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
   $$\sigma = \frac{e^2\,n_e}{m_e\,\nu_e}.$$