# Help with solution

1. Oct 5, 2008

### oasis2007

A simple and famous model of certain magnetic materials is the Ising model in
which each of N atoms is modeled as a simple spin which can point upwards or
downwards. The magnetization of the material is proportional to the number of up
spins minus the number of down spins (each spin has a certain magnetization associated
with it).
Let's consider N spins in a row (as shown below). Let the "magnetization" be M,
where M is the number of up spins minus the number of down spins. Each spin is
constantly changing its direction from up to down and vice versa according to some
complex dynamics. Assume the dynamics of one spin are independent of other spins.
↑↓↓↓↑↑↓↑↓↓↑↓↓↑↓↑↑↓↓↓↑↓
(i) Clearly state a postulate on the spin configurations which will help us calculate
probabilities of different spin configurations.
(ii) Using (i), calculate the probability of a certain M value P(M) exactly.
(iii) Show P(M) is a Gaussian in the large N limit and sketch it.
(iv) Estimate the probability that M is larger than 0.1% of N when N takes the value of