H problem in statistical mechanics

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Discussion Overview

The discussion revolves around the Hydrogen atom problem in statistical mechanics, specifically addressing the divergence of the canonical partition function and its implications for the statistical treatment of the Hydrogen atom. Participants explore various approaches to understanding this issue, including energy calculations, regularization techniques, and the effects of the Coulomb potential.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants express concern that the canonical partition function diverges for the Hydrogen atom due to the energy levels being given by E_n = -E_0/n^2 with degeneracy n^2.
  • One participant suggests that calculating energies with respect to the ground state resolves the issue, but others argue that this merely shifts the problem without resolving the divergence.
  • Another participant points out that simple regularization does not help, as summing up to a finite state N leads to a mean energy approaching zero, implying all Hydrogen atoms are in highly excited states at any temperature.
  • Some participants propose that the long-range nature of the Coulomb potential contributes to the problem, referencing Landau and Lifshitz's work on the Coulomb problem.
  • There is a discussion about the stability of isolated bound states in thermodynamics, with some asserting that they are never stable due to competing non-bound states.
  • One participant questions the sign of the exponential in the partition function, suggesting that correcting it could lead to convergence at low temperatures.
  • Concerns are raised about the implications of a divergent partition function on entropy, with discussions on whether infinite entropy is acceptable and how it relates to measurable thermodynamic quantities.
  • Participants debate the nature of the continuum of states and its contribution to the partition function, with some suggesting that the presence of an infinite number of non-bound states complicates the stability of the model.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the resolution of the Hydrogen atom problem. There are multiple competing views regarding the implications of the divergence of the partition function, the treatment of bound and non-bound states, and the validity of various approaches to regularization and energy calculations.

Contextual Notes

Participants note limitations in their discussions, including unresolved mathematical steps and the dependence on specific definitions of energy and states. The implications of the divergence of the partition function on thermodynamic properties remain a point of contention.

  • #31
genneth said:
If we are dealing with a canonical ensemble, the extensive variables are S and V; the thermodynamic limit corresponds to holding the conjugate variables T and p constant and letting S and V tend to infinity. The particle number N, is held fixed by definition. If we use a grand canonical ensemble, N is also extensive, and the thing which is held constant is the chemical potential \mu. Your statement about holding the specific volume (i.e. density) constant is equivalent to holding \mu constant in the non-interacting limit, and approximate the same in the weakly interacting limit.

I'm not really familiar with these, because all discussions about thermodynamic limit I've read are based on grand canonical ensemble, do you have any online resource about it so that I could read in more details?
 
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  • #32
kof9595995 said:
I'm not really familiar with these, because all discussions about thermodynamic limit I've read are based on grand canonical ensemble, do you have any online resource about it so that I could read in more details?

I'm afraid nothing really comes to mind at the moment. This is the sort of detail which tends to develop in people independently with time (relative to when they first learn statistical mechanics), because the usual textbook path through the topic will avoid it. A suggestion, which I learned from our local Soviet expert, is Landau and Lifschitz. The Soviet era books tended to have been written completely independently and are therefore most likely to provide orthogonal information to the usual textbooks. Didn't someone in this thread already mention that this pathology of hydrogen atom statistical mechanics is discussed in somewhere Russian?
 
  • #33
Yes, I recommended Landau Lifshetz too. I think the divergence of S and Z is unavoidable for infinite volume limit.
The thermodynamic limit is taken as V to infinity at fixed particle density.
Total energy and entropy diverge in that limit but energy density and entropy density approach a constant value.
However, the case of a single atom in an infinite volume is still special as it corresponds to vanishing energy and particle density.
 
  • #34
Ok, thank you guys, I'll have a look at it.
 

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