All that I have about the Bose-Einstein distribution is the statistics of the occupation number n of the j-th energy level,
(1) = 1/{exp[(μj - ϵj)/kBT] - 1}
where μj is the chemical potential on the j-th level, and ϵj the level energy. However, I would like to calculate the average kinetic energy in the gas, i.e.
∑j nj Ekinetic,j /∑j nj , also the most probable kinetic energy in the gas, and the similar quantities for velocity.
Does somebody know a simple way to do it?
Don't you mean
< ∑j nj Ekinetic,j /∑j nj >?
Or perhaps
< ∑j nj Ekinetic, j> /< ∑j nj >?
For a macroscopic gas one would think the last expression is sufficient (with some qualifications in the Bose condensation regime).
If you really want to calculate the probability distribution of the quantity you wrote down, it might still be possible by use of the central limit theorem (provided you have sufficiently many j-values). The numerator and denominator should both be approximately gaussian distributed (but not quite independent as probability distributions).
Yes Kåre,
of course. Thank you for the correction. I think that the most correct form is
∑j < nj > < Ekinetic, j > / ∑j < nj >?
Now, I am afraid that the number of occupied levels is not big, I speak of temperatures close to (but greater than) Tc , the temperature of transition to condensate.
But in view of your correction I realize that the issue is even more complicated. It's very late now in my country, I'd be glad to "talk" with you tomorrow.
Again, thank you.
Dear Kåre,
I have in my hands an excellent article which treats first of all the behavior of a gas at T close to Tc , though T > Tc.
First of all the authors give the Bose-Einstein statistics. But the gas is in a trap, a harmonic trap. Therefore, first of all, the energy levels ϵj are not kinetic energy. Therefore, thank you again for recommending to write < Ekinetic, j >.
And now, some questions:
A) If the gas is under the influence of a potential, the Bose-Einstein distribution is still correct? Of course, I can look at the proof of the distribution and infer by myself, but I try to economize time. If somebody is a specialist and can help, it's helpful for me.
B) The words central limit theorem look known to me, probably I learnt this a long time ago, but I either didn't use it, or I used under another name. In short, can you tell me what this theorem says? (About the Bose-Einstein statistics I am no expert, as you can see, but the Maxwell-Boltzmann one is well known to me.)
With thanks in advance,
Sofia
Igor> Attention. The sign is all wrong.
Igor is absolutely right on this: Your equation (1) should read
(1) = 1/{exp[(ϵj-μ)/kBT] - 1}
and it is further not normal to have a j-dependent chemical potential μ, as I have corrected above. Physically one must have μ / ∑j < nj >
Combined with equation (1) this gives a quite explicit expression for the quantity you want. Nevertheless, I advice you to read the section(s) about ideal quantum gases in some good, intermediate level, textbook on statistical mechanics.
Actually, I think it would be more interesting to calculate the probability distribution of the quantity you first wrote down :-)