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For the 1D Ising model with the Hamiltonian

$$H=const.-\mu h' \sum_i S_i^z$$

we can write the canonical partition sum as

$$Z_N = \sum_{ \{ S_i^z \}_N } e^{-\beta \mu h \sum_i S^z_i} = \sum_{ \{ S_i^z \}_N } \prod_i e^{-\beta \mu h S^z_i}$$

for which we then later used

$$Z_N=(Z_1)^N$$with the single particle partition sum $Z_1$

We didn't go further into the proof during the lecture, and it's not immediately obvious to me.

My attempt:

For the classical ideal gas I could use the Multinomial Theorem

$$(x_1+x_2+\dots+x_n)^k=\sum_{k_1+\dots+k_n=k} \frac{k!}{k_1!\dots k_n!} \ x_1^{k_1}x_2^{k_2} \dots x_n^{k_n}$$

but for that the factor $\frac{k!}{k_1!\dots k_n!}$ would have to be $1$.

How do I best prove $Z_N=(Z_1)^N $?

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In fact, $Z_N = (Z_1)^N$ is true for all non-interacting systems with identical components, where the Hamiltonian can be written as $H = H_1 + ... + H_N$ where $H_k$ only depends on the state of the $k$-th particle (/spin/...) and all $H_k$ are the same.

This can be seen like this.

$$Z_N = \sum_{states \ s_1...s_N} e^{- \beta H} = \sum_{s_1} \sum_{s_2} ...\sum_{s_N} e^{- \beta (H_1+...+H_N)}= \sum_{s_1} e^{- \beta (H_1)} \sum_{s_2} e^{- \beta (H_2)}...\sum_{s_N} e^{- \beta (H_N)} = ( \sum_{s_1} e^{-\beta H_1})^N = Z_1^N$$

Note: For identical particles, a factor $1/N!$ might additionally be necessary.

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First of all, the last manipulation of the partition sum is not correct. You cannot pull out a global factor $e^{-\beta\mu h}$.

To give you a hint: It might be helpful to write $$ \sum_{\{S_i\}} = \prod_i\sum_{S_i=\pm 1} = \sum_{S_1=\pm 1}\sum_{S_2=\pm 1}\cdots\sum_{S_N=\pm 1} $$ and exploit that the Boltzmann factor factorizes.

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