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

Next we want to show that [itex]\left< M \right>[/itex] 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 [itex]\sigma_1[/itex](+1 or -1):

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

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

I don't understand the last part of the section. Why is that [itex]\sum_{\sigma = \pm1} \sigma P(\sigma)[/itex] equals to [itex]\left< \sigma \right>[/itex]? And what does [itex]\left< \sigma \right>[/itex] 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.