# Something weird happens here

1. Mar 8, 2012

### markov1

I'm assuming that most of you have seen the derivation of the boltzmann distribution. My textbook (Thermal Physics, Daniel V Schroeder) derives it in this way:

Suppose we have a single atom in a resevoir. The ratio of the probabilities of finding it in two different states is equal to the ratio of the multiplicity of the resevoir, when the atom is in the two states i.e.:

p1/p2 = $\Omega$1/$\Omega$2

Now while this may be trivial I think there's something deeper to it, because with just this observation you receive fantastic results. Namely the boltzmann factors and the partition function. And at first it all seemed perfectly clear to me until I read an example, where the boltzmann factors were used. At that point it became clear to me that THE BOLTZMANN FACTORS WORK EVEN THOUGH YOU HAVE NO INFORMATION ABOUT HOW BIG YOUR SYSTEM IS!!!!!

That is counterintuitive to me. I've learned thermodynamics through combinatorics so far, and indeed the size of the system mattered here. What is it, that make the boltzmann factors independent on information of system size? i.e. how many particles etc.?

If you try to find the probability of an atom being in the zero energy state and the state with for instance energy 13eV you find that the ground state is overwhelmingly more probable!! That too is counterintuitive to me. Your resevoir contains A LOT of energy, so exciting a single atom to a higher state shouldn't in percentage reduce the energy of the resevoir significantly - i.e. the multiplicity as I have learned it should stay almost constant. But it doesn't! What's wrong with this thinking? Surely it must be connected to understand why the boltzmann factors do not care about the system size!!

A confused, but also VERY interested student :)

Last edited: Mar 8, 2012