Fundamentals of Plasma Physics

58 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD

2.8 Assignments

1. Prove Stirling’s formula. To do this first show

lnN! = ln1 + ln2 + ln3 +...lnN

=

∑N

j=1

lnj

Now assumeNis large and, using a graphical argument, show that the above expres-

sion can be expressed as an integral

lnN!≈

∫?

?

h(x)dx

where the form ofh(x)and the limits of integration are to be provided. Evaluate the

integral and obtain Stirling’s formula

lnN!≃NlnN−N for largeN

which is a way of calculating the values of factorials of large numbers. Check the

accuracy of Stirling’s formula by evaluating the left and right hand sides of Stirling’s

formula numerically and plot the results forN= 1, 10 , 100 , 1000 , 104 and higher if

possible.

2. Variational calculus and Lagrange multipliers- The entropy associated with a distrib-
   ution function is
   S=

∫∞

−∞

f(v)lnf(v)dv.

(a) Sincef(v)measures the number of particles that have velocityv, use physical

arguments to explain what valuef(±∞)must have.

(b) Suppose thatfME(v)is the distribution function having the maximum entropy

out of all distribution functions allowed for the problem at hand. Letδf(v)be

some small arbitrary deviation fromfME(v). What is the entropyS+δSasso-

ciated with the distribution functionfME(v) +δf(v)?What is the differential

of entropyδSbetween the two situations?

(c) Each particle in the distribution function has a kinetic energymv^2 / 2 and sup-

pose that there are no external forces acting on the system of particles so that

the potential energy of each particle is zero. LetEbe the total energy of all

the particles. How doesEdepend on the distribution function? If the system is

isolated and the system changes from having a distribution functionfME(v)to

having the distribution functionfME(v) +δf(v)what is the change in energy

δEbetween these two cases?

(d) By now you should have integral expressions forδSand forδE.Both of these

integrals should have what value? Make a rough sketch showing any possi-

ble, nontrivialvdependence of the integrands of these expressions (i.e., show

whether the integrands are positive definite or not).

(e) Sinceδfwas assumed to be arbitrary, what can you say about the ratio of the

integrand in the expression forδSto the integrand in the expression forδE?Is