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Q & A: Maxwell-Boltzmann

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Most recent answer: 06/20/2009
Q:
Sir I was wondering that what logic can be used to find that what fraction of molecule have what velocity - that is I mean the logic behind Maxwell-Boltzmann distribution. Secondly is there a link or book which provides step wise derivation of Maxwell's distribution equation from basics. Thanks a lot in advance .
- dipendra (age 17)
lko, india
A:
Great question.

The reasoning that Maxwell used, based on the classical physics of gases, was subtle. There are more modern arguments, based on the existence of discrete quantum states, that lead very directly to the Boltzmann factor: in an environment at temperature T, the probability of occupancy of any individual state of energy E is proportional to e-(E/kT) where k is Boltzmann's constant. The general reasoning is that nature gives equal probability to any accessible quantum state overall. But the bigger E is for the state of a little part, the less energy is left for everybody else. That means that the rest of the world can be in fewer states if our little part has bigger E. So a state of the part with bigger E has lower probability of occurring. T is just a parameter that describes, for some given environment, how rapidly that probability changes with energy.

A particularly nice book which describes the derivation is

Statistical Physics: Berkeley Physics Course, Vol. 5
by .
We have a derivation on the website for one of our courses: . It's in lecture 7, but you'll probably need to start around lecture 4 to follow it.

Once you have the Boltzmann factor, which is extremely general, it's easy to get the Maxwell-Boltzmann distribution. All you need is that for particles traveling at much less than the speed of light in three dimensions, the number of states with speeds near v is proportional to v2, which I believe is also explained in Reif's book.

Mike W.

(published on 06/20/2009)

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