How to find the partition function of the 1D Ising model?

AI Thread Summary
The discussion centers on deriving the partition function for the 1D Ising model, starting from the expression involving binomial coefficients and exponential terms. The initial derivation leads to the result Z = (2 cosh(βJ))^N, but there is confusion regarding the factor of 2 compared to the expected answer of Z = (cosh(βJ))^N. Participants are questioning whether the provided answer is incorrect or if additional steps are needed to reconcile the discrepancy. Clarification is sought on how to eliminate the factor of 2 from the final expression. The conversation highlights the importance of accurately interpreting the derivation steps in statistical mechanics.
Dom Tesilbirth
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Homework Statement
Consider a one-dimensional Ising model with ##N## spins at very low temperature. Let there be ##r## spin flips with each costing energy ##2J##. The total energy of the system is ##E=-NJ+2rJ## and the number of configurations is ##C(N, r)##, where ##r## varies from ##0## to ##N##. Find the partition function.
Relevant Equations
##E=-NJ+2rJ## and
##Z=\sum ^{N}_{r=0}C\left( N,r\right) e^{-\beta \left[ -NJ+2rJ\right] }##
Attempt at a solution:

\begin{aligned}Z=\sum ^{N}_{r=0}C\left( N,r\right) e^{-\beta \left[ -NJ+2rJ\right] }\\
\Rightarrow Z=e^{\beta NJ}\sum ^{N}_{r=0}C\left( N,r\right) e^{-2\beta rJ}\end{aligned}

Let ##e^{-2\beta J}=x##. Then ##e^{-2\beta rJ}=x^{r}##.

\begin{aligned}\therefore Z=e^{\beta NJ}\sum ^{N}_{r=0}C(N, r)x^{r}\\
\Rightarrow Z=e^{\beta NJ}\left( 1+x\right) ^{N}=\left( e^{\beta J}+e^{\beta J}e^{-2\beta J}\right) ^{N}\\
\Rightarrow Z=\left( e^{\beta J}+e^{-\beta J}\right) ^{N}\\
\Rightarrow Z=\left( 2\cosh\beta J\right) ^{N}\end{aligned}

However, the answer provided is ##Z=\left(\cosh \beta J\right) ^{N}##. How do we remove the factor ##2##? Was the given answer wrong, or is there something that I still need to do?
 
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Dom Tesilbirth said:
How do we remove the factor ##2##? Was the given answer wrong, or is there something that I still need to do?
Factor 2 seems to be right.

ref. https://en.wikipedia.org/wiki/Ising_model
 
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