I'll post a challenge that I had some fun doing, and actually has applications (in my work, at least). The original theory is found in this
awesome paper.
Consider a binary alloy. If you don't know what that is, it's the mixture of two solids. Let us call them A and B, where each of these solids has only one atomic element in its chemical composition. Although not a fancy topic like black holes and quantum gravity, most of the theory for alloys is very speculative and remains mostly unsolved.
Anyway, the alloys which we can say something very nontrivial about are those with
substitutional disorder, those that look like this:
The idea is simple. A binary alloy with substitutional disorder maintains the crystal structure of the pure solids, the only disorder comes from different atomic species occupying random lattice sites.
The only macroscopic property of the alloy is its composition, in the example it could be the ratio of black atoms to the total number. The objective of any model is to find out how the alloy's other properties (like conductivity or optical absorption) depend on its composition.
For technical reasons, it is much more simple to calculate things in a system that is periodic, which the alloy clearly is not. So, in the example above, we would approximate it as a periodic system. For example, we could pick a 2x2 square, with black and white atoms at some positions, and repeat it indefinitely in both dimensions.
We do this for all 2^4=16 possible 2x2 squares (yes, some of those are actually equivalent to each other, and I once had to develop a code to reduce the number of possibilities to test, it is a very hard problem for realistic crystal structures, so we should ignore it now), and calculate whatever we want in each of these 16 configurations. In the end, we do an average to obtain the value in the mixture.
Let us denote the alloy composition (fraction of black atoms) by x. The property you have calculated in the i-th configuration has a value P_i. If we have the probability of occurrence x_i of the i-th configuration, which depends on x (that should be evident, if x is very close to 1, the probability of a configuration with lots of black atoms should be much higher than in the case where x is very close to 0), then we can calculate the average in the alloy at composition x as:
P(x) = sum_i x_i(x) P_i
The thing is, it is not so simple to calculate x_i(x). Some alloys really like mixing together, in this case the values of x_i would very close to random. Others are like water and oil and don't like to mix at all. In this case, the probability for mixed configurations would be very close to 0.
The paper I linked to in the beginning of the post tells you how to calculate x_i(x). The idea is the following: you need to provide some energy to create a given configuration, which we call the
formation energy E_i. If this energy is very large, the probability of the configuration occurring is very low, if this energy is small, the probability is high. Alloys also have a growth temperature T. It varies a lot depending on the alloy, some are as high as 1000 K, others as low as 100 C (you can cook them). If the growth temperature is very large, differences in energy shouldn't matter at all and the alloy should be completely random.
Suppose the total number of atoms in any configuration is N (in the 2x2 square example, N=4), and consider that the i-th has n_i black atoms. The formula for calculating x_i(x) is (define b=1/kT, where k is Boltzmann's constant):
x_i = (a^n_i)*exp(-b*E_i)/ (sum_j (a^n_j)*exp(-b*E_j)) (1)
a is a positive non-dimensional constant. To find it, you must solve a polynomial equation (which is easy to do numerically), that you obtain by imposing the constraint
sum_i x_i*n_i = N*x, (2)
this just means that the average number of black atoms should equal the number specified in the composition x.
Now, to the actual challenge, which is something that helped me a lot when I figured out (I can tell the story if you guys want):
Using the system of equations (1) and (2), prove that, if you shift all energies by a linear function of the number of atoms, that is, E_i -> E_i + A + B*n_i, the probabilities x_i obtained from the equations are the same.