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## 物理代写|统计力学代写Statistical mechanics代考|Loop grand potential and derivatives

The independent harmonic oscillators analyzed in the preceding section belong to a general class of systems wherein the energy states are single-particle states. The occupancy statistics of such states has been analyzed by Pathria (1972, section 6.3), which treatment is largely followed here.

Equation (3.59) expresses the grand potential for single-particle state systems as a series of loop potentials. As mentioned above, following the analysis of singleparticle energy states in section 3.4.3, the loop grand potential, equation (3.58), is given by

$$-\beta \Omega_{l}^{\pm}=\frac{(\pm 1)^{l-1} z^{l}}{l} \sum_{n} e^{-\beta l e_{n}} .$$
Here $n$ labels a single-particle state, and $\varepsilon_{n}$ is its energy, which is the energy of a particle in that state. The fugacity is related to the chemical potential as $z=e^{\beta \mu}$. The grand potential itself is the sum of these loop potentials
\begin{aligned} -\beta \Omega^{\pm} &=\sum_{l=1}^{\infty} \frac{(\pm 1)^{l-1} z^{l}}{l} \sum_{n} e^{-l \beta e_{n}} \ &=\mp \sum_{n} \ln \left[1 \mp z e^{-\beta e_{n}}\right] . \end{aligned}
As derived in equation (4.26), the final equality is the expression for the grand potential of single-particle state systems more usually to be found in textbooks (Pathria 1972, section 6.2).

## 物理代写|统计力学代写Statistical mechanics代考|Occupancy fluctuations

It is of interest to analyze the fluctuations in the occupancy of the single-particle states. In general fluctuations are given by the second derivative of the grand potential (Pathria 1972, Attard 2002). In the present case we have (dropping the subscripts on the averages)
\begin{aligned} \left\langle N_{j}^{2}\right\rangle^{\pm}-\left\langle N_{j}\right\rangle^{\pm 2} &=\beta^{-2}\left[\frac{\partial^{2}\left(-\beta \Omega_{1}^{\pm}\right)}{\partial \varepsilon_{j}^{2}}\right]{z, V, e{k+j}} \ &=-\beta^{-1}\left[\frac{\partial\left\langle N_{j}\right\rangle^{\pm}}{\partial \varepsilon_{j}}\right]{z, V} \end{aligned} Hence the mean square relative fluctuation is \begin{aligned} \frac{\left\langle N{j}^{2}\right\rangle^{\pm}-\left\langle N_{j}\right\rangle^{\pm 2}}{\left\langle N_{j}\right\rangle^{\pm 2}} &=\frac{-1}{\beta\left\langle N_{j}\right\rangle^{\pm}}\left[\frac{\partial \ln \left\langle N_{j}\right\rangle^{\pm}}{\partial \varepsilon_{j}}\right]{z, V} \ &=\frac{1}{\left\langle N{j}\right\rangle^{\pm}} \pm \frac{1}{\left\langle N_{j}\right\rangle^{\pm}} \frac{z e^{-\beta \varepsilon_{j}}}{1 \mp z e^{-\beta \varepsilon_{j}}} \ &=\frac{1}{\left\langle N_{j}\right\rangle^{\pm}} \pm 1 \end{aligned}
The first term on the right-hand side is the classical result, (without the superscript $+$ ). At low occupancy, $\left\langle N_{j}\right\rangle^{\pm} \rightarrow 0$, this dominates, as is expected. It shows that the fluctuations in the occupancy of any single-particle state are relatively large in this regime, again as expected.

The quantum correction for bosons, $+1$, shows that the fluctuations are of the same order as the occupancy number itself in the high occupancy regime, $\mu \rightarrow \varepsilon_{j}^{-}$, $\left\langle N_{j}\right\rangle^{\pm} \rightarrow \infty$. In contrast, the quantum correction for fermions, $-1$, cancels with the classical contribution in the full occupancy regime, $\mu \gg \varepsilon_{j},\left\langle N_{j}\right\rangle^{-} \rightarrow 1$, which is to say that the fluctuations vanish. In this regime there is a high probability that the singleparticle state is always occupied by a fermion, and a correspondingly low probability that it is ever unoccupied.

# 统计力学代考

## 物理代写|统计力学代写Statistical mechanics代考|Loop grand potential and derivatives

$$-\beta \Omega_{l}^{\pm}=\frac{(\pm 1)^{l-1} z^{l}}{l} \sum_{n} e^{-\beta l e_{n}} .$$

$$-\beta \Omega^{\pm}=\sum_{l=1}^{\infty} \frac{(\pm 1)^{l-1} z^{l}}{l} \sum_{n} e^{-l \beta e_{n}} \quad=\mp \sum_{n} \ln \left[1 \mp z e^{-\beta e_{n}}\right] .$$

## 物理代写|统计力学代写Statistical mechanics代考|Occupancy fluctuations

$$\left\langle N_{j}^{2}\right\rangle^{\pm}-\left\langle N_{j}\right\rangle^{\pm 2}=\beta^{-2}\left[\frac{\partial^{2}\left(-\beta \Omega_{1}^{\pm}\right)}{\partial \varepsilon_{j}^{2}}\right] z, V, e k+j \quad=-\beta^{-1}\left[\frac{\partial\left\langle N_{j}\right\rangle^{\pm}}{\partial \varepsilon_{j}}\right] z, V$$

$$\frac{\left\langle N j^{2}\right\rangle^{\pm}-\left\langle N_{j}\right\rangle^{\pm 2}}{\left\langle N_{j}\right\rangle^{\pm 2}}=\frac{-1}{\beta\left\langle N_{j}\right\rangle^{\pm}}\left[\frac{\partial \ln \left\langle N_{j}\right\rangle^{\pm}}{\partial \varepsilon_{j}}\right] z, V \quad=\frac{1}{\langle N j\rangle^{\pm}} \pm \frac{1}{\left\langle N_{j}\right\rangle^{\pm}} \frac{z e^{-\beta \varepsilon_{j}}}{1 \mp z e^{-\beta \varepsilon_{j}}}=\frac{1}{\left\langle N_{j}\right\rangle^{\pm}} \pm 1$$

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