Exercises

1. Consider the Maxwellian distribution
   $$f ({\bf v})= n\left(\frac{m}{2\pi\,T}\right)^{3/2}\,\exp\left[-\frac{m\,({\bf v}-{\bf V})^{\,2}}{2\,T}\right].$$
   Demonstrate that

   $$n = \int f\,d^3{\bf v},$$

   $$n\,{\bf V} = \int {\bf v}\,f\,d^3{\bf v},$$

   $$\frac{3}{2}\,n\,T = \int \frac{1}{2}\,m\,v^2\,f\,d^3{\bf v}.$$

   
   

2. The species-$s$ entropy per unit volume is conventionally defined as
   $$s_s= -\int f_s\,\ln f_s\,d^3{\bf v}_s.$$
   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,
   $$f_s= n_s\left(\frac{m_s}{2\pi\,T_s}\right)^{3/2}\,\exp\left(-\frac{m\,v_s^{\,2}}{2\,T_s}\right),$$
   the entropy per unit volume takes the form
   $$s_s= n_s\left[\ln\left(\frac{T_s^{\,3/2}}{n_s}\right) +\frac{3}{2}\ln\left(\frac{2\pi}{m_s}\right)+\frac{3}{2}\right].$$

3. The Landau collision operator is written
   $$C_{12}(f_1,f_2) = \frac{\gamma_{12}}{m_1}\,\frac{\partial}{\partial{\bf v}_1}\cdot\int {\bf w}_{12}\cdot {\bf J}_{12}\,d^3{\bf v}_2,$$
   where

   $$\gamma_{12} = \left(\frac{e_1\,e_2}{4\pi\,\epsilon_0}\right)^2 2\pi\,\ln{\mit\Lambda}_c,$$

   $${\bf w}_{12} = \frac{u_{12}^{\,2}\,{\bf I} - {\bf u}_{12}{\bf u}_{12}}{u_{12}^{\,3}},$$

   $$u_{12} = \vert{\bf v}_1-{\bf v}_2\vert,$$

   $${\bf J}_{12} = \frac{\partial f_1}{\partial{\bf v}_1}\,\frac{f_2}{m_1} -\frac{f_1}{m_2}\,\frac{\partial f_2}{\partial {\bf v}_2}.$$

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

   $$\int C_{12}\,d^3{\bf v}_1 =0,$$

   $$\int m_1\,{\bf v}_1\,C_{12}\,d^3{\bf v}_1 = -\int m_2\,{\bf v}_2\,C_{21}\,d^3{\bf v}_2,$$

   $$\int\frac{1}{2}\,m_1\,v_1^{\,2}\,C_{12}\,d^3{\bf v}_1 =- \int\frac{1}{2}\,m_2\,v_2^{\,2}\,C_{21}\,d^3{\bf v}_2.$$

   
   

4. The net heating rate per unit volume of type $1$ particles due to Coulomb collisions with type $2$ particles is
   $$W_{12}=\int \frac{1}{2}\,m_1\,v^2\,C_{12}\,d^3{\bf v},$$
   where $C_{12}$ is the Landau collision operator. Suppose that both species have Maxwellian distribution functions with zero mean velocities:

   $$f_1 = n_1\left(\frac{m_1}{2\pi\,T_1}\right)^{3/2}\,\exp\left(-\frac{m_1\,v_1^{\,2}}{2\,T_1}\right),$$

   $$f_2 = n_2\left(\frac{m_2}{2\pi\,T_2}\right)^{3/2}\,\exp\left(-\frac{m_1\,v_2^{\,2}}{2\,T_2}\right).$$

   
   
   Suppose, further, that the kinetic temperatures of the two species are almost the same (i.e., $T_1\simeq T_2$ ). Demonstrate that
   $$W_{12} = 3\,\frac{\mu_{12}}{m_2}\,\frac{n_1}{\tau_{12}}\,(T_2-T_1),$$
   where the collision time, $\tau_{12}$ , is defined in Equation (3.181). In particular, show that in an electron-ion plasma
   $$W_{ei} \simeq 3\,\frac{m_e}{m_i}\,\frac{n_e}{\tau_{ei}}\,(T_i-T_e).$$