Partition Function in Thermal Physics: Overcounting States?

[ak416]

This is a question about thermal physics. There's this partition function Z = sum over all states s of the system ( exp(-E_s/T)). And its just used to calculate the probability of any state by taking the Boltzmann factor exp(-E_s/T) of that state and dividing over the partition function. there's one question that asks to show that the partition function for a combined system, Z(1and2) = Z(1)*Z(2). I understand the way its proved, you just take a double sum and say that E(1and2) =E(1)+E(2), so you can separate the sums. But by using a double sum arent you possibly overcounting some states? For example if E_s1 + E_s2 = 1 + 3 and E_s1 + E_s2 = 3 + 1, also 2+2... Shouldnt this just count as one state of the system, call it E_s = 4. Or would it be better to just keep it this way and then whenever you want to count the probability of observing an E = 4 of the double system, you would have to add all the possible Boltzmann factors corresponding to E = 4. I think i just answered my question..., but I am just wondering what's the right way to think about it, because there's another part in the book about ideal gases talking about how when you have a system with distinct particles you can overcount, but when you have a system with identical particles, you have to multiply the partition function by 1/N! They also say at the end that in our argument we have assumed that all N occupied orbitals (i guess they mean energy levels) are always different orbitals. How does this change anything?

 

[Physics Monkey]

The assumptions here are independence and distinguishability. It isn't true that that the partition function of a composite system is simply the product of the partition functions of its subsystems if the subsystems interact or if they are identical. The $$1/N!$$ in the classical partition function of an ideal gas is the classic example of this subtlety.

 

[Gokul43201]

ak : It sounds like your doubt is just another way of stating the Gibbs Paradox. You might want to look that up too.

 

[katewhitney]

Do you suppose you could explain how you arrived at the answer in the first place?

Z = sum[exp(e/T)]

so Z(1+2) = Z(1)Z(2)

...

a double sum -- as in sum[ exp(e1/t) + exp(e2/t) ] = sum[exp(s1/t)*sum[sum[exp(s2/t)]
??

I understand the partition function idea - but I'm poor with sums :|