Reader Request: What's the Matter with Stat Mech?

Gaaah, flashback. I have a BSc in chemistry, and at the time our required courses leaned heavily toward the physical and theoretical side of things (I have always suspected that was the only way they could actually get enough people to take those classes). So all the chemistry majors had to take stat mech at 8:30 followed by Molecular Properties and Structure II (basically spectroscopy) at 9:30, 3 times a week. It was truly painful. (This may be why it took me 3 tries to get through them, although I eventually got quite fond of spectroscopy -- I like manipulating objects in my head.)

*blink* You found heat capacity to be the worst of the fuzzy, abstract, incomprehensible macro quantities? Maybe my experience was different because I took a bio-flavored thermo class (of which stat mech was only a part). But heat capacity always seemed refreshingly concrete to me after spending yet another hour wondering what exactly was the difference between Internal Energy, Enthalpy, Helmholtz Free Energy, and Gibbs Free Energy (some of which decrease when you increase the temperature... gah).

when you're talking about systems of 1023 particles, you're not always going to be able to boil it down to a simple picture of the motion of individual particles

I think you've just hit one of the nails on the head for why students struggle with subjects like this. I'm a physical chemist who mostly teaches general chemistry and the thermo/kinetics portion of pchem at a small liberal arts college. While none of my classes start at 8:30 am (I'm awake then, but my students never are), the problems you mention being able to visualize how large collections of particles behave, along with the mathematical abstractions, are some of what students in these classes really struggle with.

I don't have a solution, but I keep trying to relate what they are learning to what they care about - beer, their majors, medicine, health... I don't think any one thing will work for all students, but hopefully being specific about why they should care at different times will help them make some of their own connections.

Wow. That describes very closely my experience with thermodynamics as a required engineering undergrad course, although as with Alioth, it was enthalpy that really seemed divorced from reality.

Except without the good professor. (Our guy had an annoying tendency to make people come up and do problems on the blackboard. Seriously, old man? What, are we in third grade, here?)

Also, with steam tables-- my God, the steam tables!!

Also, since it was a semester long class crammed into a five week summer session, it was three or four hours a day... every fucking day.

I second Alioth's comment; I felt like I finished both of my stat mech classes with less of an understanding of free energy and macroscopic properties of elements than I had when I started the classes.

But if you just add an i to the exponent in your QM professor's equation, you get all of QM and QFT.

I've taught stat mech and I STILL don't really understand it. I mean, I understand all of the problems that we solve and models that we explore, but as far as the fundamental difference between various free energies, or what enthalpy is, well, I'm a physicist, not a chemist. I start with a system, construct a model, and deduce its consequences. I get that process. I just don't get the various free energies.

And the fundamental idea is not the partition function. It's the Boltzmann distribution. Which for some reason is always derived from a thermodynamic identity. Schroeder's book strives mightily to do this in a physical way, but it still seems a bit abstract at the end of the day.

I think that I'd rather start from the Maxwell-Boltzmann distribution, which has the physical picture of actual objects colliding with each other, and which can be derived in a few different ways (some more physical than others) and then argue that (1) this comes from an underlying Boltzmann distribution if you look at this in terms of density of states and (2) anything in equilibrium with an ideal gas must also obey the Boltzmann distribution. That approach is at least more physical.

I'm trying to remember which stat mech book it is that begins by mentioning Boltzmann's suicide (and a few other stat mech suicides, IIRC) and then suggests that some caution is in order :)

I'm happy to say that I had awesome stat mech courses. A good reference is Chandler's _Introduction to Modern Statistical Mechanics_, although it's a bit terse to be a great first text.

Internal energy, enthalpy, Gibbs free energy, and Helmholtz free energy can all be used to describe the stability (or equilibrium) of a thermodynamic system. Here's the rules:

E = E(S, V, n), E is a minimum at equilibrium
H = H(S, P, n), H is a minimum at equilibrium
G = G(T, P, n), G is a minimum at equilibrium
A = A(T, V, n), A is a minimum at equilibrium

where E is the internal energy, H is the enthalpy, G is
the Gibbs free energy, A is the Helmholtz free energy,
S is the entropy, V is the volume, n is a vector of
mole fractions, T is the temperature, and P is the pressure. If the variables that you control are the temperature and the pressure, then you want to work with the Gibbs free energy. If the variables that you control are the temperature and the volume, then Helmholtz is your friend. One of the fundamental results of statistical mechanics is that it doesn't matter which of these representations (ensembles) you work with--if you have enough particles in the system you'll get the same answer for each of them.

Incidentally, that first condition is more commonly stated as:

S = S(E, V, n), S is a maximum at equilibrium

It's the same condition, but it's considerably easier to imagine experimental conditions where the total energy is fixed, rather than one where the entropy is fixed.

Alex,
It's Gooldstein's "States of Matter", a truly awesome text that connects the statistical mechanics of gases, solids, liquids, and phase transitions. The liquid theory is a little dated, but the whole book is a really nice intro.

Oops, that's Goodstein, David L. Goodstein.

Here's the actual quote:

"Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics."

Page 1, 1st paragraph, David L. Goodstein, States of Matter.

I still remember the first question on the first test (and I took this class 26 years ago):

Given a particle in a a box, derive the ideal gas law.

The professor was also one of the few that I still remember: Dr. Uzi Landman. The class was taught as more of a lead in to quantum mechanics and worked very well that way.

My other favorite undergrad stat mech moment: Nearly every example discussed in class involved a gas in a box in some way, and every time, the professor drew a rectangle on the board to represent the box. Nothing else was ever done with the rectangle.

About three-quarters of the way through the course, he started a problem by saying "We have a gas in a box," and drawing the rectangle. Then he stopped, turned around, and said "Because, of course, you wouldn't know it was a gas in a box if I didn't draw the rectangle." Then he continued with the problem.

But heat capacity always seemed refreshingly concrete to me after spending yet another hour wondering what exactly was the difference between Internal Energy, Enthalpy, Helmholtz Free Energy, and Gibbs Free Energy (some of which decrease when you increase the temperature... gah).

Heat capacity was one of the more concrete quantities, you're right, and I even have a reasonable idea of what it is, and how it varies between classes of substances, but we were always deriving things like the temperature dependence of the specific heat at constant volume, and the only way I had to know if I was on the right track was if the problem said "Show that the specific heat at constant volume has the form..." I've blocked most of the even-more-abstract quantities out of my memory entirely.

(Thermo in grad school was even worse-- all I remember of that was that there were four (?) "Maxwell relations" connecting the partial derivative of one thing to a different partial derivative of a different thing. If you picked the right one to start with, the problem took maybe 5 lines. If you picked the wrong one, you could noodle around for pages and pages of algebra, always looking like you were getting close to whatever it was you were supposed to show, but never getting there. It was maddening.)

One of the problems with stat. mech. is what Chad was describing as an "abstract game". It's great that the commenter at #8 got it, but when I hear "Gibbs" and "Helmholtz" and think back to taking partial derivatives of various quantities with respect to various other quantities, my eyese glaze over. I remember not feeling any real appreciation for it when I was an undergrad. The only thing I retained from the class was the basic idea of entropy/temperature and the Boltzmann distribution.

The other issue with stat. mech. is that I don't think there's a good consensus on what should be taught. There's such a wide variety of interesting and important things that could be covered (Random walk, Diffusion, Fermi/Bose gas, Fluctuation-Dissipation theorem, Density Matrices, transport, etc. etc.) that there's less common experience. And a great deal of time is spent on the partial derivatives, which I find much less interesting than the above list, so there's little time to cover that other stuff.

i am with you Chad. frakin' Maxwell relations. drove me nuts. partial derivatives with pressure constant or temp constant. or frustration constant.

thank you Grant, for your explanation. i eventually figured that out on my own, but it frustrated me while i was learning it. in class the theory was never connected to experiment explicitly, so it left me in a lot of confusion. i think you may have just saved a whole bunch of irritated thermo students.

All I remember from statistical mechanics at Cambridge (UK) in ~ 1980 was a long calculation of something and the final comment that "the factor of 10**23 is irrelevant" :-)

I actually kinda liked stat mech precisely because it was an abstract game. I didn't have as good an intuitive feel for physics as would have been helpful, so classes like stat mech and quantam were easier for me. Fortunately, I figured this out and chose to go to grad school in computer science rather than in physics!

Stat Mech seems to be more physics. Thermo is almost masturbation with partial derivatives. Not a fundamental science, IMHO. But often even stat mech then turns into math with sums and integrals for partition functions. Math can illuminate physics, but oftentimes obscures it too!

You describe my experience exactly. The whole course seemed to be the partial derivative of something with respect to something else and . . blah blah blah.

As a side note, shouldn't scienceblogs have some sort of error that pops up when you're missing a parenthesis in your post?

Statistical Mechanics and Thermodynamics are both wonderful subjects. I have taken them both in physics and chemistry departments, undergraduate and graduate, just to wallow in the many distinct ways to elucidate the material.

I have since taught a class in statistical thermodynamics. I can definitely say: (1) to fully appreciate it, you need a solid background in classical thermodynamics, where all the terminology you find confusing, originates and (2) your arrogant professor is exactly correct in that the partition function (embodying the concept of the ensemble and averaging) is the only new idea not contained in classical thermodynamics.

The main problem is that almost all physics students DO NOT get any kind of good classical thermodynamics background. Chemists do it better. Engineers do it way better. There just isn't an enshrined place for it in the physics program, nor a standard textbook, unlike for Griffiths or Goldstein. This propagates through the generations: physicists take, perhaps, one-quarter's worth of classical thermo (if at all) and don't really get it in a good way. Enough time to hate Maxwell relations, but not enough to get it straight, then --zip!-- its time for sexy QM. Then when they grow up and need to teach their one month thermo unit to kids who learned partial differentiation the week prior... they flub it, since, really, they don't know any better, and next month is QM which has cats and greek letters and spherical Neumann functions and all the cool stuff.

/ranty mcrant, done.

I had to look pretty far back, but I found that one time I did blog about teaching thermo in freshman physics. I am 100% convinced that the problem starts with freshman thermo, which is much more in need of a major curriculum development than quantum mechanics.

I tried a major pedagogical experiment with it last year and was extremely happy with the result, but need to see more (including talking to them after they take an engineering class in thermo) before I will say more than I will here.

Let me put it this way. IMHO it is wrong to teach the Bohr model as if it contains quantum mechanics. It doesn't. Real quantum mechanics does not have "quantum jumps" as Bohr described them. Starting there will hide the fact that transitions only occur when two wave functions overlap with an operator. The physics is all very local (in classical QM), despite all the "weirdness". Similarly, I now think it is wrong to teach thermodynamics by starting with thermodynamics. The language itself is wrong. Specific Heat is not heat! It is a property (essentially the derivative) of the internal energy. The rate of change of internal energy tells us how much energy has to flow (heat) to change the temperature. "Latent heat" is needed when you can't take the derivative because of the phase transition. Most of the other odd details of language are just ways of saying what is being held constant while we move energy in and out of the system.

We should get the basic ideas of statistical mechanics in as early as possible, even if it has to be done crudely. I now go from the internal energy of an ideal gas to its "specific heat", then simply assert a similar (phenomenological) result for solids and liquids before taking up the problems that are normally at the start of the topic. I take up the ideal gas right along with temperature, riffing off of the idea of a gas thermometer.

The underlying philosophy is that students frame their learning with the first thing you teach them. I want that to be physics, not a bunch of problem-specific rules using language from the 18th century.

Maybe this reflects my own anomalous education (I learned graduate stat mech before I got any exposure to real thermodynamics), but all of the language of thermo reflects a total abject ignorance of what was going on by the people trying to quantify what was going on, leading to language that implies that heat is a substance, and the most important substance at that. They were building steam engines long before Clausius, let alone Boltzmann, came along. Maxwell's ideas about gases seemed to get swamped by the success of thermo, maybe because atoms and molecules weren't fully accepted at that time. In contrast, radio was developed long after Maxwell came along. Can you imagine teaching E+M using all of that electrical fluid nonsense you can read from circa 1800?

Let me second that last comment. Too many subjects are taught in historical, rather than logical order. In my own education, I did not understand any thermodynamics until a year later when I learned stat mech, and things started making sense. Sure, I could manipulate all the symbols, but they did not mean much, because well - they don't. They were invented while people had very crude ideas on what is going on, and were replaced with sharper concepts. We should start with those sharper concepts, the subject should not be all that complicated when taught the right way. The comment cited about the partition function being the only real idea is pretty much right on target.

I'm currently in the process of reading Kadanoff's "Statistical Physics - Statics, Dynamics and Renormalization", and it seems to take a very different approach from most Stat Mech books I've met along my undergrad and grad school studies.

He seems to present a more modern, and thus big-picture, view of the subject, without keeping it leashed to ye olde Thermo as strongly as most books do.

I'm not sure I'd recommend it as a first text to read, at least not without the support of a good course instructor, but for someone who's taken Stat Mech at some level, it seems to really go a long way in explaining the why's and the deeper meaning.

I'd suggest you take a look, if you get the chance.

As a theorist, stat mech always frustrated me for seeming to lack coherent theoretical structure... in the underlying intuitive picture kind of sense, which I guess may just come back to the difficult-to-visualize-10^32-particles problem.

If you look at it that way, there's only one idea in any area of physics:
Classical: F=mA
Quantum: H Psi = E Psi
Relativity: what happens if time is a component of the space vector?
Electrodynamics, optics, etc, etc etc

In thermodynamics, the idea is simply, if I measure something in a system with constant variables, say number, volume, and temperature, what comparable thing should I measure if I have a system that has instead constant number, pressure, and temperature? exact differentials, in other words, and if I recall correctly, Einstein called thermodynamics the most perfect example of a physical theory imaginable.

in stat mech, the question is what happens when we have more than one? Whatever your physics of one or two is, what does it mean when we have 3, 4, an Avogadro's Number of such things? Yes, everything else is just algebra, but like all the other examples in physics, what wonderful algebra, huh?

And, can I also point out that, given that you've had recent posts talking about superconductivity, and yet you're talking about only the stat mech of gases in this post? How 'bout lasers and population inversion? Or Einstein's crystal results? Stat mech even pops up in traffic control studies, avalanches, protein folding, and on and on.

Will, @20:

Engineers do it way better.

I am, to put it mildly, not convinced that this is true.

Also, Chad, here's a suggestion for your next conversational-type topic: Mangled topics.

I can't figure out an elegant or pithy way to phrase it as a question, but there are a handful of things from my education (both engineering and CS) that were just painful... either because they were being taught wrong, or because some key piece of requisite knowledge slipped through the curricular cracks and was never required.

And when I say painful, I mean much more painful than they needed to be-- invariably, once I struggle through on my own to the key insight(s), my internal response is an aggrieved, "Well why the fuck didn't they just tell me that?!"

Engineering thermo is called "heat transfer". It is often nothing like what physicist think of as thermodynamics, aside from the diffusion equation.

CCPhysicist nailed it on the head: "Most of the other odd details of language are just ways of saying what is being held constant while we move energy in and out of the system." This needs to be said at the outset, hammered on, and demonstrated ad nauseum. It is abstract in that something is held constant (even though the constant variable depends on which type of ensemble, i.e., what are your system's boundary conditions). But it is also very very concrete in that each type of ensemble has a thermodynamic or system variable held constant. Invariants are always powerful ideas, touchstones, and here they are added to grand (canonical) ideas.

I was very unhappy with all my thermo and stat mech references and I found this great website: http://stp.clarku.edu/. If you click on the link for the "lecture notes," you can read those for free (though they supposedly being published as a book right now). The notes clear up a lot of confusion about stat mech and thermo. They spend a lot more time on "why" rather than "let's do some algebra". If you click on the link for "concept inventory," you can see how well you know thermo and stat mech.

I took quantum many-body theory last spring, and we used everyone's favorite Fetter & Walecka as the text. Well, there's a review of statistical mechanics at the beginning and I remember all of my grad student brethren groaning as we got started. Thermodynamics and stat mech are my favorite subjects in physics, so I was happy to talk about them again.

I think part of the reason that we (physics students) are not exposed enough to these topics. The typical grad student goes through something like 5-6 semesters of classical electrodynamics, not to mention whatever they do in QFT or their research. In contrast, one might typically have 1-2 semesters of thermo + stat mech. One just has to go back to the subject over and over. It's conceptually trickier than other parts of physics, but it has being hard in common with every other part of physics.