A special case of the grand canonical ensemble

[Ted Ali]

Homework Statement:

Can we consider the internal energy $U$ constant and only allow for the number of particles $N$ to vary, in the grand canonical ensemble?

Relevant Equations:

$$\mu = -T \left(\frac{\partial S}{\partial N}\right)_{U,V} \hspace{1cm} (1)$$
$$\frac{1}{T} = \left(\frac{\partial S}{\partial U} \right) \hspace{1cm} (2)$$

In addition to the homework statement and considering only the case where $U= constant$ and $N = large$ : Can we also consider the definition of chemical potential $\mu$ and temperature $T$ as in equations $(1)$ and $(2)$, and use them in the grand partition function?

More specifically, we can take the case of an Einstein Solid and the Schroeder's definition of internal energy $U = qhf$. Assuming that $U$ is depending only on a constant number of energy quanta $q$ and allowing only $N$ to vary. Can we use the grand partition function and $(1), (2)$, for calculating the average value of $N$? In this case of an Einstein Solid, we also assume that $N \gg q$.

The grand partition function is given by: $$Q_{(\alpha, \beta)} = \sum_{N=0}^{\infty} e^{\alpha N} Z_{N}(\alpha, \beta) \hspace{1cm} (3)$$
And: $$\bar{N} =\left( \frac{\partial\ln{Q}}{\partial \alpha}\right)_\beta \hspace{1cm} (4)$$
Where: $$\beta = \frac{1}{kT} \hspace{1cm} \alpha = \frac{\mu}{kT} \hspace{1cm} (5)$$
$$Z_1 = 1/ (1 - e^{-\frac{hf}{kT}})\hspace{1cm}(6)$$
And: $$Z_N = (Z_1)^N \hspace{1cm}(7)$$
Under the above conditions and equations:
$$\mu = -kT\frac{q}{N} = \frac{kT}{1 - e^{hf/kT}}\hspace{1cm} (8)$$

 

[Ted Ali]

Temperature $T$ is given by equation $(2)$, in the microcanonical ensemble and calculated in Wikipedia (https://en.wikipedia.org/wiki/Einstein_solid). The final result is: $$\frac{q}{N} = \frac{1}{e^{hf/kT} - 1}\hspace{1cm} (9)$$
The chemical potential $\mu$ is given by $(1)$, when $U, V$ are held constant. The calculation of $(1)$ can be found in the solutions manual of D. Schroeder's book: "An Introduction to Thermal Physics". The final result is $$\mu = -kT\ln(1+ \frac{q}{N})\hspace{1cm} (10)$$ (exercise 3.36).
In the case we examine it is assumed that $N \gg q$ so $(10)$ becomes: $$\mu = -kT\frac{q}{N}\hspace{1cm} (11)$$
From $(9)$ and $(11)$ we have equation $(8)$, in its final form.
Finally let's comment that in $(6)$, we have neglected the ground state energy of each one quantum harmonic oscillator of the Einstein solid, in the calculation of the canonical partition function (https://en.wikipedia.org/wiki/Einstein_solid). And that in $(7)$ we have assumed that the oscillators are distinguishable.
Equations $(3), (4), (5)$ come from M. Bellac's book: "Equilibrium and non-Equilibrium Statistical Thermodynamics", pg. 148.
So the "sum" of the questions still remains open: Can we use $(8)$ in the grand partition function?