Thermodynamics and Statistical Mechanics I - Home Exercise 2

Thermodynamics and Statistical Mechanics I - Home Exercise 2
1. It has been said that ”six monkeys set to strum unintelligently on typewriters for millions of years would be bound in time to produce all the books in the British Museum.”
Suppose that 1010 monkeys(∼ the number of monkeys in the world) have been seated
at typewriters for the entire age of the universe, ∼ 1018 s. Suppose that a monkey can
hit 10 keys on the typewriter per second. Assume the typewriter has 44 keys (we wont
distinguish between upper and lower case). Assume that Shakespeares Hamlet has 105
characters. Will the monkeys hit upon Hamlet?
(a) Calculate the probability that any given sequence of 105 characters typed at random will come out in the correct sequence (the sequence of Hamlet).
(b) Calculate the probability that one of the monkeys will type Hamlet in the age of
the universe.
(c) Judging by these results, could the statement in the beginning of the question be
true?
(d) Would the monkeys even produce to be or not to be (18 characters including
spaces)?
2. Consider N non-interacting spins. Each spin can be either UP or DOWN with equal
probability.
(a) What is the difference between a microscopic state of the system and a macroscopic
state?
(b) What is the number of possible arrangements of each macroscopic state and what
is the probability of each state?
(c) What is the average magnetization of the system?
(d) What is the variance of the magnetization?
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3. Given two systems of N1 ≈ N2 ≈ 1022 spins with multiplicity functions g1 (N1 , s1 ) and
g2 (N2 , s−s1 ), the product g1 g2 as a function of s1 is relatively sharply peaked at s1 = ŝ1 .
For s1 = ŝ1 + 1012 , the product g1 g2 is reduced by 10−174 from its peak value. Use the
Gaussian approximation of the multiplicity function.
(a) Compute g1 g2 /(g1 g2 )max for s1 = ŝ1 + 1011 and s = 0.
(b) For s = 1020 , by what factor must you multiply (g1 g2 )max to make it equal to
P
g1 (N1 , s1 )g2 (N2 , s − s1 )?
s1
Give the factor to the nearest order of magnitude.
(c) How large is the fractional error in the entropy when you ignore this factor?
4. A quantum harmonic oscillator has energy levels of the form En = ~ω(n + 21 ). Consider
N
P
a set of N fixed oscillators. The total energy of the system
Eni = E is fixed. Find
ni =1
the multiplicity function for the set of oscillators.
Note that a set of N oscillators is physically identical to a single N -dimentional oscillator.
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