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April 13th, 2009

The Metaphysics of Statistical Thermodynamics

I recently picked up Leonard Nash’s 1973 Elements of Statistical Thermodynamics as some light airplane reading (well, it’s light in the sense of being a 138 page paperback printed on thin paper), because I’ve been interested in figuring out more about the applications of the Principle of Indifference (roughly, that one’s credences in various propositions should be proportional to the number of ways that the propositions can be true). From considering the first example discussed on page 4, I’m already starting to consider connections between this epistemic principle and the underlying metaphysics. In particular, I think that there may be ways in which thermodynamics can be used to give arguments for or against the existence of tropes (or substrata or bundles or other metaphysical posits).

Consider a submicroscopic simple harmonic oscillator with a characteristic frequency. (I believe a natural example would be a molecule of H2, where the bond between the two hydrogen atoms can roughly be seen as a vibrating spring, so the two atoms are bouncing closer together and farther apart – if this example is bad for some reason, please let me know in the comments so I can understand this stuff better!) An important discovery in quantum mechanics is the fact that the oscillations in such a system can’t be of just any size, but that the sizes are separated by a constant amount. The energy of the system is proportional to the size and frequency of the oscillations, and thus it too is quantized. By adjusting the units of energy if necessary, we can arrange that the possible states of such an oscillator have exactly 0, 1, 2, … units of energy.

Now consider a set of three identical such oscillators, arranged in a fixed lattice, so that the three can be distinguished by their position in the lattice. The first calculation, on p. 4 of the book, is figuring out how many different states such a system can be in if the total energy of the oscillations is 3.

With a bit of counting, one can see that the total number of different available states of such a system is 10. If you don’t want to try to write them all out for yourself, note that there are 3 possible states where one of the oscillators has energy 3 and the other two have energy 0; 6 possible states where one of the oscillators has energy 2, one has energy 1, and the other has energy 0; and 1 possible state where all three of the oscillators have energy 1. It’s easy to see that these are all the possibilities, because there are no ways for any oscillator to have negative energy, or non-integer energy, and we’re ignoring states in which there is any more energy, or some other aspect of the system has any relevant energy.

Now, there are other ways one might have tried to calculate this value, and gotten the wrong answer. For instance, one might have supposed the following – there are three units of energy, and three possibilities for the location of each of them, so there are 33=27 possible states of the system. But it’s clear what’s gone wrong in this calculation – we’ve counted the state where the first oscillator has the first unit of energy and the second oscillator has the second and third as distinct from the state where the first oscillator has the second unit of energy and the second oscillator has the first and third, when in actuality these are just the same state – the first oscillator has oscillations that are slightly larger than the base state, the second has oscillations that are twice as large, and the third oscillator is in the base state. The units of energy aren’t really distinct objects that can be located differently in the system – thinking in terms of units of energy is just a useful metaphor that helps draw attention to the fact that total energy can be conserved by an interaction in which one oscillator switches smaller vibrations while another oscillator switches to larger vibrations.

Now, if the three oscillators weren’t in a fixed lattice, then even the count of 10 states would be too many. If the oscillators were free to change position arbitrarily, then there would really only be 3 states – the one where one of them has energy 3 and the ohter two have energy 0; the one where one has energy 2, one has energy 1, and one has energy 0; and the one where all three have energy 1. In this case, one might say that the only reason 3 is the appropriate count rather than 10 is that we can’t tell apart the situation where oscillator A is the one with energy 3 and the one where oscillator B is the one with energy 3, because there’s no way for us to tell which is oscillator A and which is oscillator B. However, such an argument makes use of a particular metaphysical assumption, namely the idea that there is a fact of the matter as to which of the three oscillators is which. If we’re willing to help ourselves to this assumption, then we can get the answer to be 10 rather than 3.

But once we make this assumption, we might wonder again about the count of 27 – our only argument that 10 was the right answer rather than 27 is that there’s only one way for all three oscillators to have the same energy. But we could only justify this by the fact that we can’t detect any distinctions among states of this sort. Perhaps the units of energy really are distinct (if unobservable) objects, in which case 27 would be the appropriate count, even though we can only distinguish 10 of them, just as we standardly think that 10 is the appropriate count in the case of mobile oscillators, even though we can only distinguish 3 of them.

So at this point we are stumped – if we want to count the possible states of such a system, we can get the answers 3, 10, or 27 (and maybe others), but knowing which is right depends on knowing some facts about the metaphysics of the situation, and whether there are states that differ in unobservable ways, and if so, what sorts of unobservable ways they can differ.

Fortunately, there is a well-developed theory of statistical thermodynamics that uses these numbers of states to explain observational properties of very large systems of this sort. For instance (roughly speaking), the particular wavelength distribution of light emitted by a glowing-hot object tells us whether the energy is all concentrated in a few particles in the object (like the states in which one of the three oscillators has all three units of energy) or scattered evenly (like the states in which each oscillator has exactly one unit of energy). If there are really only 3 states for us to be counting, then systems should be equally likely to have each of the 3 possible emission spectra. If there are 10 states, then one spectrum should account for 30% of the systems, one for 60%, and one for 10%. If there are 27 states, then one spectrum should account for 3/27 of the systems, one for 18/27 of them, and the other for 6/27 of them. Thus, the statistics of large systems can tell us what the underlying states are like, so by counting the frequencies of large systems with various emission spectra (or other properties that can be correlated with the different macroscopically distinguishable states) we can figure out what the microstates are like, and thus figure out what the metaphysics of these systems is really like, even though there are no directly observable differences between the microstates, and the metaphysics is not directly observable.

Now strictly speaking, this inference is mediated by a further assumption, namely that an equal frequency of large systems will be in each of the microstates, whatever the microstates really are. If we grant this assumption, then frequencies of macrostates can tell us something about the microstates. And this assumption seems like a perfectly natural one – it seems to be the assumption of the Principle of Indifference, which many people have taken to be a basic principle of rationality. (Well, these are only the same assumption if we elide the distinction between frequency of systems of a certain type and one’s credence that a given individual system will be of that type.)

However, the Principle of Indifference has come under lots of fire over the past century or so. I won’t go into the details here, but suffice to say, most (or at least a very large minority of) people who think there are credences that obey rational constraints think that the Principle of Indifference is at best licensed in special cases, if ever.

In the case of thermodynamics though, we have a nice test – if we count the frequencies of systems in various macrostates, then we should either get frequencies in ratios of 1:1:1 or 3:6:1 or 3:18:6 (or whatever the relevant numbers are if we’re dealing with systems larger than three oscillators with three units of energy). If we regularly get frequencies that aren’t proportional to any of the ways we can think of counting distinct microstates that go with a system, then we’ll have to give up the relevant instances of the Principle of Indifference. Just as there doesn’t seem to be any conceptual necessity to a particular one of these three distributions being the empirically observed one, there doesn’t seem to be any conceptual necessity that the empirically observed distribution will be one of them, rather than, say, 4:183:29.

But as a matter of fact, in every situation where this has been tested (I believe) we actually do observe distributions that correspond to one of the ways of specifying microstates. Since the number of conceivable distributions is astronomical, the fact that we always get distributions that correspond to one of the small number we can conceive of seems to give strong evidence both that the relevant instances of the Principle of Indifference are correct (at least in their frequency form, if not as principles of rationality) as well as that the metaphysical assumptions made in carrying out that particular count are correct.

Thermodynamics, as a result, seems to let us use merely statistical evidence to support both some broad assumptions about the role of frequencies in observation, and to support metaphysical principles about unobservable distinctions that exist in the world.

The really interesting idea about this that came to me when I was considering the examples in the book is that we might use this to test certain traditional positions in metaphysics. Consider a system with three oscillators and five units of energy, and the macrostate where two of the oscillators each have two units of energy, and the third has one unit. If there really are distinct units of energy and distinct oscillators, then we get 90 states of this type. If there are distinct units of energy but no distinct oscillators, then we get 30 states. If there are distinct oscillators but no distinct units of energy then we get 3 states. If there are no distinct oscillators or units of energy then we get 1 state. (I believe there are types of actual systems where each of these accounts ends up being the appropriate one, depending on whether they’re made of fermions or bosons, and whether it’s units of energy or something else that are being shared.)

But the answer of 3 states in the case of distinct oscillators but no distinct units of energy depended on the thought that all that matters is which oscillator has the property of having one unit of energy, and which ones have the property of having two units of energy. A metaphysician who believes in tropes will have to say that this is the wrong way of thinking about things – there’s not just one property of having two units of energy, but two particular tropes of having two units of energy. So if we believe in tropes, then the proper count here may turn out to be 6, because either of the two high energy oscillators could have either of the two high energy tropes. So theoretically, it seems that thermodynamics should be able to give evidence for or against the existence of tropes, and possibly other theoretical posits of analytic metaphysics.

Now perhaps if I were to work out the full set of counts, the trope theorist would double the count for all the other macrostates as well – in such a case, the trope theorist and the non-trope theorist would get different counts, but the same proportions, and thus no different predictions. However, I don’t think this will be the case for all systems, so there should theoretically be some test cases available.

Of course, different trope theorists will also endorse different ways of counting – perhaps this system shouldn’t be thought of as having only six possibilities, because there’s no reason to limit consideration to just the two high energy tropes there actually are in any realization of the system. There are infinitely many possible distinct high energy tropes that could have been used, and not just two, so there are really infinitely many possible microstates corresponding to a given macrostate. This would be an unpleasant result, because it would make the frequencies impossible to calculate (they’d be ratios of infinity to infinity), but the trope theorist of this sort may be able to save things by considering ratios of “3 times infinity to 6 times infinity to 1 times infinity” to be the same in some sense as 3:6:1, so that she gets the same predicted frequencies as the non-trope theorist.

Thus, it might turn out that people with different metaphysical theories would have several different ways to come up with the same predicted ratios, but they’d need to work out what the math actually looks like. Prima facie, it looks like there might actually be different empirical predictions the different claims might make, even though the individual distinctions are unobservable, just as in the case of the units of energy and the distinct oscillators.

Posted by Kenny Easwaran in Uncategorized

5 Comments »

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5 Responses to “The Metaphysics of Statistical Thermodynamics”

  1. SFrench says:

    Hey Kenny,
    What you’ve described is a general approach first articulated by Reichenbach and subsequently pursued by a number of us in the philosophy of physics (eg Bas van Fraassen, Nick Huggett, myself), particularly with regard to the metaphysics of individuality and identity. So, I have 3 quick things to note;
    1. The ‘received’ view for many years was, to put it crudely, that counting permutations supported the claim that particles obeying classical statistical mechanics should be regarded as individuals, whereas those covered by quantum stats (whether Fermi-Dirac or Bose-Einstein) had to be understood as non-individuals in some sense. Again, the argument, put very simply goes something like this: consider two particles distributed over two energy states – to get the probabilities right, classical stats count this arrangement twice, whereas q, stats count it only once and this is accounted for by the classical particles being individuals, the quantum ones not. Now this assumes the principle of indifference but Bach, I dimly recall, questioned this assumption and showed how one can recover the right stats without p.o.i. I can dig up the reference if you’re interested.
    2. This received view has been rejected: by Huggett, who argues that classical stats can be accounted for without assuming particles as individuals; and by van Fraassen and myself, who have separately argued that q. stats does not imply non-individuality – one can maintain they are individuals but explain the non-classical probabilities in terms of the relevant symmetry principles (symmetrisation for B-E; anti-symm for F-D). In effect the particles’ individuality – understood via haecceity or whatever – becomes ‘veiled’ by the relations they enter into. Now, maybe the trope theorist can do something similar with your example (which kinda mixes quantum and semi-classical considerations, as often seems to be the case with textbooks!), effectively ensuring we get the right ‘count’ by pushing the metaphysical implications off onto the symmetrisation.
    3. The negative conclusion of the above work was that, again putting it crudely, its not so straightforward to draw metaphysical conclusions from the relevant physics. One way to express this is through a kind of ‘metaphysical underdetermination’ wrt individuality and non-individuality in QM. I used to think that some positions do get ruled out, such as those that appeal to some form of the Identity of Indiscernibles, but Saunders has argued that a form of weak discernibility can be articulated in certain quantum situations (eg singlet states of spin) which effectively preserves the PII (although see Katherine Hawley’s recent piece in Mind).
    Hope that helps some.
    cheers,
    Steven

  2. SFrench says:

    Sorry, me again! So, here’s an obvious way the trope theorist can respond:
    usually we think of permuting objects – particles – over states, but we can always permute the states around the objects, as it were. Think of balls and boxes – we can switch balls between boxes, or boxes ‘under’ balls. Indeed, in QM not only can we define particle permutation operators that operate on the particle labels, which is what we usually invoke in the standard treatment (permuting all particles over states and then taking either symmetric or anti-symmetric combinations) but we can also define ‘place’ or state permutation operators, that operate on the ‘state’ labels (Dirac has a beautiful presentation of all this in his Principles of QM; under normal circs the two kinds of operator have the same effect but they do come apart for so-called ‘parastatistics’ which yield q. stats others than B-E or F-D and were big in the 60s and early 70s, but lest not go there).
    So, now consider the permutation of states ‘under’ particles – the important thing for the physics is that the ‘right’ count comes out – that is, for ex., the arrangement of two states ‘under’ two particles gets counted twice for Maxwell-Boltzmann stats and once only for both of B-E and F-D. But the trope theorist can surely always guarantee that by simply insisting that each state = two tropes, say (using Kenny’s example), rather than one state, one property, as it were. Now, that requires some consideration of how a physical state decomposes into tropes, but that doesn’t seem a big deal (but then I know precious little trope theory!).
    So, the count for the physics will always come out right, as long as we pay attention to trope composition. Does that sound right?
    cheers,
    Steven

  3. jeffrey ketland says:

    Kenny: “… we can arrange that the possible states of such an oscillator have exactly 0, 1, 2, … units of energy.

    Strictly, a quantized oscillator must have non-zero energy. The lowest energy eigenstate has energy 1/2 \hbar \omega (where \omega is the characteristic frequency). Google “zero-point energy” if you’re curious.

  4. Kenny Easwaran says:

    Hi Steven,

    That’s useful to know more about the history of this sort of discussion! I’m not sure I’ve entirely followed all of the steps you mention, but it sounds like, even though the statistics might not determine a unique correct metaphysics, it puts some constraints on what (say) the trope theorist can say, and also has spurred some discussion of what the relevant symmetry principles will have to be like.

    Anyway, I’d love to see that Bach piece you mention that shows what can be done without the principle of indifference – that would help make all this make a lot more sense to me, because I’m quite skeptical of indifference principles.

    Jeffrey – The book I was looking at pointed out that the energies had to be of the form (1/2+n)hv, where n is an integer (or something similar), so I was a bit surprised when it made the move to talking about the total energy being an integer, but that’s what I followed here. I suppose what they must have been doing is taking the total of the energy, changing the units so that the differences were 1, and then counting energy in terms of difference from the lowest possible energy state for the system, rather than total energy. Anyway, hopefully I’ll figure out a bit more about this zero point energy as I work further through the book, or else I’ll do some googling as you suggest (and wikipedia-ing, assuming that the relevant wikipedia articles don’t presuppose much more background knowledge than I have).

  5. SFrench says:

    Hi Kenny,
    Here are some refs (although its been years since I looked at this):

    A, Bach, The Concept of Indistinguishable Particles in Classical and Quantum Physics, Foundations of Physics 18 (1988), 639-649

    A Bach, Indistinguishable Classical Particles, Springer 1997

    Also,
    Classical and quantum statistics as finite random processes
    Journal Foundations of Physics
    Publisher Springer Netherlands
    ISSN 0015-9018 (Print) 1572-9516 (Online)
    Issue Volume 19, Number 6 / June, 1989
    Pages 743-754
    D. Costantini1 and U. Garibaldi2

    cheers,
    Steven

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