Strange feature of BE, FD and Boltzmann distributions

[Philip Koeck]

Hi.

I've just come across something rather strange, I believe, about the micro-canonical derivation of the BE-distribution (as well as the Boltzmann and FD-distributions).
See for example https://en.wikipedia.org/wiki/Bose–Einstein_statistics#Derivation_from_the_microcanonical_ensemble

The starting assumption is that the system is isolated from the surroundings and has a fixed energy, which I'll call E.
On the other hand an integral over the resulting distribution function taken from energy E as lower integration boundary to infinity as upper boundary is not zero. This seems to indicate that there is a certain non-zero probability that an individual particle in the system can have an energy higher than the total energy of the system.

Am I misunderstanding something, or is there a way to explain this?

 

[vanhees71]

That's tricky, because what Wikipedia describes what you can derive in this way if you work out this description is the distribution function from the grand-canonical ensemble starting with the microcanonical arguments and then doing the approximations using Lagrange multipliers and assuming $n_i \gg 1$ and $g_i \gg 1$ while $n_i/g_i = \mathcal{O}(1)$.

 

[Philip Koeck]

That's tricky, because what Wikipedia describes what you can derive in this way if you work out this description is the distribution function from the grand-canonical ensemble starting with the microcanonical arguments and then doing the approximations using Lagrange multipliers and assuming $n_i \gg 1$ and $g_i \gg 1$ while $n_i/g_i = \mathcal{O}(1)$.

I wouldn't say this approach is unique for this Wikipedia page. You'll find similar derivations in many textbooks, for example Alonso-Finn.

The chemical potential occurs in the derivation because of the constraint on the total particle number. That's why you arrive at the same distribution as you get in the grand-canonical approach.

If you don't assume the particle number to be constant then the microcanonical derivation gives the same result as the canonical one. (This might not make sense for helium atoms, but maybe for photons.)

 

[vanhees71]

The point is that the grand-canonical ensemble describes the situation of a system coupled to both a "heat bath" and a "particle bath", i.e., it is open concerning the possibility to exchange both energy and particles with these "baths". In the thermodynamic limit, i.e., for all extensive quantities going to infinity and keeping the corresponding intensive quantities/densities constant, all the ensembles are equivalent, because the fluctuations are negligible.

 

[Philip Koeck]

The point is that the grand-canonical ensemble describes the situation of a system coupled to both a "heat bath" and a "particle bath", i.e., it is open concerning the possibility to exchange both energy and particles with these "baths". In the thermodynamic limit, i.e., for all extensive quantities going to infinity and keeping the corresponding intensive quantities/densities constant, all the ensembles are equivalent, because the fluctuations are negligible.

I completely agree with you.

Interestingly, in the micro-canonical approach sort of the opposite is happening.
Fixing the total energy of the closed system leads to a Lagrange multiplier which turns out to be 1/kT and fixing the particle number leads to a Lagrange multiplier which turns out to be -μ/kT.

Anyway, in the microcanonical approach the system has a finite total energy, but the resulting distribution allows for particles to have an energy that is higher than this value.
That's pretty strange to my way of thinking.

In the canonical and grand-canonical picture this contradiction doesn't occur since the heat bath has an infinite amount of energy, which the particles of the system can "borrow from", I guess.

How should we react to this? Should we discard the microcanonical picture?

 

[hutchphd]

That's pretty strange to my way of thinking.

It certainly requires consideration. In all of these arguments some approximation to the various factorials is applied to get to the continuous distribution, so this is not a shocking result. In fact you will (I think) find that the probability of one particle having more than the total energy of the N particle system to go like $\frac 1 N$

How should we react to this? Should we discard the microcanonical picture?

I'm not inclined to do so, but I do find most of thermodynamics vaguely metaphysical already.

 

[Haborix]

I think what makes thermodynamics seem metaphysical is the willy-nilly use of limit taking combined with the fact that thermodynamics is the subject whose sole purpose is to relate physical variables; i.e. if you take a limit here you're probably taking it for any number of other variables. Sometimes after taking the thermodynamic limit, people will try to study limiting cases by essentially undoing the thermodynamic limit only in their already approximate result. That very quickly starts to look like voodoo.