Friday, January 4, 2019

Solutions To Statistical Mechanics (2)

1) For the plasma bifurcation, when: F(xm, y) F >  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) =   (me/mi)/ (  w / w e)2 + 1/ [(w / w )  2 - (k Vow e)2  ]


Let:  x =  w/ w e       And:    y = k Vo/ w e

Then:

1= F(x, y) = (me/mi)/ x 2+ 1/ (x2 - y2)
   
Now focus on  Fm    for F(x,y)  whereby  Fm    >   1 and there will be two real roots  (See e.g. Fm   defined in the graph below and  xm   in relation to it)

No photo description available.


Now, take:

 x  F(x,y)  =   -2(me/mi)/ x3   -   2/ (x - y) 

Or:   x  F(xm, y) =  0

And we note:      x  F(xm, y) =   6 (me/mi)/ x4   -  6/ (x - y)  4     is always +ve


Whence:   Fm   ~   (me/mi1/3 /  y 2+   1/  y2     

So   Fm   >  1 requires:

y   <     [ (me/mi1/3  +   1 ] 1/2

Or:

y   <     [( 0.081/3  +   1 ] 1/2


i.e.   y   <   1.03

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   w ) :

w  =
 k Vo   w e


k Vo   -    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 directions relative to the coordinate axes.  "Instability" then occurs when the (dispersion) equation has complex roots, i.e. .when the local minimum (  Fm  ). is greater than 1,

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