Average magnetic moment of the system

[WeiShan Ng]

I was reading the statistical physics textbook and was really confused with the notation:

We denoted the probability of the state (\sigma_1,...,\sigma_N) by P(\sigma_1,...,\sigma_N), and the average of an observable A(\sigma_1,...,\sigma_N) is just
\left< A \right> = \sum_{{\sigma}} P(\sigma_1,...,\sigma_N)A(\sigma_1,...\sigma_N)
The average magnetic moment of a paramagnetic system, in an ensemble defined by P, is determined by the average of A=M=\mu_B(\sigma_1+\sigma_2+...+\sigma_N), so that
\left< M \right> = \mu_B \sum_{\{\sigma\}} \left( \sum_{i} \sigma_i \right) P(\sigma_1,..., \sigma_N)

Next we want to show that \left< M \right> is proportional to the average magnetic moment of a single spin. First we define the probability for a given spin - with i=1 for example-to have the moment \sigma_1(+1 or -1):

P(\sigma_1)=\sum_{\{\sigma\}}' P(\sigma_1,...,\sigma_N)
where \sum' denotes that we are summing over microscopic states with fixed \sigma_1.

We can therefore rewrite \left< M \right> in the form
\left< M \right> = \mu_B N \sum_{\sigma \pm 1} \sigma P(\sigma) = \mu_B N\left< \sigma \right>
where \left< \sigma \right> is the average of the observable A=\sigma_i

I don't understand the last part of the section. Why is that \sum_{\sigma = \pm1} \sigma P(\sigma) equals to \left< \sigma \right>? And what does \left< \sigma \right> actually mean? Is it the average value of the total spin in the paramagnetic system?

 

[kuruman]

Is it the average value of the total spin in the paramagnetic system?

Yes, $<\sigma>$ means the average value of the spin. The general equation for the average is
$$\left< M \right> = \mu_B N \frac{ \sum \sigma P(\sigma)}{\sum{ P(\sigma)}} $$
When the probability is normalized, the denominator is 1.