CHM 5414 Lecture Notes
19 September 1996

Ludwig Boltzmann made the connection in 1896 for Thermodynamics. Claude Shannon made the connection in 1948 for Information Theory. But Boltzmann died first; so it was his privilege to use as his epitaph the expression of his life's work, S=kln(W), on his Vienna tombstone. (Natural logarithm was "log" then and became "ln" later.) This photo by Dieter Flamm appears in Atkins and Beran's General Chemistry text.

We show in the next lecture that the connection between statistics and entropy is intimately linked to mixing of all kinds.

In our last episode ("Meanwhile back at the ranch" sounded like a stretch), we showed (at left) how Lagrange Undetermined multipliers, and , are used to incorporate the dependencies of the population values {n} on the fixed number of particles, N, and the fixed total energy, E. With those dependencies treated explicitly, as it were, we were free to presume the independence of the {dn} and require all their coefficients be zero to maximize lnW.

Now we're left with determining the Undetermined Multipliers and . The expressions which determine them are the same as those which prompted their inclusion, the constancy of both N and E. So we have but to sum the optimal sequence {n*} set to N to find not only an expression to eliminate but into the bargain the single most powerful connection between the statistical and thermodynamic worlds: Z, the "sum over states!"

That Z will measure the population availability of states as we'll see soon. But for now, we can make some hay out of the expressions we have even before we determine . Note that if the zeroth energy level _o is taken to be the zero of the energy scale (we're free to choose one), then its exponential term in the Boltzmann distribution becomes exp(0)=1; so the ground state population must be merely N/Z. It appears as if we might have to modify this if the ground energy level were degenerate, but in fact we don't, since the Z sums over all the degeneracies as well; it's index is of wavefunctions not energy levels! (Of course the ground state population will be the population of all the wavefunction states that are degenerate at the ground state energy level.)

Since the zeroth population is N/Z, we've a simpler expression for the Boltzmann Distribution is given at left. Henceforth, we'll drop the *; it will be assumed that the only sequence {n} in which we're interested is the one which maximizes W.

Strictly speaking, we must include all the sequences { n } within the width of the W vs { n } distribution about { n* }. W* is just the peak, but all { n } within a couple of standard deviations of { n* } contribute. Fortunately, we've shown that distribution to be pathologically narrow! So all those { n } in what Nash calls "the Predominant Group" (PG) will be indistinguishable from one another.

Now we're left with .

Boltzmann to the rescue! S = k ln(W)

As the photo caption describes, Boltzmann proposed that entropy, S, well known to his predecessors as reversible heat scaled by temperature (Q_rev/T), a mysterious measure of unavailable work, be equated to k ln(W) . . . well, k ln(W*) . . . if we were retaining the asterisks . . . which we're not. This accorded with the classical notion that dS(universe), the global changes in entropy, never decline. As the universe moves inexorably toward equilibrium (where dS=0), it should be moving toward that W* maximum, attaining it only at equilibrium. In other words, the universe is maximizing the number of ways it might be found consistent with the conservation of matter/energy.

But note that dS is associated with (reversible) heat flow. (That means heat flow near equilibrium conditions.) So we can't preserve the total energy E if there's net heat flow. Fortunately, near equilibrium, reversible heat flow is a trivial perturbation on the system . . . which is why it's specified that way. So infinitesimal energy flow will not disturb the equilibrium sequence {n} of populations, and we can use the expression we have derived above but allow dE to be something near but not at zero. We will, however, insist that the system is closed, i.e., no matter enters or leaves; so dN=0 still.

This split-brain experiment is conducted in what Stat. Mechers call a canonical ensemble. The name isn't important but the concept is. The ensemble is an infinite collection of replicas of the system. Their ensemble energy (the sum of all replica energies) is fixed, but they can share it among themselves. It's as if the system is its own heat bath! This way we don't have to wring our hands over what's happening in the surroundings. As Pogo might have said, "We have met the surroundings, and they are us!"

You'll note that in a fit of pedagogic disregard, we've made the optimal (Boltzmann) population sequence {}, using the Greek "eta," instead of {n*}. This was done to be consistent with the Nash notation.

The "approximately equal" sign comes when we use Stirling's approximation, but we know that for Avogadro's Number, for example, that approximation is phenomenally good. The terms disappears due to constancy of N, and the term would've vanished too if we hadn't permitted Q_rev energy flows.

Now dS = k d(lnW) by Boltzmann hence dS = k dE (above), but we also want dS = Q_rev/T from thermodynamics. Since Q_rev is the only kind of energy flow we're permitting in this ensemble, dE=Q_rev. So kdE from Boltzmann equals (1/T)dE from thermodynamics. That is (but) one way of discovering that:

There are other satisfying aspects to Boltzmann's statistical entropy formula. As shown in Nash, if the universe consists of two subsystems, a and b, which can exchange energy, the one gaining it will increase it's W at the expense of the one losing it, but that can only happen because the total number of ways of finding the universe W = W_a * W_b is increasing. In other words, dW(universe) > 0 until W* is reached at which point dW=0.

Since dS=kd(lnW)=(k/W)dW, it too will increase until W=W* (equilibrium), since (k/W) is positive.

There are systems with T < 0 (and thus k < 0) such as lasers just before discharge, but they are far from equilibrium.

Chris Parr
University of Texas at Dallas
Programs in Chemistry, Room BE3.506
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Richardson, TX  75083-0688

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