Arithmetic mean Fermi Dirac & Bose Einstein

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Discussion Overview

The discussion revolves around the relationship between the arithmetic mean of Fermi-Dirac (FD) and Bose-Einstein (BE) distributions and the Maxwell-Boltzmann (MB) distribution for indistinguishable particles. Participants explore the implications of this relationship across different temperature regimes and the underlying theoretical principles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asserts that the arithmetic mean of FD and BE distributions equals the MB distribution for indistinguishable particles, seeking an explanation for this phenomenon.
  • Another participant suggests that this finding indicates that quantum mechanical averaged values behave classically, noting the classical analogue of the distributions.
  • A different participant questions whether this holds true for all temperatures, highlighting that Fermi-Dirac distributions account for energies that MB distributions do not at low temperatures.
  • One participant mentions their limited theoretical background, explaining their analysis was based on a simple system with two particles and three energy states, confirming their calculations without deep theoretical understanding.
  • Another participant expresses skepticism about the validity of the relationship across all temperatures, although they acknowledge it may hold in the limit of high temperatures.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the relationship across all temperature ranges, indicating a lack of consensus on this point.

Contextual Notes

Some assumptions regarding temperature limits and the nature of the systems being discussed may not be fully articulated, leading to potential gaps in understanding the broader implications of the findings.

Frank Einstein
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Hi everybody, I was doing one asignment form class, I was tasked to prove that in one system, the arimetic mean of FD and BE distributions is equal to MB's distribution for undishtingable particles.
After doing the numbers I found out that it actually was, but I don't know why this happens, can someone explain this to me or point me to a place where I can learn that?
Thanks.
 
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I think this is a demonstration that the quantum mechanical averaged values behave classically.
Mind that the classical analogue of the Bose Einstein/Fermi Dirac distributions is the Maxwell Boltzmann distribution (give the number density as a function of the energy of the system).
 
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For all temperatures? That would surprise me. Fermi-Dirac includes particles at an energy way above the energies Maxwell-Boltzmann at low temperatures would give, and I don't see how MB would reproduce the step coming from Fermi-Dirac at the Fermi energy.
 
Well, to be honest I have not advanced that much in theory lessons. All I know is that I have a system with two particles and three possible enrgies for them and I had to study the partiton function for each of the three distributions, I checked that the math were right and saw that it acturally worked so I asked the quastion.

By the way thank you very much for your anwsers.
 
Ah, for only two particles it might look different.
I'm not convinced that it works out for all temperatures, however. In the limit of large temperature: sure.
 
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