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  1. Consider the Maxwellian distribution

    $\displaystyle 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

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

  2. The species-$ s$ entropy per unit volume is conventionally defined as

    $\displaystyle 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,

    $\displaystyle 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

    $\displaystyle 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

    $\displaystyle 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,


    $\displaystyle \gamma_{12}$ $\displaystyle = \left(\frac{e_1\,e_2}{4\pi\,\epsilon_0}\right)^2 2\pi\,\ln{\mit\Lambda}_c,$    
    $\displaystyle {\bf w}_{12}$ $\displaystyle = \frac{u_{12}^{\,2}\,{\bf I} - {\bf u}_{12}{\bf u}_{12}}{u_{12}^{\,3}},$    
    $\displaystyle u_{12}$ $\displaystyle = \vert{\bf v}_1-{\bf v}_2\vert,$    
    $\displaystyle {\bf J}_{12}$ $\displaystyle = \frac{\partial f_1}{\partial{\bf v}_1}\,\frac{f_2}{m_1} -\frac{f_1}{m_2}\,\frac{\partial f_2}{\partial {\bf v}_2}.$    

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

    $\displaystyle \int C_{12}\,d^3{\bf v}_1$ $\displaystyle =0,$    
    $\displaystyle \int m_1\,{\bf v}_1\,C_{12}\,d^3{\bf v}_1$ $\displaystyle = -\int m_2\,{\bf v}_2\,C_{21}\,d^3{\bf v}_2,$    
    $\displaystyle \int\frac{1}{2}\,m_1\,v_1^{\,2}\,C_{12}\,d^3{\bf v}_1$ $\displaystyle =- \int\frac{1}{2}\,m_2\,v_2^{\,2}\,C_{21}\,d^3{\bf v}_2.$    

  4. The net heating rate per unit volume of type $ 1$ particles due to Coulomb collisions with type $ 2$ particles is

    $\displaystyle 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:

    $\displaystyle f_1$ $\displaystyle = n_1\left(\frac{m_1}{2\pi\,T_1}\right)^{3/2}\,\exp\left(-\frac{m_1\,v_1^{\,2}}{2\,T_1}\right),$    
    $\displaystyle f_2$ $\displaystyle = n_2\left(\frac{m_2}{2\pi\,T_2}\right)^{3/2}\,\exp\left(-\frac{m_1\,v_2^{\,2}}{2\,T_2}\right).$    

    Suppose, further, that the kinetic temperatures of the two species are almost the same (i.e., $ T_1\simeq T_2$ ). Demonstrate that

    $\displaystyle 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

    $\displaystyle W_{ei} \simeq 3\,\frac{m_e}{m_i}\,\frac{n_e}{\tau_{ei}}\,(T_i-T_e).

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