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Partition Function
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When estimating the average energy, it becomes evident that there exists a generative function with which various parameters can be calculated. This function is known as the partition function and serves as the foundation for computing properties of diverse systems.
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Average energy
Equation
To compute the average energy, we use the weighted average of energies from various states $r$, taking into account their respective probabilities, as represented by
| $P_r=Ce^{-\beta E_r}$ |
This is done in the following manner:
$\bar{E}=\displaystyle\frac{\displaystyle\sum_rP_rE_r}{\displaystyle\sum_rP_r}$
The result is obtained by considering the values of
| $\bar{E}=\displaystyle\frac{\displaystyle\sum_rE_re^{-\beta E_r}}{\displaystyle\sum_re^{-\beta E_r}}$ |
.
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Partition function
Equation
The average energy is determined with respect to beta del sistema $1/J$, energía del estado $r$ $J$, energía media del sistema $J$ and numero del estado $-$
| $\bar{E}=\displaystyle\frac{\displaystyle\sum_rE_re^{-\beta E_r}}{\displaystyle\sum_re^{-\beta E_r}}$ |
and can be expressed as follows:
$\bar{E}=-\displaystyle\frac{1}{\sum_re^{-\beta E_r}}\displaystyle\frac{\partial}{\partial\beta}\sum_re^{-\beta E_r}$
This can be summarized as
$\bar{E}=-\displaystyle\frac{1}{Z}\displaystyle\frac{\partial Z}{\partial\beta}$
where we introduce the so-called partition function with beta del sistema $1/J$, energía del estado $r$ $J$, energía media del sistema $J$ and numero del estado $-$:
| $Z=\displaystyle\sum_Re^{-\beta E_R}$ |
The letter $Z$ originates from the German word Zustandsumme (Zustand=State, Summe=sum).
The partition function is a generating function, meaning it generates other functions that have physical significance.
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Average Energy calculated with the partition function
Equation
As it is evident that
$\displaystyle\frac{\partial\ln Z}{\partial\beta} =\displaystyle\frac{1}{Z}\displaystyle\frac{\partial Z}{\partial\beta}$
and
$\bar{E}=-\displaystyle\frac{1}{Z}\displaystyle\frac{\partial Z}{\partial\beta}$
this implies that, with
| $\bar{E}=-\displaystyle\frac{\partial\ln Z}{\partial\beta}$ |
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Video: Partition Function