bf v}\cdot \nabla f = C(f).$$
   We can crudely approximate the collision operator as
   $$C = -\nu\,(f-f_0)$$
   where $\nu$ is the effective collision frequency, and
   $$f_0 = \frac{n}{\pi^{3/2}\,v_t^{\,3}}\exp\left[-\frac{({\bf v}-{\bf V})^{\,2}}{v_t^{\,2}}\right].$$
   Here, $v_t=\sqrt{2\,T/m}$ . Suppose that the mean-free-path $l=v_t/\nu$ is much less than the typical variation lengthscale of equilibrium quantities (such as $n$ , $T$ , and ${\bf V}$ ). Demonstrate that it is a good approximation to write
   $$f =f_0 - \nu^{-1}\,{\bf v}\cdot\nabla f_0.$$
   1. Suppose that $n$ and $T$ are uniform, but that ${\bf V} = V_y(x)\,{\bf e}_y$ . Demonstrate that the only non-zero components of the viscosity tensor are
      $$\pi_{xy}=\pi_{yx} = -\eta\,\frac{dV_y}{dx},$$
      where
      $$\eta = \frac{1}{2}\,m\,n\,\nu\,l^{\,2}.$$

   2. Suppose that $n$ is uniform, and ${\bf V}={\bf0}$ , but that $T=T(x)$ . Demonstrate that the only non-zero component of the heat flux density is
      $$q_x = - \kappa\,\frac{dT}{dx},$$
      where
      $$\kappa = \frac{5}{2}\,n\,\nu\,l^{\,2}.$$

   3. Suppose that ${\bf V}={\bf0}$ , and $n=n(x)$ and $T=T(x)$ , but that $p=n\,T$ is constant. Demonstrate that the only non-zero component of the heat flux density is
      $$q_x = - \kappa\,\frac{dT}{dx},$$
      where
      $$\kappa = \frac{5}{4}\,n\,\nu\,l^{\,2}.$$

8. Consider a spatially uniform, unmagnetized plasma in which both species have zero mean flow velocity. Let $n_e$ and $T_e$ be the electron number density and temperature, respectively. Let ${\bf E}$ be the ambient electric field. The electron distribution function $f_e$ satisfies the simplified kinetic equation
   $$-\frac{e}{m_e}\,{\bf E}\cdot \nabla_v f_e = C_e.$$
   We can crudely approximate the electron collision operator as
   $$C_e = -\nu_e\,(f_e-f_0)$$
   where $\nu_e$ is the effective electron-ion collision frequency, and
   $$f_0 = \frac{n_e}{\pi^{3/2}\,v_{t\,e}^{\,3}}\exp\left(-\frac{v^2}{v_{t\,e}^{\,2}}\right).$$
   Here, $v_{t\,e}=\sqrt{2\,T_e/m_e}$ . Suppose that $E\ll m_e\,\nu_e\,v_{t\,e}/e$ . Demonstrate that it is a good approximation to write
   $$f_e =f_0 + \frac{e}{m_e\,\nu_e}\,{\bf E}\cdot\nabla_v f_0.$$
   Hence, show that
   $${\bf j} = \sigma\,{\bf E},$$
   where
   $$\sigma = \frac{e^2\,n_e}{m_e\,\nu_e}.$$