All questions are worth the same number of points.
Note (21st Jan): a typo in 2(d) has just been fixed.
Note: don't be put off or scared if some of the questions involve concepts that we haven't covered in lecture yet. We will have done all that you need by the end of next Wednesday's lecture, so don't worry!
Question 1:
give a succinct definition (with a very brief illustrative
example, if applicable) for the following six things:
Question 2: the meaning of "never"
Consider a "two-state paramagnet", a system which has N
elementary dipoles which can point either up or down. Suppose an
up-pointing dipole costs energy E, and a down-pointing one
costs energy -E. Fix the total energy to be zero, and consider
N= 1023 elementary dipoles.
(a) How many microstates are accessible to the system?
(b) Suppose that the microstate of this system changes a billion times
per second. How many microstates will it explore in ten billion years
(roughly the age of the universe)?
(c) Is it correct to say that, if you wait long enough, a system will
eventually be found in every "accessible" microstate? Explain your
answer.
(d) What is different if we allow the whole system to have
overall energy +2E instead?
Question 3: physics of a walking drunk.
Suppose you were
drunk, walking randomly, in one dimension (i.e. you can move
only to the right, or to the left). The journey consists of N
steps, all the same size, each chosen randomly to be either forward or
backward.
(a) Explain in detail why this setup is equivalent to a spin system
(remember to explain the analogues of all features of the
spin system).
(b) Where are you most likely to find yourself, after the end
of a long random walk?
(c) Suppose you take a random walk of 10,000 steps, each 1m long.
About how far from your starting point would you expect to be at the
end? Call this position x. How probable is it that, instead,
you end up 50 metres away from x?
(d) A good example of a random walk in nature is the
diffusion of a single molecule through a gas. The average
step length, for the motion of a molecule in between collisions with
other molecules, is then called the mean free path. For
nitrogen at room temperature and atmospheric pressure, this is about
150nm. At room temperature, the average speed of a nitrogen molecule
is about 500m/s. Find the probability that, after one second, the
molecule has moved 1mm away from where it started. (Assume the box of
nitrogen is one-dimensional.) This calculation helps explain why
diffusion is so darn slow; convection is a much faster method of
moving molecules around!
Question 4.
Consider a system with two different energy levels,
-E/2 and +E/2. The higher-energy state has multiplicity
3, while the lower-energy state has multiplicity 1.
Find the
partition function Z. Calculate the average energy U
and the quantity Cv. Explain intuitively the
high-temperature limit of your expression for U.
Question 5: theoretical zipperdynamics (hard)
Do KK question
3.7 (in the text). In addition, make and explain a graph of the
average number of open links, as a function of temperature.