I Can the Exact Boltzmann Distribution Yield Specific Quantum State Populations?

AI Thread Summary
The discussion centers on the application of the exact Boltzmann distribution to derive specific quantum state populations, questioning the feasibility of obtaining integer occupancy numbers. The user expresses difficulty in achieving integer solutions for occupancy numbers given certain parameters, suggesting that an additional constraint may be necessary. It is noted that non-integer occupancy numbers might represent time-averaged states rather than actual physical configurations. The conversation highlights the complexity of defining a physical configuration in a dynamic system and the limitations of relying solely on mathematical functions like the Gamma function. Ultimately, the need for a method to derive integral solutions for occupancy numbers is emphasized, alongside the utility of non-integer averages for practical calculations.
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Exact Boltzmann distribution
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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?
 
There is no reason for the ni*s to be integral. n is treated as a continuous variable to permit differentiation. (See section D.) The ni*s are the values that maximise the probability function. Of course non-integral nis 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 nis) with the highest probability, other than trial and error. Do you need to do this?
 
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
 
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 ni are very small compared to ni.) 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".)
 
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