Talk:Maxwell–Boltzmann statistics

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[edit] Derivation

I think this derivation is more complete than the one in the "derivation of the partition function" article. I also think that the derivation belongs here rather than in the latter article because the Bose-Einstein statistics and Fermi-Dirac statistics articles each contain their own derivation, with all three articles now being very similar in development. Eventually I would like to remove or reduce the derivation from the partition function article and link to this one instead. This would not affect the canonical and grand canonical material in the partition function article. PAR 07:16, 29 October 2005 (UTC)

more complete? in what sense? because it's more tedious? the longer derivations in the FD, BE, and MB statistics articles are of the physically un-illuminating and unnecessarily contorted variety. partition functions are, more or less, the multiplicity of the system. by exponentiating entropy and use partition functions, dealing with multiplicities directly can be avoided entirely. no graduate textbook i've seen takes this unappealing approach. Mct mht 18:00, 27 April 2006 (UTC)
Calling the derivation "tedious", "contorted", and "unappealing" implies that you know of a derivation that is less so, without sacrificing rigor. Please modify this page accordingly. PAR 23:39, 13 May 2006 (UTC)

I AGREE WITH USER :PAR|PAR

let me qualify my comment a bit. IMHO, the article derivation of the partition function needs to go or a rewrite. i did not mean to compare with that article. Mct mht 23:42, 18 May 2006 (UTC)

[edit] disagree with combining Maxwell-Boltzmann distribution with this article

In general, probability distributions have their own pages. I'd say the Maxwell-Boltzmann distribution should be no different. Also, the theorem presented on this page CAN be applied to moledular motion, but it really applies to all chemical (physical) systems and is the basis for a heck of a lot more than the Maxwell-Boltzmann distribution. Pdbailey 02:35, 18 April 2006 (UTC)

      • I disagree with combining [Maxwell-Boltzmann distribution] with this article ***

It is a mater of audience and communication not aethestics of MAth. People who want to understand a simple introduction to this subject will read the first few paragraphs. e.g. my usage was for a 16-year who wanted a layer more depth than "collision theory". Communication is about context as well as content therefore embedding a simple introduction in a more complex context will reduce he access and usefulness. (Stefan@wasilewski.com)

[edit] mistake in Boltzmann counting

the reasoning given in Boltzmann counting is not quite right. a direct calculation shows that the "corrected" multiplicity W still fails to given additive entropy. for example, multiply each Ni by 2 and the entropy fails to double. something is wrong.

the context in which this problem is brough up also seems to be unusual. the common approach is calculate the entropy for ideal gas, by computing directly the available volume of the phase space, then seeing that an ad hoc reduction by a factor of N! is required. it is called ad hoc precisely because it is exact only if the expected value of the distribution numbers Ni is much less then 1. this defines the classical limit, and Gibb's reduction factor only fixes the counting in the classical limit. this could be the problem here, as no such assumptions, that the system is in the classical limit, was made in the first derivation in the article. therefore using the expression obtained in the derivation leads to the incorrectness on Boltzmann counting.

if no justification is given or changes made, that section should be deleted. Mct mht 05:09, 24 May 2006 (UTC)

in fact, it obviously doesn't make sense to use the Gibbs reduction factor here. let's assume there's no degeneracy and gi = 1 for all i. if you divide by the Gibbs factor N!, this leads to an entropy that's always non-positive, clearly nonsense. section will be deleted. Mct mht 05:52, 24 May 2006 (UTC)

Question: is the final term in the W equation actually correct? Should it be (N - N1 - . . . - Nk-1)! in the numerator? [12:31, 11 July 2006 (AGS)]

[edit] Main equation error?

Is the main equation really correct?

\frac{N_i}{N} = \frac {g_i} {e^{(\epsilon_i-\mu)/kT}} = \frac{g_i e^{-\epsilon_i/kT}}{Z}

Shouldn't it be:

N_i = \frac {g_i} {e^{(\epsilon_i-\mu)/kT}} = \frac{N}{Z} g_i e^{-\epsilon_i/kT}

In the first derivation one line reads:

N_i = \frac{g_i}{e^{(\epsilon_i-\mu)/kT}}

This line directly conflicts with the main equation.

Reply: the eqn you're referring to from the introduction seems ok. Ni is the occupation number for the i-th state, and N is the total number of particles. so Ni/N is essentially the probability that a member of the ensemble chosen at random would be in the i-th state, which should be g_i e^{-\epsilon_i/kT} divided by the normalizing constant Z. on the other hand, as you pointed out, something doesn't quite make sense with the eqn N_i = \frac{g_i}{e^{(\epsilon_i-\mu)/kT}}. that section needs a careful look-over. perhaps a clean-up tag should be attached to it. Mct mht 15:29, 21 February 2007 (UTC)

[edit] Not so fast

<Traumantischer 18:33, 26 February 2007 (UTC)> I didn't like that after a lot of detail in the beginning of the presentation, the Maxwell-Boltzmann statistics is pulled out of the hat. So I propose to somehow mangle the below derivation, replacing the short form:

Using Stirling's approximation for the factorials and taking the derivative with respect to Ni, and setting the result to zero and solving for Ni yields the Maxwell-Boltzmann population numbers.:

by the following (please revise):

With Stirling's approximation in the form

\ln N! \approx N \ln N - N

and neglecting some constants, we obtain

f(N_i) \approx N \ln N - \sum N_i \ln N_i + \sum N_i \ln g_i + \alpha (N-\sum N_i)+ \beta(E-\sum N_i E_i)

We seek now extrema by setting

\frac{\partial f(N_i)}{\partial N_i} = \ln{N_i} + \ln{g_i} -1 - \alpha - \beta E_i = 0

For convenience we can replace the expression ( − 1 − α) by α and obtain solutions

N_i = g_i e^{\alpha-\beta E_i}

However, α and β still need to be determined. Since we have

N = \sum N_i = e^{\alpha} \sum g_i e^{-\beta E_i}

it is easy to see that

e^{\alpha} = \frac{N}{\sum g_i e^{-\beta E_i}}

So one gets:

\frac{N_i}{N} = \frac{g_i e^{-\beta E_i}}{\sum_j g_j e^{-\beta E_j}}

end of insertion

<Traumantischer 18:33, 26 February 2007 (UTC)>

[edit] Quantization?

Suppose we have a number of energy levels, labelled by index i , each level having energy εi and containing a total of Ni particles. To begin with, let's ignore the degeneracy problem. Assume that there is only one way to put Ni particles into energy level i.

Does a number of energy levels mean that energy isn't continuous. But rather quantized? For instances, restricting to energy levels of 1.2, 2.4, 3.6, 4.8, 6.0 ... would imply that a quanta of thermal energy carries 1.2 unit of energy.

If it is continuous it would mean I can always find another state between states. But that would lead to the integration of the probability of all states to go to infinite, would it not?

And there's another problem: it's states that we put distinguishable particles into energy levels.

But if the quantization of thermal energy is correct. Why not put distinguishable energy quanta into distinguishable particles instead and than count their number of ways and than find the distribution with the highest value?

I was wondering if I've put this question into the right place. Do inform me where to post this question if there is one. I remember there was a help desk, but couldn't find it. Is it more appropriate? User:Tikai 16:22, 2 May 2008 (UTC)