Re: Basic Statistical Mechanics Questions

mahdiarnt_at_yahoo.com
Date: 01/27/05 

Date: Thu, 27 Jan 2005 21:01:45 +0000 (UTC)

> 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.

yes, it can be; as far as I know :-)

> 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.?

There *are* other parameters for a macroscopic system other than T, P,
V, N... But it depends on the nature of the system. For one with two
different types of molecules, say, there are two distinct N's and mu's
and the 1st law reads dE = T dS + mu_1 dN_1 + mu_2 dN_2 + ... The same
can be for a system of magnets and so on. Basically, there are
extensive parameters like E, V, N, etc. which directly enter in the
stat mech description, and intensive ones like T, P, mu, etc. which do
not appear directly in the stat mech picture (in fact nor in the thermo
one. They're called potentials and are defined as derviatives of E wrt
the corresponding extensive param's, see below). For any system you
should first find the complete set of extensive param's describing the
*macroscopic* system, and then go on with the def's of intensive ones.

> 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?

The first law, dE = T dS - P dV + ... is used in stat mech because it
gives a definition for T, P, etc., that's all. In other words, T is
defined to be dE/dS at constant V...; P is -dE/dV in constant S.... Now
all you need in stat mech is to find S in terms of extensive param's
and that's S = k log Omega since Omega can be evaluated by
combinatories... from E, V, N....

> 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?

It is called Euler relation and follows from the homogeniety of E (or
S) in terms of the extensive param's: E(aS, aV, aN...) = a E(S, V,
N...) (or S(aE, aV, aN...) = a S(E, V, N...)). For more details cf H.
Callen chapter 3. There you'll find much more about the previous
questions as well. I think you'll love thermodynamics then! Read the
first few chapters about thermo and the go on to the stat mech part.
It's not as much elaborate as Pathria or Huang are, but more
fundamental about thermal concepts.

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

Hope it helps!

Mahdiyar

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