Moore and Schroeder proposed an effective approach to introducing entropy and the second law through computational study of models with easily countable states at fixed energy. However, such systems are rare: the only familiar examples are the Einstein solid and the two-state paramagnet, which limits the available questions for assignment or discussion. This work considers the more general p-state paramagnet and describes the modestly more complicated counting of its microstates. An instructor can draw on this family of systems to assign a variety of new problems or open-ended projects that students can complete with the help of a spreadsheet program or analytic calculation.

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Because of the horizontal scaling in the top panel of Fig. 1, the higher-p graphs have denser data points: this leads to lower multiplicity in any given data point and thus to lower peak heights, which can be visually misleading. So rather than directly graphing the relative multiplicity Ω / Ω total for each state, this graph shows a fractional density of states, normalizing by the (relative) size of the energy steps for each system. This makes the graphs easy to compare by ensuring that the area under each curve equals one.
11.
It is important when using Excel to program multiplicity formulas using the =COMBIN(N,q) function for the binomial coefficients rather than explicitly using factorials, because that function in Excel can handle larger arguments than the factorials can: it is coded to perform the cancellation of terms between numerator and denominator before evaluating the result. Some spreadsheet programs other than Excel do not implement =COMBIN(N,q) carefully in this way, and are limited to smaller systems as a result.
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See supplementary material at https://www.scitation.org/doi/suppl/10.1119/5.0061383 for the sample code.
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Keeping the next-leading term from Neuschel's expansion is equivalent to including the 1 / 12 N terms in Stirling's approximation for N!, as can be verified explicitly for the p = 2 case.
16.
For the p = 3 case, Eq. (4) can be rewritten in terms of a hypergeometric function whose N limit falls into a class studied by Cvitković et al. (Ref. 17). However, the challenging calculation involved leads to exactly the same result found (far more easily) with Stirling's approximation for p = 2.
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Supplementary Material

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