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## Homework Statement

Hi guys,

I'm going nuts on a problem. Out of memory the problem was a system of N pairs of atoms that interact between each others so that the Hamiltonian of the whole systeme is ##H=-J \sum _{n=1}^N \sigma _i \tau _i## where the only possible values of the ##\tau _i##'s and ##\sigma _i##'s are -1 and 1.

1)Calculate ##\langle \sigma _i \tau _i \rangle## (I think this is called the correlation function, since it seems to be the correlation between the spins of 2 adjacent pair of atoms or so).

2)Calculate ##\langle \varepsilon \rangle##. The problem goes on but I don't remember the next questions.

## Homework Equations

Mean value of A: ##\frac{\sum _{\text{over all states}} \exp (-\beta H) \cdot A }{Z}## where Z is the partition function of the system.

## The Attempt at a Solution

I am stuck on part 1). I know the answer is ##\tanh (\beta J)## but I'm unable to show it.

First I notice that the partition function ##Z=Z_1^N##, in other words it is the partition function of a single pair of spins to the N'th power.

Where Z_1=##\sum _{\text{over all states}} \exp (-\beta H)=\sum _{\text{over all states}} \exp (\beta J \sum _{i=1}^N \sigma _i \tau _i )##. Stuck here, I don't know how to write out these sums.

Now maybe the numerator of ##

\frac{\sum _{\text{over all states}} \exp (-\beta H) \cdot \sigma _i \tau _i }{Z}

## is worth the derivative of the denominator with respect to ##\beta J## and if Z is worth ##\cosh (\beta J)## then I would be done with the calculations but I am really not able to show that Z is indeed worth cosh(beta J). Also I am not even sure that this is true because I have that Z is Z_1 to the N'th power and I see no way how to obtain a hyperbolic cosine.

Thanks for any tip.

Edit: Ok I considered N=3, I know how to write the terms of the sum for the partition function but there are already a lot of them. I really don't see how to reach the result.

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