Re: Basic Statistical Mechanics Questions

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

Date: Thu, 27 Jan 2005 15:44:46 +0000 (UTC)

[I tried replying before -- have problems connecting with SMTP server
occasionally.]

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.?

A common derivation is to posit a state function S = S(E, V, N) or
any other variable that would be sufficient to describe the state of
a system. This defines the variables.

A standard derivation starts by considering a box with an impermeable
piston (start with it fixed). Energy is allowed to move by thermal
conduction from box 1 to box 2 through the piston. Then the
total change in entropy is

dS = dS1 + dS2 = (dS1/dE1)dE1 + (dS2/dE2)dE2, where dE1 + dE2 = 0.

The (dS1/dE1) are partial derivatives holding V and N const. At this
point, 1/T = dS1/dE1 is identified. The total entropy change dS >= 0.
This yields an inequality indicating energy flows from high T to low T
until T1=T2. Similar arguments can be made for V and N by allowing the
piston to move and then to become permeable. It is possible to play
with the derivatives to show that p = T(dS/dV) at const E,N.

In this argument, the core of the assumption is that there is SOME state
function S that must increase and which is a function only of the other
state variables that are sufficient to define the state of the system.

>
> 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?

A standard argument is that you can let one of the systems in the above
box be a micro system. The other side of the box is a "heat bath." The
idea behind a heat bath is that its macroscopic behavior is pretty much
unaffected by the details of its treatment... you can heat it in a
number of ways, but its macroscopic behavior won't be particularly
sensitive to how it is heated. It won't matter how you stir it for the
most part, it will absorb dissipative work, etc. To the point, the
macroscopic variables are not sensitive to the microscopic trajectory.
So, you can replace the details of particular trajectories with a kind
of stochastic noisy trajectory. (There's a lot of argument over how
this transition -- emergence of this "emergent phenomena" actually
happens, and when it doesn't happen, but this is the primary crux of the
process.)

A second assumption is that you are looking for some distribution
function Omega(E)dE that will give you the number of ways to find a
system between E and E+dE. If two systems are exchanging energy, then
the total number of ways they will be in equilibrium is

integral Omega1(E1)Omega2(E-E1)dE1

This integral will be dominated by the maximum of the product of
Omega1(E1) Omega(E-E1). This maximizes at the same place that
log Omega1(E1) + log Omega(E-E1) maximizes. Further, log Omega, being
additive, will produce an extensive variable like S. Boltzmann posited
that the relationship between number of states and entropy was

S = k log Omega

(its on his tombstone).

Therefore,

Omega = exp(S/k).

Then you can ask what the number of ways of finding a microsystem in
some particular state in contact with a heat bath will be:

Omega proportional to exp(S(E - E1, N - N1)//k).

[NOTE!!! V is tied up with the structure of E1!!!] This gives you the
Boltzmann factors, and ultimately the grand canonical 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?

E = E(S, V, N)

is homogeneous of order 1 -- which is to say, all the variables are
extrinsic. If you double the size of the system, E, V, N, and S all
double. More particularly, increasing the system by a factor of r will
yield:

rE = E(rS, rV, rN).

Take the partial derivative with respect to r, and set r=1.

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

If this is for homework....

Dan