I remember sitting through an undergrad stat mech lecture, I think in my senior year, when the professor wrote down the classical partition function for an ideal gas, and made the comment that despite popular misconception this N! appearing in the denominator really didn't have anything to do with the quantum mechanics of identical particles.
Now this sounded like a good philosophic kind of question to sink my teeth into, and I kind of enjoyed the bravely reactionary nature of his comment. I mean if you have an easy 1/N! from QM, why not use it, eh? He must have had a profound reason, although one he didn't seem able to share with us at the time.
The nasty part of this question is that, even forgetting about QM for a second, it seems to put a hard-nosed stat mech result square in the middle of a grand metaphysical problem. Gibbs justified the 1/N! by asserting that the particles are identical (albeit in a merely Luddite, classical way) and that two configurations which differ merely by, say, an exchange of two particle positions aren't "really" distinct. Well why not? I mean they are two different particles. Suppose that we probe a bit deeper and found that no two electrons are really identical, that at a very small scale they had little scratches or barbs or something that were unique little fingerprints. Doesn't matter how small the scratches -- if there are scratches -- blammo! the 1/N! has to go away. Apparently then even the approximate validity of classical stat mech is a metaphysical beacon signaling to us that every single goddam electron is exactly the same.
However, there is a less metaphysical possibility: the particles aren't particles, they're excitations of some kind of field. Then it's not really the case that particle1 here and particle2 there is distinct from particle2 here and particle1 there. What we really have is just field excited both here and there, symmetrically. Now I won't bother here about the fact that this is precisely the state of affairs described by QFT. Could just as well have been a classical field, same result. Bottom line is, stat mech isn't subtley warning us about the inherently quantum nature of our universe, and it isn't telling us something about the metaphysical sameness of all electrons. It's telling us, it ain't particles at all -- it's fields.
Thursday, November 22, 2007
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8 comments:
Hi Eric,
I agree with your conclusion, but I'm confused about what you mean when you say that the field excitation idea is "less metaphysical". Isn't that a very metaphysical issue, whether electrons are billiard ball type objects or excitations of a field?
I have wondered about the relation of stat mech and thermodynamics. Does the distribution derived from the particles do no more than provide a way to move from a theory of micro-particles to macro-features like pressure and temperature? I think most physicists believe that it is ontological rather than epistemic. But if probability distributions are epistemic (as Bayesians seem to think), how could it be ontological? What's your take?
Gilliatt,
I'd say that's a physical vs. metaphysical issue. I was referring to the idea of the total, perfect sameness of a class particles (down to every single detail) as a metaphysical issue, although the line does become thin.
Ron,
Well, I don't think anyone really regards the stat-mech distribution as ontological, rather it reflects our ignorance of all the particles' exact positions/velocities.
There is a contentious issue about whether entropy is epistemic or ontological, but I don't think you're referring to that.
Ron,
It occurs to me now what you're asking about is e.g. the single particle distribution function (so, for an ideal gas, a Gaussian in the velocity). The right way to view this is velocity distribution for a particle picked at random, in which case it does simply reflect our uncertainty about that velocity. Now you can also regard it as a relative frequency of particles with a given velocity, which is more an ontological statement.
But the Bayesians are right that relative frequency is not the fundamental concept. It merely follows from the many-particle joint distribution. In fact, the relative frequency is a kind of highly accurate approximation, and it's the joint distribution that tells us how accurate.
Thanks for your responses to what was probably not a well-posed question. I have thought about it some more and I'll try this:
First, I cannot believe that chance is a real, objective feature of the universe. Rather it reflects ignorance or generality. Probability is an estimate of that ignorance or generality by arguments based, for example, on symmetries (particles of the same kind are identical except for velocity, etc.).
Second, in stat mech there is the use of probability distributions to derive what are objective features of gasses like temperature and pressure.
So I have wondered how it is that some objective features of the macro-world can be derived from subjective probability estimates about the micro-world.
Instead of changing my mind and believing that chance is real, or believe that the universe is mind-dependent, my thought is that we are relating ideas - theories. Maybe we are not relating non-objective beliefs about how features of particles might be distributed with objective features of collections of these particles.
I concluded that thermodynamical _theory_ is derived from _theories_ about how certain features of particles might be distributed.
I wondered what you think might resolve (or dismiss) this seeming problem. Perehaps it was in your initial statements and I didn't see it.
Ron Harris
Ron,
I think there's a simpler case than stat mech that will answer your question. Take one million fair coins and flip them. Almost exactly 50% of them will turn up heads, which is an objective, physical property of this group of coins.
So did this objective property flow from the fact that we were maximally ignorant about the outcome of each flip? No, it flowed from the facts (i) that there is a physical symmetry in each coin, and (ii) that the causal factors in the outcome of each toss are independent of those in each other toss.
Hi Eric,
I guess there is something significant about our not being "maximally ignorant" in the coin toss.
In any case, the coin toss is a deterministic system but we are ignorant of the initial conditions and other details of the toss. I understand that it is even possible for people to learn to control the coin toss to the extent that they can do considerably better than 50/50.
So I keep coming back to the notion that it is ignorance (and generality) that account for so-called "chance".
Oh I absolutely agree that chance is always the product of our own ignorance, and does not refer to some metaphysical fuzziness out there in the world.
What I'm saying is that certain objective properties (like the relative frequency of heads in a sequence of tosses or the temperature of a gas) can be understood through an analysis that is probabilistic in nature. That the analysis is probabilistic does not imply that the physical causes (of those properties coming into being) are themselves probabilistic in some metaphysical way.
It occurs to me that the basic error here is in attributing an aspect of consciousness to the objects we are conscious of.
The specific trick here I think is that the inputs of our analysis include both the nature of our own ignorance (e.g. about how each toss will come out) and certain objective aspects of the objects involved (e.g. the symmetry and independence of the coins). We combine these to get probabilities for statements about certain objective properties (e.g. the relative frequency of heads), but it's the objective inputs, and not the inputs concerning our ignorance, that encode the physical causes of these properties' coming into being.
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