Stars and bars (combinatorics): Difference between revisions

Revision as of 12:30, 5 June 2024

Revision as of 19:13, 16 June 2024

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The graphical method was used by [[Paul Ehrenfest]] and [[Heike Kamerlingh Onnes]]—with symbol '''ε''' (quantum energy element) in place of a star and the symbol '''0''' in place of a bar—as a simple derivation of [[Max Planck]]'s expression for the number of "complexions" for a system of "resonators" of a single frequency.<ref>{{cite journal |last1=Ehrenfest |first1=Paul |last2=Kamerlingh Onnes |first2=Heike |title=Simplified deduction of the formula from the theory of combinations which Planck uses as the basis of his radiation theory | journal = Proceedings of the KNAW | volume=17 | date = 1914 | pages = 870–873 | url = https://dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publDetail&pId=PU00012735 | access-date = 16 May 2024}}</ref><ref>{{cite journal |last1=Ehrenfest |first1=Paul |last2=Kamerlingh Onnes |first2=Heike |title=Simplified deduction of the formula from the theory of combinations which Planck uses as the basis of his radiation theory |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |series=Series 6 |date=1915 |volume=29 |issue=170 |pages=297–301 |doi=10.1080/14786440208635308 |url=http://dx.doi.org/10.1080/14786440208635308 |access-date=5 December 2020}}</ref>

By complexions ([[Microstate (statistical mechanics)|microstates]]) Planck meant distributions of {{mvar|P}} energy elements '''ε''' over {{mvar|N}} resonators.<ref>{{cite journal |last1=Planck |first1=Max |title=Ueber das Gesetz der Energieverteilung im Normalspectrum |journal=Annalen der Physik |date=1901 |volume=309 |issue=3 |pages=553–563 |doi=10.1002/andp.19013090310 |bibcode=1901AnP...309..553P |doi-access=free }}</ref><ref>{{cite journal | last = Gearhart | first = C. | title = Planck, the Quantum, and the Historians | journal = Phys. perspectPerspect. | volume = 4 |pages = 170–215 | date = 2002 | issue = 2 |doi = 10.1007/s00016-002-8363-7 | bibcode = 2002PhP.....4..170G | url = https://employees.csbsju.edu/cgearhart/planck/pqh.pdf | access-date = 16 May 2024}}</ref> The number {{mvar|R}} of complexions is

:<math>R=\frac {(N+P-1)!}{P!(N-1)!}. \ </math>