A special case of the grand canonical ensemble

In summary, the conversation discusses the possibility of using the definition of chemical potential and temperature in the grand partition function for calculating the average value of N in the case of an Einstein Solid. The grand partition function is given by equation (3) and the calculation of N is given by equation (4). Equations (5) and (6) provide the values for beta and alpha, respectively. Under certain conditions, equation (8) can be used to calculate the chemical potential. The conversation also mentions the neglect of ground state energy in the calculation of the canonical partition function and the assumption of distinguishable oscillators in equation (7). Additionally, equations (3), (4), and (5) come from M. Bellac's book.
  • #1
Ted Ali
12
1
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:
$$\mu = -kT\frac{q}{N} = \frac{kT}{1 - e^{hf/kT}}\hspace{1cm} (8)$$
 
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  • #2
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?
 
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What is a special case of the grand canonical ensemble?

A special case of the grand canonical ensemble is a statistical mechanics model that describes the behavior of a system in equilibrium with a reservoir of particles and energy, where the number of particles in the system is fixed and the temperature and chemical potential are allowed to fluctuate.

How is a special case of the grand canonical ensemble different from the general grand canonical ensemble?

In a special case of the grand canonical ensemble, the number of particles in the system is fixed, while in the general grand canonical ensemble, the number of particles is allowed to fluctuate. This means that the special case is more suitable for studying systems with a fixed number of particles, such as solids or condensed matter systems.

What types of systems can be described using a special case of the grand canonical ensemble?

A special case of the grand canonical ensemble can be used to describe systems that are in equilibrium with a reservoir of particles and energy, and have a fixed number of particles. This includes systems such as solids, liquids, and condensed matter systems.

How is the grand canonical ensemble related to other statistical mechanics ensembles?

The grand canonical ensemble is related to other statistical mechanics ensembles, such as the canonical ensemble and the microcanonical ensemble, through mathematical transformations. These ensembles are all used to describe the behavior of systems in equilibrium, but they differ in the parameters that are held constant.

What are the main applications of the special case of the grand canonical ensemble?

The special case of the grand canonical ensemble has many applications in the study of physical systems, particularly in condensed matter physics. It is often used to study phase transitions, critical phenomena, and the behavior of materials at different temperatures and chemical potentials.

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