Talk:Maxwell-Boltzmann distribution

The way it is presented at first it seems to refer to the general Boltzmann distribution? - then it can be used for any, arbitrarily strongly interacting system, as long as the subsystem considered is large enough. user:FlorianMarquardt

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Equation (7) in the article looks positively wrong. The author states Substituting Equation 6 into Equation 4 and using p_i=mv_i for each component of momentum gives:

<math>

f_p (p_x, p_y, p_x) = \left( \frac{\pi mkT}{2} \right) ^ {3/2} \exp \left[ \frac{-m}{2kT(v_x^2 + v_y^2 + v_z^2)} \right] </math> (7)

Equation 4:

<math>

f_p (p_x, p_y, p_x) = cq^{-1} \exp \left[ \frac{-1}{2mkT(p_x^2 + p_y^2 + p_z^2)} \right] </math> (4)

Equation 6:

<math>

c = q (2 \pi mkT)^{3/2} </math> (6)

If I perform the act of substituting 6 into 4 and substituting p_i=mv_i I get:

<math>

f_p (p_x, p_y, p_x) = ( 2 \pi mkT ) ^ {3/2} \exp \left[ \frac{-1}{2kT(mv_x^2 + mv_y^2 + mv_z^2)} \right] </math>

Please comment. --snoyes 22:17 Mar 9, 2003 (UTC)

You haven't texified (6) correctly. It's supposed to be

<math>

c = q (2 \pi mkT)^{-3/2} </math>

not

<math>

c = q (2 \pi mkT)^{3/2} </math>

-- Derek Ross 22:56 Mar 9, 2003 (UTC)

Thanks a lot for pointing that out ! I really must be more careful. But what about the second part of (7), where one substitutes p_i = mv_i ? --snoyes 23:03 Mar 9, 2003 (UTC)

I'm still lookin at it but I think the (4) is wrongly texified too. It makes much more sense if the sum of p's is on top of the fraction. But I'm just doing a little research to ensure that that's the right thing to do. -- Derek Ross

Which would mean that (3) is also wrongly texified. The problem is that:

exp[-1/2mkT(p_x² + p_y² + p_z²)]

is just so damn interpretable. That is partly the reason I'm going to all the trouble of texifying all these articles; To disambiguate them. --snoyes 23:21 Mar 9, 2003 (UTC)

Yep, (3) should be

exp[-(p_x² + p_y² + p_z²)/(2mkT)]

and the others should changed analogously. -- Derek Ross

Excellent. However, there remain problems with (7); it would now have to be:

<math>

f_p (p_x, p_y, p_x) = ( 2 \pi mkT) ^ {-3/2} \exp \left[ \frac{-(v_x^2 + v_y^2 + v_z^2)}{2kT} \right] </math> (7)

(ie. the ms cancel and the stuff in the first bracket is all to the power ^(-3/2)) ? --snoyes 23:42 Mar 9, 2003 (UTC)

And why is cq^(-1) not written as (c/q) in (4) ? (style?)--snoyes 23:44 Mar 9, 2003 (UTC)

Yep, style. They both mean the same thing. -- Derek Ross

Substituting p² with p=mv gives m²v². You can then pull all the m²'s out with the distributive law and cancel the m in the denominator leaving an m in the numerator which is what you want.

As for the other point ...

<math>

( 2 \pi mkT) ^ {-3/2} = (1 / 2 \pi mkT) ^ {3/2} </math>

... so there's no problem there, just a matter of personal taste about how you want to write the formula. -- Derek Ross 23:52 Mar 9, 2003 (UTC)

I stand corrected again, thank you Derek. As for the personal taste, I don't care which one - do you have a personal preference? I shall use that. --snoyes 23:57 Mar 9, 2003 (UTC)

Some people get confused by the q^-n notation. I think that it's often a better idea to change it to 1/(qⁿ) instead. Also I would separate out the relatively constant parts to make something like... well if I knew Tex I would do it myself. Sadly I don't. Guess I'll have to learn ! -- Derek Ross

I learnt it in a couple of hours solely for changeing all the stuff on wikipedia ;-) --snoyes 00:08 Mar 10, 2003 (UTC)

One thing I would like. exp[x] is actually supposed to be e^x. It would be nice if you could change that -- Derek Ross

Hmm, unfortunately it looks like this:

<math>

e^{\left( \frac{-(p_x^2 + p_y^2 + p_z^2)}{2mkT} \right)} </math>--snoyes 00:27 Mar 10, 2003 (UTC)

Too bad, exp it will have to be then. -- Derek Ross

I see someone has learnt some tex. ;-) Couple of small things with (8). Corrected it is:

<math>

f_v (v_x, v_y, v_z) = \left( \frac{m}{2 \pi kT} \right) ^ {3/2} \exp \left[ \frac{-m(v_x^2 + v_y^2 + v_z^2)}{2kT} \right] </math> (8) --snoyes 05:34 Mar 10, 2003 (UTC)

Not at all, just monkey see, monkey do, plus good ole cut'n'paste -- hence the mistakes. One other change which I think needs making is that the integral signs should actually be partial integral signs and likewise the differentiation operators should be partial differentiation operators but as I said, if only I knew Tex... -- Derek Ross