Solutions To Statistical Mechanics (2)

1) For the plasma bifurcation, when: F(x_m, y) = F_m  >  1 there will be two real and two complex roots. Find the two real roots. 

Solution:    We are given the  original dispersion equation:

F(w) =   (m_e/m_i)/ (  w / w _e)² + 1/ [(w / w _e )  ² - (k V_o/ w _e)²  ]


Let:  x =  w/ w _e       And:    y = k V_o/ w _e

Then:

1= F(x, y) = (m_e/m_i)/ x ²+ 1/ (x² - y²)
   
Now focus on  F_m    for F(x,y)  whereby  F_m    >   1 and there will be two real roots  (See e.g. F_m   defined in the graph below and  x_m   in relation to it, i.e. for the dotted midline from x-axis crossing the F=1 line).



Now, take:

¶ x  F(x,y)  =   -2(m_e/m_i)/ x³   -   2/ (x - y) ³ 

Or:  ¶ x  F(x_m, y) =  0

And we note:     ¶ ² x  F(x_m, y) =   6 (m_e/m_i)/ x⁴   -  6/ (x - y)  ⁴     is always +ve


Whence:   F_m   ~   (m_e/m_i) ^1/3 /  y ²+   1/  y²     

So   F_m   >  1 requires:

y   <     [ (m_e/m_i) ^1/3  +   1 ] <sup>1/2

So   in the long wavelength limit, the two real roots can be deduced to approach

x  =   y +  1    and   x =   y  - 1    Or  ( in terms of</sup>   w ⁾ :

^{w  =}
 k V_o   +  w _e


k V_{o   -}    w _e


2) Explain how the plasma bifurcation above occurs mathematically

The plasma bifurcation occurs because of a counter streaming plasma flow (i.e. for ions and electrons) in velocity space, which leads to a split symmetrical solution as well as a symmetrical one - but in a different direction relative to the coordinate axes.  "Instability" then occurs when the (dispersion) equation has complex roots, i.e. .when the local minimum (   F_m    ). is greater than 1,

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