Where Is the Error in My Understanding of the Boltzmann Distribution?

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Discussion Overview

The discussion revolves around the Boltzmann distribution and the Maxwell-Boltzmann distribution, specifically addressing the expected proportionality of the distribution to energy (E) and the factors influencing this relationship. Participants explore the mathematical derivation and integration measures involved in these distributions.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions their understanding of the proportionality in the Maxwell-Boltzmann distribution, expecting it to be proportional to E^-1/2 and exp(-E), contrary to the formula presented in a reference.
  • Another participant asks for clarification on the expectation of proportionality to E^-1/2 and requests the full expression for the Maxwell distribution.
  • A participant explains their reasoning that relates momentum (p) to energy (E) and suggests that the derivative of momentum with respect to energy leads to the expected proportionality.
  • One participant proposes that the distribution used in the calculation is over the magnitude of momentum rather than its components, leading to an additional factor that alters the expected proportionality.
  • A later reply reiterates the previous point about the distribution of momentum and discusses the implications of integrating over spherical coordinates, which introduces an extra factor affecting the relationship with energy.
  • The same participant reflects on the distribution of potential energy in a gas and questions whether the absence of the E^1/2 factor in that context is appropriate or if a better comparison exists.

Areas of Agreement / Disagreement

Participants express differing views on the expected proportionality in the Maxwell-Boltzmann distribution and the factors influencing it. There is no consensus on the correct interpretation of the distribution or the implications of the additional factors discussed.

Contextual Notes

The discussion includes assumptions about the integration measures and the dimensionality of the distributions, which may not be fully resolved. The implications of these assumptions on the expected relationships are not conclusively established.

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Why do you expect proportionality to E^-1/2 ? What would be your full expression for the Maxwell distribution and why?
 
I expect that p^2/(2m)=E, so p=(2Em)^1/2 so
dp/dE = proportional to E^-1/2
Maybe px, py, pz, demand a different formula?
 
I suspect the following, although i haven't really analyzed it in detail:

The distribution f_p used in that calculation is the distribution over the magnitude of p, not the distribution over the three components px, py, pz. So there is an extra factor p^2 due to the integration measure p^2 dp dphi dtheta, and then you integrates over phi/theta. The extra factor p^2 causes an extra factor E , that turns your E^-1/2 into the E^1/2 in the formula.
 
torquil said:
I suspect the following, although i haven't really analyzed it in detail:

The distribution f_p used in that calculation is the distribution over the magnitude of p, not the distribution over the three components px, py, pz. So there is an extra factor p^2 due to the integration measure p^2 dp dphi dtheta, and then you integrates over phi/theta. The extra factor p^2 causes an extra factor E , that turns your E^-1/2 into the E^1/2 in the formula.
Yes, now I see that it is so. Above in my link is calculation with distrubution of speed v. If I use dn/dv and dv/dE, and f(v) the above is clear.

It was also unclear to me, that distribution of potential energy of gas in vertical tube has not factor E^1/2. Because this factor is a consequence of three dimensions.
Is it OK comparision, or it should be something better for a distribution without additional factor E^1/2?
 

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