Problems and Solutions on Thermodynamics and Statistical Mechanics

162 Problems €4 SollLtioru on Thermodynamics €4 Statistical Mechanica

(b) About how long must one wait for this answer to be true?
(c) What would be the effect of a 1% preference of the predator for
eating red bacteria on (a) and (b)?
(Princeton)
Solution:
(a) After a sufficiently long time, the bacteria will amount to a huge
number N >> 10,000 without the existence of a predator. That the
predator eats bacteria at random is mathematically equivalent to select-
ing n = 10,000 bacteria out of N bacteria as survivors. N ,> n means that
in every selection the probabilities of surviving “red” and “green” E. coli
are the same. There are 2n ways of selection, and there are Cg ways to
survive m “red” ones. Therefore the probability distribution of the number
of “red” E. coli is
1 1 fl!
-C” = -. , m=0,1, ..., n
2n 2n m!(n - m)!

(b) We require N >> n. In practice it suffices to have N/n = lo2. As

N = 2tn,t = 6 to 7 hours would be sufficient.

(c) If the probability of eating red bacteria is

eating green is (i - p) , the result in (a) becomes

c: (;+?I)” (;-P)n-m

n! n-m

m!(n - m)!

The result in (b) is unchanged.

200s

(a) What are the reduced density matrices in position and momentum

(b) Let us denote the reduced density matrix in momentum space by

spaces?

q5(pl, pz). Show that if q5 is diagonal, that is,

d4Pl)PZ) = f(Pl)Lm >