Re: Basic Statistical Mechanics Questions

qmagick_at_yahoo.com
Date: 01/27/05 

Date: Thu, 27 Jan 2005 21:01:22 +0000 (UTC)

First I wanna say I am not an expert on these matters , but here
goes...

lost.and.lonely.physicist@gmail.com wrote:
> 1) Can thermodynamics be completely derived from statistical
mechanics?
> For instance, the first law of thermodynamics states
>
> dE = T dS - P dV + mu dN

I was reading Feynmann's book on Statistical Thermo. and he derived

dE = T dS - p dV

purely from the Partition function. I think the inclusion of mu dN can
also be done but you might wanna check out that book to see how it is
done.

> a) How does one know there aren't any more terms in this "1-form"
> expansion? How do we know a system can be described by T, P, V, and N
> only; no other possible variables other than re-expressing them in
> terms of S, E, mu, etc.?

You don't know. If things don't come out right experimentally you find
where you made your mistake by not including some variable. The
question you have is the same as asking how do we know that
conservation of energy is actually true for any particular case. Well,
everytime the numbers did not add up it was allways that some energy
term was not taken into consideration. The good thing is that dE = T dS
... applies to very specific systems that have been very well examined.
That is about it I guesse...

> b) Can this first law be "derived" from statistical mechanical
> arguments?

No, I don't think so. But once it is taken as an axium the terms can be
related to the Partition function by various steps. I hope I am not
contradicting myself here.

> 2) I'm also trying to follow why E = T S - P V + mu N, as I've seen
in
> R.K.Pathria's book in deriving PV/T = ln[grand partition function].
Is
> it legitimate to integrate the first law by treating T, P and mu as
> constant? Or how else can one obtain this relation?

This trick is due to two people. Gibbs and Legendre. If I remember
right you can perform a Legendre transform on dE to get E = ... I am
not so sure about your second part to this question. I do know that one
theme of stat. thermo. is in setting certain variables as constant and
integrating or finding the derivative with respect to one of the other
variables which physically do not change much. That is a confusing
sentence but the best I can do :)
                      -- NPC

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