Examining Aspects Of The Klimontovich Equation Used In Plasma Physics

Note 1: The Dirac delta form of the Klimontovich equation 

N_i  (x, v, t) =  d  [x_i   -  x_i  (t)]  d  [v_i    -  v_i  (t)] 

refers to the density at a given time for a single particle, using the  d - function.

Recall here:  

An important property of this function is:

ad (a  - b) =   bd (a  - b)  

Note 2: In the 6D phase space, the Klimontovich equation reformulated as conservation of N (e.g. dN/dt = 0) is written:

¶  N(x, v, t)/ ¶ t   +  v  Ñ x N + F/m · Ñ v  N  =  0

Where:  Ñ x  =  (¶ x , ¶ y , ¶ z ),   Ñ v = (¶ vx , ¶ vy , ¶ vz )

This equation is made equivalent to Newton's equations for all particles in the plasma, i.e. by the appearance of  F/m   in the 3rd term.   This is just Newton's 2nd law in terms of v' (e.g. dv/dt), the acceleration.

Note 3. To derive the Boltzmann equation from the Klimontovich equation, we first do an ensemble average of N:

<N>  =  f (x, v, t)   and N  =   f  +    dN

E =   <E> +  dE   And:   B =   <B> +  dB

Now substitute N  =   f  +    dN  and  F/m = (E + v x B) into the 6D phase space   Klimontovich equation to get:

¶ f / ¶ t  + ¶ < dN>/ ¶ t +   v · Ñ x  f _o +  v X dB · Ñ x < dN> +

 q/ m ( <E> +  dE + v x <B> ) · Ñ v  f  + Ñ v  dN  =  0

Þ   

¶ f / ¶ t  +  v Ñ x  f  + q/ m ( <E>  + v x <B>) · Ñ v  f =

- q/ m ( <dE · Ñ < dN>  - q/ m <v x dB Ñ dN>

I.e. the Boltzmann eqn. in plasma physics

Note the presence of the perturbed quantities:

N  =   f  +    dN

E =   <E> +  dE

<B> +  dB

gives rise to the collision term in the Boltzmann equation.    

The Klimontovich equation is basically an exact equation for the time evolution of a plasma obtained by taking the time derivative of the density N x. Written out in full:

¶  N x(x, v, t)/ ¶ t =

  - å ^No _i=1   X_i ·Ñ x d [x_i   -    X_i (t)]d [x_i   -   V_i (t)] -

å ^No _i=1   V_i ·Ñ v  d [x_i   -    X_i (t)]d [x_i   -   V_i (t)]