On the thermodynamic limit
Good afternoon professor,
I have a question about the thermodynamic limit and how it should be applied to any general problem in statistical physics.
For example, in T05.1(b) we were asked to calculate the TD limit of ln(partition function). If my understanding is correct, there are two things we could have done:
1) N * lim as N -> infinity of (1/N) * ln(partition function)
2) lim as N -> infinity of ln(partition function) WITHOUT dividing or multiplying by N at all
A fellow student did option 1) and published it on this blog. This is also how we had done it in class before and I can see it could be seen as consistent since it's just multiplying by N/N, but why can we multiply the logarithm by N outside of the limit and by 1/N inside of it and use that instead of the original logarithm to calculate the internal energy and other quantities? How is this formally acceptable?
I'm sorry if this is just a basic calculus question, but the sources I've read say that the TD limit by definition just means that as V and N approach infinity the quantity V/N is a constant and don't explain the validity of procedure 1.
I have a question about the thermodynamic limit and how it should be applied to any general problem in statistical physics.
For example, in T05.1(b) we were asked to calculate the TD limit of ln(partition function). If my understanding is correct, there are two things we could have done:
1) N * lim as N -> infinity of (1/N) * ln(partition function)
2) lim as N -> infinity of ln(partition function) WITHOUT dividing or multiplying by N at all
A fellow student did option 1) and published it on this blog. This is also how we had done it in class before and I can see it could be seen as consistent since it's just multiplying by N/N, but why can we multiply the logarithm by N outside of the limit and by 1/N inside of it and use that instead of the original logarithm to calculate the internal energy and other quantities? How is this formally acceptable?
I'm sorry if this is just a basic calculus question, but the sources I've read say that the TD limit by definition just means that as V and N approach infinity the quantity V/N is a constant and don't explain the validity of procedure 1.
Sorry, I forgot to answer this one. It is a very good and difficult question. From my point of view this question is connected to the existance of extensive and intensive quantities. It is a bit strange, because the fundamental postulates of thermodynamics do not make any statements on that. In statistical mechanics, we start with the microcanonical ensemble and that is where we first make use of the thermodynamic limit and the expectation that certain variables are extrinsic and others intrinsic.
ResponderBorrar* So there, we have on the one hand the mechanical variables, like for instance the volume in the case of a gas in a container, or the trap frequency in the case of atoms in a harmonic trap). It easy to realize that these variables may be intrinsic, extrinsic, or anything in between. For instance, in the case of an ideal gas, nobody can prevent me from choosing sqrt(V) as my mechanical variable. And we learned that the trap frequencies are also neither intrinsic nor extrinsic.
* And we have the genuinely thermodynamic variables, internal energy, entropy and temperature.
Only those are the variables which are definitely extrinsic or intrinsic.
* How should one then take the correct thermodynamic limit? In the microcanonical ensemble, for instance, we know that we have to increase the particle number and the energy such that E/N remains constant. But what about the mechanical parameters which appear in the Hamiltonian function? I think that there is no rigorous answer to that question. You have to do it in a sensible way. But most importantly, one has to do it in such a way that the entropy becomes extensive.
* Supose we know how to take the thermodynamic limit, why then do we calculate the entropy as
N * lim_{N -> infty} N^{-1} Ln( partition funcion) . That is because we require Ln( partition function) to become extensive in the thermodynamic limit. Then, dividing it by N makes it intensive. That implies that taking the limit lim_{N -> infty} yields a finite value -- the average entropy per particle, for instance. Finally,we multiply by N to obtain the well behaved extensive entropy we all love.
* The same applies for the other partition functions. In the canonical ensemble ln(partition function) yields the free energy, A = U - TS, which must be extensive, also.
Thank you very much for the detailed answer professor! I see it's a complicated subject
Borrar