SM:Parastatistics, grand canonical ensemble function

In summary, the conversation discusses a question regarding the grand canonical ensemble and the calculation of <N> and <n_r>. The individual steps for calculating these values are shown, but there are questions about convergence and the validity of the expressions, particularly for bosons. The conversation also mentions the recovery of the fermion distribution, which is found to be correct. The question is expected to verify the final, simplified expression for <N> and <n_r>.
  • #1
binbagsss
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Homework Statement


Hi
I am looking at the question attached.

parastatistics.jpg


Parts c and d, see below

Homework Equations

The Attempt at a Solution

First of all showing that ##<N> ## and ##<n_r>## agree

I have ##Z=\Pi_r z_r ##, where ##Z## here denotes the grand canonical ensemble.
So therefore we have ## \log(Z) = \sum\limits_r log (z) ## and since ##<N>=\frac{1}{\beta}\frac{\partial}{\partial \mu} \log Z ## the calculation seems pretty trivial unless there are some subtleties I am missing. The only subtlety I can think of however is the requirement of convergence such that the derivative can be taken inside the summation and then the derivative of the summation= sum of the derivative of each term, please see below.

So I have:
##z_r= \sum\limits_{n_r=0}^{n_r=p} x_r^{n_r } ##
so ## <n_r> =\frac{1}{\beta}\frac{\partial}{\partial \mu} \log ( \sum\limits_{n_r=0}^{n_r=p} x_r^{n_r } ) = \frac{1}{\beta} \frac{\sum\limits_{n_r=0}^{n_r=p} n_r \beta x_r^{n_r }}{\sum\limits_{n_r=0}^{n_r=p} x_r^{n_r } } ## (1)

And

##<N>= \frac{1}{\beta}\frac{\partial}{\partial \mu} \sum\limits_r \log ( \sum\limits_{n_r=0}^{n_r=p} x_r^{n_r } ) = \frac{1}{\beta} \sum\limits_r \frac{\partial}{\partial \mu} \log ( \sum\limits_{n_r=0}^{n_r=p} x_r^{n_r } ) = \frac{1}{\beta} \sum\limits_r \frac{\sum\limits_{n_r=0}^{n_r=p} n_r \beta x_r^{n_r }}{\sum\limits_{n_r=0}^{n_r=p} x_r^{n_r } } ## (2)

Where the 2nd equality has assumed convergence over the summation ## \sum\limits_r ##.

QUESTION 1- 2ND EQUALITY USING CONVERGENCE
So the question asks to use (1) and ## <N>= \sum\limits_r <n_r> ## to show that (2) is consistent with the latter. It is clear that the two agree. HOWEVER, I am unsure how to justify convergence over the summation of ##r##. Here ##r## is a single particle state, as far as I'm aware, (although most likely physically unlikely) this can be infinite, an ##n_r ## is the number of particles occupying a single ## r ## state. And so the summation over ##r## is infinite. If I ##E_r > \mu ## then this is an infinite summation of the logarithm of a summation which I know converges...but I don't think this really helps me out? (since i have ## \sum\limits_r \log (\sum\limits_{n_r} x_r^{n_r }) ## where ##x_r = \exp^{-\beta(E_r-\mu)}##).

QUESTION 2- UNABLE TO USE GENERAL EXPRESSION TO RECOVER BOSON DISTRUBUTION
so for fermions ##p=1## for bosons ##p=\infty##.
Looking at (1), the numerator multiplies ##n_r ## (dropped down from the exponentiial derivative) and so clearly when ##n_r=\infty ## this expression is dodgy/invalid- I am unsure what is wrong, however, with my above working.
The best I can seem to do is to recover the boson distribution from the expression at an earlier step, more generalised, which was:

## \frac{1}{\beta} \frac{\partial}{\partial \mu}( \log (\frac{1}{1-x_r})) ## it its easy to show that I attain ## \frac{1}{x_r^{-1}-1} ## as expected. (where I have use infinite summation formula for this and the assumption ## E_r > \mu ##
However I imagine the question wants you to be verifying your final, simplified expression. I am unsure what is wrong with (1)...the fermion distribution recovery is ok:
## <n_r> = \frac{1}{\beta} \frac{\sum\limits_{n_r=0}^{n_r=1} n_r \beta x_r^{n_r }}{\sum\limits_{n_r=0}^{n_r=1} x_r^{n_r } } ##
this gives:
## \frac{1}{\beta}( 0 + \frac{\beta x_r}{1+x_r} )= \frac{1}{x_r^{-1}+1}## as expected.

Many thanks in advance
 

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  • #2
I can't say for sure, but I suspect that for part (b) you are expected to evaluate the sum in ##z_r= \sum\limits_{n_r=0}^{n_r=p} x_r^{n_r } ## and get a fairly simple result for ##z_r## in terms of ##x_r## and ##p##. Then you can use this in later parts of the problem.
 
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FAQ: SM:Parastatistics, grand canonical ensemble function

1. What is SM:Parastatistics?

SM:Parastatistics is a statistical model used to describe the behavior of systems with particles that are indistinguishable from one another.

2. How does the grand canonical ensemble function in SM:Parastatistics?

The grand canonical ensemble function in SM:Parastatistics is used to describe the distribution of particles in a system with a fixed chemical potential and temperature. It takes into account the possibility of particles entering and leaving the system, unlike the canonical ensemble which assumes a fixed number of particles.

3. What is the difference between SM:Parastatistics and Bose-Einstein statistics?

SM:Parastatistics and Bose-Einstein statistics both apply to systems with indistinguishable particles, but they differ in their treatment of the particles' exchange symmetry. SM:Parastatistics allows for any exchange symmetry, while Bose-Einstein statistics only allows for symmetric exchange.

4. How is SM:Parastatistics related to quantum mechanics?

SM:Parastatistics is closely related to quantum mechanics as it describes the behavior of systems with identical particles, which is a key concept in quantum mechanics. It also takes into account the indistinguishability of particles, which is a fundamental principle in quantum mechanics.

5. What are some applications of SM:Parastatistics?

SM:Parastatistics has been used to study a wide range of systems, including gases, liquids, and solids. It has also been applied in various fields such as quantum physics, thermodynamics, and statistical mechanics. Additionally, it has been used in the development of new materials and technologies, such as in the study of superconductivity and the behavior of quantum dots.

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