The Necessity of Statistical Mechanics for Getting Macro From Micro

This is the first part in a three-part series on the foundations of statistical mechanics.

1. The Necessity of Statistical Mechanics for Getting Macro From Micro
2. Is The Fundamental Postulate of Statistical Mechanics A Priori?
3. The Central Paradox of Statistical Mechanics: The Problem of The Past

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Let’s start this out with a thought experiment. Imagine that you have access to the exact fundamental laws of physics. Suppose further that you have unlimited computing power, for instance, you have an oracle that can instantly complete any computable task. What then do you know about the world?

The tempting answer: Everything! But of course, upon further consideration, you are missing a crucial ingredient: the initial conditions of the universe. The laws themselves aren’t enough to tell you about your universe, as many different universes are compatible with the laws. By specifying the state of the universe at any one time (which incidentally does not have to be an “initial” time), though, you should be able to narrow down this set of compatible universes. So let’s amend our question:

Suppose that you have unlimited computing power, that you know the exact microphysical laws, and that you know the state of the universe at some moment. Then what do you know about the world?

The answer is: It depends! What exactly do you know about the state of the universe? Do you know it’s exact microstate? As in, do you know the position and momentum of every single particle in the universe? If so, then yes, the entire past and future of the universe are accessible to you. But suppose that instead of knowing the exact microstate, you only have access to a macroscopic description of the universe. For example, maybe you have a temperature map as well as a particle density function over the universe. Or perhaps you know the exact states of some particles, just not all of them.

Well, if you only have access to the macrostate of the system (which, notice, is the epistemic situation that we find ourselves in, being that full access to the exact microstate of the universe is as technologically remote as can be), then it should be clear that you can’t specify the exact microstate at all other times. This is nothing too surprising or interesting… starting with imperfect knowledge you will not arrive at perfect knowledge. But we might hope that in the absence of a full description of the microstate of the universe at all other times, you could at least give a detailed macroscopic description of the universe at other times.

That is, here’s what seems like a reasonable expectation: If I had infinite computational power, knew the exact microphysical laws, and knew, say, that a closed box was occupied by a cloud of noninteracting gas in its corner, then I should be able to draw the conclusion that “The gas will disperse.” Or, if I knew that an ice cube was sitting outdoors on a table in the sun, then I should be able to apply my knowledge of microphysics to conclude that “The ice cube will melt”. And we’d hope that in addition to being able to make statements like these, we’d also be able to produce precise predictions for how long it would take for the gas to become uniformly distributed over the box, or for how long it would take for the ice cube to melt.

Here is the interesting and surprising bit. It turns out that this is in principle impossible to do. Just the exact microphysical laws and an infinity of computing power is not enough to do the job! In fact, the microphysical laws will in general tell us almost nothing about the future evolution or past history of macroscopic systems!

Take this in for a moment. You might not believe me (especially if you’re a physicist). For one thing, we don’t know the exact form of the microphysical laws. It would seem that such a bold statement about their insufficiencies would require us to at least first know what they are, right? No, it turns out that the statement that microphysics is is far too weak to tell us about the behavior of macroscopic systems holds for an enormously large class of possible laws of physics, a class that we are very sure that our universe belongs to.

Let’s prove this. We start out with the following observation that will be familiar to physicists: the microphysical laws appear to be time-reversible. That is, it appears to be the case that for every possible evolution of a system compatible with the laws of physics, the time-reverse of that evolution (obtained by simply reversing the trajectories of all particles) is also perfectly compatible with the laws of physics.*

This is surprising! Doesn’t it seem like there are trajectories that are physically possible for particles to take, such that their time reverse is physically impossible? Doesn’t it seem like classical mechanics would say that a ball sitting on the ground couldn’t suddenly bounce up to your hand? An egg unscramble? A gas collect in the corner of a room? The answer to all of the above is no. Classical mechanics, and fundamental physics in general, admits the possibilities of all these things. A fun puzzle for you is to think about why the first example (the ball initially at rest on the ground bouncing up higher and higher until it comes to rest in your hand) is not a violation of the conservation of energy.

Now here’s the argument: Suppose that you have a box that you know is filled with an ideal gas at equilibrium (uniformly spread through the volume). There are many many (infinitely many) microstates that are compatible with this description. We can conclusively say that in 15 minutes the gas will still be dispersed only if all of these microstates, when evolved forward 15 minutes, end up dispersed.

But, and here’s the crucial step, we also know that there exist very peculiar states (such as the macrostate in which all the gas particles have come together to form a perfect statuette of Michael Jackson) such that these states will in 15 minutes evolve to the dispersed state. And by time reversibility, this tells us that there is another perfectly valid history of the gas that starts uniformly dispersed and evolves over 15 minutes into a perfect statuette of Michael Jackson. That is, if we believe that complicated configurations of gases disperse, and believe that physics is time-reversible, then you must also believe that there are microstates compatible with dispersed states of gas that will in the next moment coalesce into some complicated configuration.

1. A collection of gas shaped exactly like Michael Jackson will disperse uniformly across its container.
2. Physics is time reversible.
3. So uniformly dispersed gases can coalesce into a collection of gases shaped exactly like Michael Jackson.

At this point you might be thinking “yeah, sure, microphysics doesn’t in principle rule out the possibility that a uniformly dispersed gas will coalesce into Michael Jackson, or any other crazy configuration. But who cares? It’s so incredibly unlikely!” To which the response is: Yes, exactly, it’s extremely unlikely. But nothing in the microphysical laws says this! Look as hard as you can at the laws of motion, you will not find a probability distribution over the likelihood of the different microstates compatible with a given macrostate. And indeed, different initial conditions of the universe will give different such frequencies distributions! To make any statements about the relative likelihood of some microstates over others, you need some principle above and beyond the microphysical laws.

To summarize. All that microphysics + infinite computing power allows you to say about a macrostate is the following: Here are all the microstates that are compatible with that macrostate, and here are all the past and future histories of each of these microstates. And given time reversibility, these future histories cover an enormously diverse set of predictions about the future, from “the gas will disperse” to “the gas will form into a statuette of Michael Jackson”. To get reasonable predictions about how the world will actually behave, we need some other principle, a principle that allows us to disregard these “perverse” microstates. And microphysics contains no such principle.

Statistical mechanics is thus the study of the necessary augmentation to a fundamental theory of physics that allows us to make predictions about the world, given that we are not in the position to know its exact microstate. This necessary augmentation is known as the fundamental postulate of statistical mechanics, and it takes the form of a probability distribution over microstates. Some people describe the postulate as saying “all microstates being equally likely”, but that phrasing is a big mistake, as the sentence “all states are equally likely” is not well defined over a continuous set of states. (More on that in a bit.) To really understand the fundamental postulate, we have to introduce the notion of phase space.

The phase space for a system is a mathematical space in which every point represents a full specification of the positions and momenta of all particles in the system. So, for example, a system consisting of 1000 classical particles swimming around in an infinite universe would have 6000 degrees of freedom (three position coordinates and three momentum coordinates per particle). Each of these degrees of freedom is isomorphic to the real numbers. So phase space for this system must be ℝ⁶⁰⁰⁰, and a point in phase space is a specification of the values of all 6000 degrees of freedom. In general, for N classical particles, phase space is ℝ^6N.

With the concept of phase space in hand, we can define the fundamental postulate of statistical mechanics. This is: the probability distribution over microstates compatible with a given macrostate is uniform over the corresponding volume of phase space.

It turns out that if you just measure the volume of the “perverse states” in phase space, you end up finding that it composes approximately 0% of the volume of compatible microstates in phase space. This of course allows us to say of perverse states, “Sure they’re there, and technically it’s possible that my system is in such a state, but it’s so incredibly unlikely that it makes virtually no impact on my prediction of the future behavior of my system.” And indeed, when you start going through the math and seeing the way that systems most likely evolve given the fundamental postulate, you see that the predictions you get match beautifully with our observations of nature.

Next time: What is the epistemic status of the fundamental postulate? Do we have good a priori reasons to believe it?

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* There are some subtleties here. For one, we think that there actually is a very small time asymmetry in the weak nuclear force. And some collapse interpretations of quantum mechanics have the collapse of the wave function as an irreversible process, although Everettian quantum mechanics denies this. For the moment, let’s disregard all of that. The time asymmetry in the weak nuclear force is not going to have any relevant effect on the proof made here, besides making it uglier and more complicated. What we need is technically not exact time-reversibility, but very-approximate time-reversibility. And that we have. Collapsing wave functions are a more troubling business, and are a genuine way out of the argument made in this post.