Can the Exact Boltzmann Distribution Yield Specific Quantum State Populations?

[rabbed]

TL;DR Summary

Exact Boltzmann distribution

Hi

With the exact Boltzmann distribution, ni = InverseDigamma(-α-β*εi)-1:
https://studyres.com/doc/269738/revision-of-boltzmann-statistics-for-a-finite-number-of-p...

Shouldn't I be able to get (n0, n1, n2, n3, n4, n5, n6, n7) = (6, 3, 2, 0, 0, 0, 0, 0) for some α and β, if N=11, E=7 and Δε=1?
It doesn't seem possible.

 

[rabbed]

I think the issue is that there should really be another constraint - that the ni’s should be integers. Otherwise the answers won’t make much physical sense (non-integer occupancy numbers). We’re relying on the Digamma function coming from a Gamma function which can be defined in multiple ways as long as its integer inputs give integer outputs..
Maybe the Gamma function can be redefined so that we get integer solutions for the occupancy numbers?

 

[mjc123]

There is no reason for the n_i*s to be integral. n is treated as a continuous variable to permit differentiation. (See section D.) The n_i*s are the values that maximise the probability function. Of course non-integral n_is don't describe an actual physical configuration. If they have a physical meaning, I suggest it is as a time-average, as molecules are constantly being bumped up and down between levels. I'm not sure there's a simple way to determine the physical configuration (integral n_is) with the highest probability, other than trial and error. Do you need to do this?

 

[rabbed]

Thank you for answering!
Seen as a strictly mathematical problem (N distinct balls in K distinct boxes labeled with increasing integer scores) there is an actual right answer/distribution with a probability, so why wouldn’t we want that instead of relying on some random answer that depend on how the gamma function was derived when there in fact are several correct ”Gamma”-functions which extends the factorial? See: http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html

 

[mjc123]

I suppose the issue is, at least partly, that in a dynamic system the configuration is constantly changing. Each configuration has a probability, and one of them is the most probable, but none is "the configuration" of the system. (Unlike the Boltzmann distribution for large N, where fluctuations in n_i are very small compared to n_i.) But you can define a time-average, with non-integral average occupancy numbers, which is useful for calculating things like heat capacity, as in the paper. (I guess for the physicist it isn't "seen as a strictly mathematical problem".)

 

[hutchphd]

As a lousy mathematician I find the wikipedia article to be useful
https://en.wikipedia.org/wiki/Digamma_function
(also Abramowiitz and Stegun )
Can't you show this using the recursion relation directly?