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Open AccessArticle

Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box

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Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
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Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Via P. Giura 1, I-10125 Torino, Italy
*
Author to whom correspondence should be addressed.
Entropy 2018, 20(9), 645; https://doi.org/10.3390/e20090645
Received: 25 June 2018 / Revised: 24 August 2018 / Accepted: 24 August 2018 / Published: 28 August 2018
From basic principles, we review some fundamentals of entropy calculations, some of which are implicit in the literature. We mainly deal with microcanonical ensembles to effectively compare the counting of states in continuous and discrete settings. When dealing with non-interacting elements, this effectively reduces the calculation of the microcanonical entropy to counting the number of certain partitions, or compositions of a number. This is true in the literal sense, when quantization is assumed, even in the classical limit. Thus, we build on a moderately dated, ingenuous mathematical work of Haselgrove and Temperley on counting the partitions of an arbitrarily large positive integer into a fixed (but still large) number of summands, and show that it allows us to exactly calculate the low energy/temperature entropy of a one-dimensional Bose–Einstein gas in a box. Next, aided by the asymptotic analysis of the number of compositions of an integer as the sum of three squares, we estimate the entropy of the three-dimensional problem. For each selection of the total energy, there is a very sharp optimal number of particles to realize that energy. Therefore, the entropy is ‘large’ and almost independent of the particles, when the particles exceed that number. This number scales as the energy to the power of ( 2 / 3 ) -rds in one dimension, and ( 3 / 5 ) -ths in three dimensions. In the one-dimensional case, the threshold approaches zero temperature in the thermodynamic limit, but it is finite for mesoscopic systems. Below that value, we studied the intermediate stage, before the number of particles becomes a strong limiting factor for entropy optimization. We apply the results of moments of partitions of Coons and Kirsten to calculate the relative fluctuations of the ground state and excited states occupation numbers. At much lower temperatures than threshold, they vanish in all dimensions. We briefly review some of the same results in the grand canonical ensemble to show to what extents they differ. View Full-Text
Keywords: time correlation functions; time reversal; parity; quantum symmetry transformations time correlation functions; time reversal; parity; quantum symmetry transformations
MDPI and ACS Style

De Gregorio, P.; Rondoni, L. Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box. Entropy 2018, 20, 645.

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