Expectation value of the occupation number in the FD and BE distributions

In summary: Therefore, the first equation can be deduced from the given relations.In summary, the conversation discusses the computation of the Grand Partition Function and its relation to the expectation value of the occupation number. The first equation, which is provided in the book being read, can be derived from the given relations.
  • #1
Lebnm
31
1
In the derivation of the Fermi-Dirac and Bose-Einstein distributions, we compute the Grand Partion Function ##Q##. With ##Q##, we can compute the espection value of the occupation number ##n_{l}##. This is the number of particles in the same energy level ##\varepsilon _{l}##. The book I am reading write $$\langle n_{l} \rangle = - \frac{1}{\beta} \frac{\partial \ }{\partial \varepsilon _{l}}\ ln \ Q,$$ but the book dosen't deduce this equation. I know that the expection value of the number operator ##\hat{N}## is given by $$\langle \hat{N} \rangle = z \frac{\partial \ }{\partial z}\ ln \ Q,$$ where ##z = e^{\beta \mu}## ( ##\mu## is the chemical potential, and I am using the Grand Canonical Ensemble), and that the eigenvalues of ##\hat{N}## are ##N = \sum _{l} n_{l}##. How the first equation follow from these relations?
 
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  • #2
The grand partition function is given by
$$
Q = \sum_l e^{- \beta n_l (\varepsilon_l - \mu)}
$$
therefore
$$
\begin{align*}
-\frac{1}{\beta} \frac{\partial}{\partial \varepsilon_l} \ln Q &= -\frac{1}{\beta} \left[ \frac{1}{Q} \frac{\partial Q}{\partial \varepsilon_l} \right]
\\
&= -\frac{1}{\beta} \left[ \frac{1}{Q} \sum_l (- \beta n_l ) e^{- \beta n_l (\varepsilon_l - \mu)} \right] \\
&= \frac{1}{Q} \sum_l n_l e^{- \beta n_l (\varepsilon_l - \mu)} \\
&= \langle n_l \rangle
\end{align*}
$$
where the last equality is obtained from the formula for calculating expectation values.
 
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Related to Expectation value of the occupation number in the FD and BE distributions

What is the expectation value of the occupation number in the Fermi-Dirac (FD) distribution?

The expectation value of the occupation number in the FD distribution is the average number of particles occupying a specific energy state at a given temperature. It takes into account the probability of each energy state being occupied by a particle, weighted by the Fermi-Dirac distribution function.

What is the expectation value of the occupation number in the Bose-Einstein (BE) distribution?

The expectation value of the occupation number in the BE distribution is the average number of particles occupying a specific energy state at a given temperature. It takes into account the probability of each energy state being occupied by a particle, weighted by the Bose-Einstein distribution function.

How is the expectation value of the occupation number calculated in the FD and BE distributions?

The expectation value of the occupation number is calculated by summing the product of the probability of each energy state being occupied by a particle and the number of particles in that state, for all energy states. In the FD distribution, this sum is limited to energy states below the Fermi energy, while in the BE distribution, it includes all energy states.

What is the significance of the expectation value of the occupation number in the FD and BE distributions?

The expectation value of the occupation number is a key quantity in understanding the behavior of particles in a system at a given temperature. It provides information about the average number of particles occupying each energy state and is used to calculate other important thermodynamic quantities such as the specific heat and entropy.

How does the expectation value of the occupation number change with temperature in the FD and BE distributions?

In the FD distribution, the expectation value of the occupation number increases with temperature, as more energy states become available for particles to occupy. In the BE distribution, the expectation value decreases with temperature, as more particles occupy the lowest energy state due to Bose-Einstein condensation.

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