Fundamentals of Plasma Physics

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

subsequent holes, we see there areN!ways of putting all the pegs in all the holes.
Let us complicate things further. Suppose that we arrange the holes inMgroups, say
groupG 1 has the first 10 holes, groupG 2 has the next 19 holes, groupG 3 has the next 4
holes and so on, up to groupM. We will usefto denote the number of holes in a group,
thusf(1) = 10,f(2) = 19,f(3) = 4,etc. The number of ways of arranging pegs within
a group is just the factorial of the number of pegs in the group, e.g., the number of ways
of arranging the pegs within group 1 is just10!and so in general the number of ways of
arranging the pegs in thejthgroup is[f(j)]!.
Let us denoteCas the number of ways for putting all the pegs in all the groupswithout
caringabout the internal arrangement within groups. The number of ways of putting the
pegs in all the groupscaringabout the internal arrangements in all the groups isC×f(1)!×
f(2)!×...f(M)!,but this is just the number of ways of putting all the pegs in all the holes,
i.e.,

C×f(1)!×f(2)!×...f(M)! =N!
or

C=

N!

f(1)!×f(2)!×...f(M)!

.

NowCis just the number of microscopic states corresponding to the macroscopic state
of the prescribed groupingf(1) = 10, f(2) = 19, f(3) = 4,etc. so the entropy is just
S= lnCor

S = ln

(

N!

f(1)!×f(2)!×...f(M)!

)

= lnN!−lnf(1)!−lnf(2)!−...−lnf(M)!

(2.38)

Stirling’s formula shows that the large argument asymptotic limit of the factorial function
is
lim
k→large
lnk! =klnk−k. (2.39)

Noting thatf(1) +f(2) +...f(M) =Nthe entropy becomes

S = NlnN−f(1) lnf(1)−f(2)lnf(2)−...−f(M) lnf(M)

= NlnN−

∑M

j=1

f(j)lnf(j) (2.40)

The constantNlnNis often dropped, giving

S=−

∑M

j=1

f(j)lnf(j). (2.41)

Ifjis made into a continuous variable say,j→vso thatf(v)dvis the number of items in
the group labeled byv,then the entropy can be written as

S=−

∫

dvf(v)lnf(v). (2.42)