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Magnetization in terms of the partition function

Why do we need to take the derivative of Log(Z_N) with respect to beta when computing the magnetization?

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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[- ...

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