Basic Statistical Mechanics Questions

lost.and.lonely.physicist_at_gmail.com
Date: 01/25/05 

Date: Tue, 25 Jan 2005 16:34:41 +0000 (UTC)

I'm a grad student in physics but I did not learn my statistical
mechanics and thermodynamics very well, and I hope to get some help in
clarifying some basic concepts.

1) Can thermodynamics be completely derived from statistical mechanics?
For instance, the first law of thermodynamics states

dE = T dS - P dV + mu dN

where E is energy, T temperature with the Boltzmann's constant = 1, S
entropy, P pressure, V volume, mu the chemical potential, and N the
number of particles.

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.?

b) Can this first law be "derived" from statistical mechanical
arguments? I often see stat mech books introducing the partition
function et al and then out of the blue use the first law in the above
form (or transformed with the introduction of the Free energy, Gibbs
energy, Helmholtz, etc.). I just looked at R.K.Pathria's book and
he/she uses the above law to identify the lagrange multipliers in the
partition function with T, mu, etc. I'm confused what the first
principles are: isn't everything supposed to follow from the partition
function?

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?

Thanks for the help and I hope this group will tolerate my elementary
questions - I may have more to add later.

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