Solutions To Phase Space Plasma Physics Problems

The  Problems:

1. (a)Write the phase space dimension for a 3-particle system.

(b) Write out the number density for this system.

Solutions:

a) We need the phase space formula: 6No  - D where No is the number of particles in the system.  In this case  No   = 3, so:

 6No  - D =   6 (3) -D  or 18 dimensions

b)  The number density for the system is:

N( X_i.... X_n ,V_{i ....}V_n ) = å ^No _i=1      d (X_i   -   X_i (t)) d (V_i  -   V_i (t))

Where  No    = 3,  so:

N( X₁, X₂ , X₃ ,V₁ , V_2., V₃ ) =

d (X₁  -  X₁ (t)) d (V₁   -  V₁ (t))d (X₂  -  X₂ (t)) d (V₂   -   V₂ (t)) d (X₃  -   X₃ (t)) d (V₃   -  V₃ (t)

2. We saw  (x₁, v₁)  = ( _x1 , y₁ , z₁, _vxi, _vyi,  _vzi)  is a 6-dimensional phase space.  What would be the number of dimensions for the 12-D  phase space:

 (x₁, v₁, x₂, v₂ )?

Write them all out in a bracket.

Solution:

(x₁, v₁, x₂, v₂ ) =  ( _x1 , y₁ , z₁, _vxi, _vyi,  _vzi, _x2 , y₂ , z₂, _vx2, _vy2,  _vz2,) 

3.  Show how V'_i  =   q/ m [ E (X_i ) + V_i   x  B( X_i )

translates to Newton's 2nd law of motion.

(Hint: You need to incorporate the Lorentz force.)

Answer:

Given the nature of the plasma, charged particles of the system moving in magnetic field (B) the Lorentz force is:  

F = q ( E  + v x   B)

Or, in terms of Newton's 2nd law:

m (dv/dt) = ma = q ( E  + v x   B)

Or, to conform with the plasma expression:

V'_i  =   q/ m ( E  + v x   B)

Where: E =  E (X_i )       and: v x   B  = V_i   x  B( X_i )

Where the  X_i   are position dependent in the coordinate system.