Why do we need to take the derivative of Log(Z_N) with respect to beta when computing the magnetization?
The magnetisation is the thermal average of the sum of all the N magnetic moments. In general the average of a function f(x1,x2,...) is given by:
Integral over x1,x2,.. of [f(x1,x2,..) times P(x1,x2,...)] (1)
where p(x1,x2) is the probability density. In our case the x's are the magnetic moments. The probability density in the canonical ensemble is given by:
P(x1,x2,...) = Exp[- ...
A detailed solution is given.