Re: Basic Statistical Mechanics Questions

From: Dan Platt (DanP57_at_ispwest.com)
Date: 01/27/05 

Date: Thu, 27 Jan 2005 15:44:49 +0000 (UTC)

lost.and.lonely.physicist@gmail.com wrote:
> 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.?

Usually, this correspondence is made from a classical argument,
starting with

S = S(E, V, N),

and playing with a box with a partition that first lets energy
move by conduction (V, N fixed), then allows V to move (movable piston),
then allows masses to move (N moves), with the total on both sides
being constant. Movement occurs until the entropy is maximized.

>
> 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 argument track to follow from classical towards statistical is that
the fraction of states (Omega) looks like

        S = k log Omega

(on Boltzmann's tombstone), so that

        Omega = exp(S/k).

Then just use the expansion as for the box partitioned with the barrier
to get the Boltzmann factors you're familiar with. One side of the box
is a "heat bath," the other is a microsystem in any of a number of
determined states. In this case, you're counting the number of states
of the composite system: how many ways can the heat bath be in
equilibrium with the microsystem in some particular state?

The more physical question to ask at that point is: what type of
properties must be true about exchanges of energy with the heat sink in
order for the above to be a reasonable estimate of the number of states?
[HINT: the macroscopic behavior of the bath doesn't care about the
details of how it is stirred, how a flame is applied, etc -- it is
pretty well characterized by volume, density, temperature, etc;
macroscopic behavior is pretty much insensitive to its microscopic
trajectories]

>
> 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?

Consider the expression of E = E(S, V, N). All of these variables are
"extrinsic." Double the size of the system, and you double the size of
all of the variables E, S, V, and N. So, you can write:

r E(S, V, N) = E(rS, rV, rN).

Then you can take the partial derivative of this with respect to r, and
then set r = 1.

(this type of function is called homogenous of degree 1; if E(rS, rV,
rN) = r^k E(S,V,N), it would be homogenous of degree k.)

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

Dan

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