Here are the slides from the physics colloquium I gave at UC Santa Cruz last week, entitled “Why is the Past Different from the Future? The Origin of the Universe and the Arrow of Time.” (Also in pdf.)
The real reason I’m sharing this with you is because this talk provoked one of the best responses I’ve ever received, which the provokee felt moved to share with me:
Finally, the magnitude of the entropy of the universe as a function of time is a very interesting problem for cosmology, but to suggest that a law of physics depends on it is sheer nonsense. Carroll’s statement that the second law owes its existence to cosmology is one of the dummest [sic] remarks I heard in any of our physics colloquia, apart from [redacted]’s earlier remarks about consciousness in quantum mechanics. I am astounded that physicists in the audience always listen politely to such nonsense. Afterwards, I had dinner with some graduate students who readily understood my objections, but Carroll remained adamant.
My powers of persuasion are apparently not always fully efficacious.
Also, that marvelous illustration of entropy in the bottom right of the above slide? Alan Guth’s office.
Update: Originally added as a comment, but I’m moving it up here–
The point of the “objection” is extremely simple, as is the reason why it is irrelevant. Suppose we had a thermodynamic system, described by certain macroscopic variables, not quite in equilibrium. Suppose further that we chose a random microstate compatible with the macroscopic variables (as you do, for example, in a numerical simulation). Then, following the evolution of that microstate into the future, it is overwhelmingly likely that the entropy will increase. Voila, we have “derived” the Second Law.
However, it is also overwhelmingly likely that evolving that microstate into the past will lead to an increase in entropy. Which is not true of the universe in which we live. So the above exercise, while it gets the right answer for the future, is not actually “right,” if what we care about is describing the real world. Which I do. If we want to understand the distribution function on microstates that is actually true, we need to impose a low-entropy condition in the past; there is no way to get it from purely time-symmetric assumptions.
Boltzmann’s H-theorem, while interesting and important, is even worse. It makes an assumption that is not true (molecular chaos) to reach a conclusion that is not true (the entropy is certain, not just likely, to increase toward the future — and also to the past).
The nice thing about stat mech is that almost any distribution function will work to derive the Second Law, as long as you don’t put some constraints on the future state. That’s why textbook stat mech does a perfectly good job without talking about the Big Bang. But if you want to describe why the Second Law actually works in the real world in which we actually live, cosmology inevitably comes into play.
Speaking of model-specific derivations of the Second Law, and odd conspiracies of apparently unrelated physical laws to prevent violations of it, I’m surprised no one alluded to the so-called laws of black hole mechanics and their relationship to the laws of thermodynamics. The peculiarly intimate relationship between general relativity and thermodynamics is probably the strongest indication that the laws of thermodynamics are “wired in” to the laws of physics at a very deep level, in a way that transcends both classical and quantum statistical mechanics as presently understood.
(If this sounds like semi-mystical bullshit I’m sorry…)
Thanks for posting your slide presentations. They’re great. Best regards,
– Steve Esser
The best thing about the picture of Alan’s office is that there actually IS a mexican hat in the clutter!
PS: My comment 26 is related to the second paragraph in comment 4 (from Neil B).
Sean,
After lengthy discussion, the Critic apparently agrees that your basic argument is correct, but only in classical mechanics.
And what was that drink you claim you spilled? I somehow remember it was something with peaches and cream…
And how hard can it be to escape a Banana Slug anyway?
Good to see you.
Memories of kinetic theory come flooding back. . . .
From one perspective, going from the assumption of molecular chaos to a temporally increasing entropy sounds like an exercise in triviality: you throw away information, you get entropy. As John Baez said back in 1996 (week 80 of This Week’s Finds):
Nice description Sean, but one thing still bothers me. Presumeably the current macrostate includes all our books, artifacts, memory traces, and the correlations encoded in our theories and observations. If so, and it’s still true that vast majority of microstates compatible with that macrostate would backward evolve (devolve?) into states of higher entropy, what’s our justification for assuming that we aren’t just a fluctuation?
I don’t see how you get out of that trap without saying something like “that way madness lies.” If the past wasn’t low entropy, we can’t really do physics – oops – sounds anthropic.
Thank you for posting the slides.
My mind kept drifting to literary theory… how the interpretations of any text expand… analogical entropy.
Thinking of religious texts–they don’t expand to a vacuum state, to infinite interpretations, and therefore mean nothing… but continuously give birth to “baby universes” .. in essence, interpretations which themselves constitute new cannons of meaning for future generations to interpret.
Wonderful stuff… thank you for this blog and your valuable time away from your real work.
what’s our justification for assuming that we aren’t just a fluctuation?
Principles of mediocrity yet again. There should be vastly more brains without all that crap, so the existence of all this extraneous stuff means that we’re special somehow.
Sean, you make the same damn mistake that almost everyone discussing this subject makes, which renders your discussion incomprehensible.
The H-Theorem is not about the time evolution of a particular system. It is about the time evolution of a probability distribution function (PDF).
The time evolution of a PDF is not (directly) subject to Newton’s laws; rather it is a new type of physical entity whose behavior is (like all physics) something we can try to model mathematically, and see if the results work.
As such, all the standard complaints about the “problems” of the H-theorem eg recurrance of states or Liouville’s theorem are simply irrelevant. Those complaints refer to something different, just like your complaint
“to reach a conclusion that is not true (the entropy is certain, not just likely, to increase toward the future — and also to the past).”
You are complaining here about the behavior of a particular process drawn from
a universe of random processes. The PDF describing that universe does indeed have monotonic behavior; the fact that you might occasionally get 100 heads in a row doesn’t change the fact that the probability of a coin toss for heads is one half.
The fact that both Boltzmann and Ehrenfest were confused on this issue and thus made no sense on the subjec doesn’t change the fact that it is freaking shameful that, 100 years later, most physicists remain just as confused. Talking about PDFs is both more rigorous and a whole lot more comprehensible than this vague nonsense of microstates vs macrostates.
CIP, Aaron gives the most important answer. If fluctuations appeared with a truly thermal distribution, our whole universe would fluctuate into existence much less frequently than other anthropically-allowed (or whatever-allowed) configurations.
David Albert has a potentially stronger objection. If we are a fluctuation, it’s overwhelmingly likely that we came from a higher-entropy (near) past. All of our records etc. are just statistical flukes, not reliable traces of past historical events. But that includes all of the data we’ve ever used to invent the laws of physics, including the laws of statistical mechanics we’ve used to derive this conclusion. So the idea that we’re just a fluctuation, while not rule-outable, is cognitively unstable — there can never be any self-consistent justification for it.
HOLY BUTT. CosmicVariance MUST participate in this.
Another thread that compels one to inquire:What comes first,particles to create Entropy, or Entropy that governs particles?
The really interesting slide’s are 3-4-5
In a low entropy early Universe, is the Particle_number in phase space, born from Black Hole Production, ie Maximum number of particles contained within a minimum volume of space. When a Black hole is “full”, then and ONLY then do Particles rebound away as particle_creation?
Interestingly, this early state scenario has a counterpart for the other end of the Universe Entropic cycle, ie minimum particle number contained within a maximum volume of space?
In some inflation models, Andre Linde for instance, there is a prediction/need for a “slow-roll2 out of inflation period, this needs a small particle mumber (less particles here means less impacts, thus less entropy), so the big bang was followed by a period of little thuds!
The end of Universe phase really compels one to ask of E=mc2, is the ENERGY a particle energy or a Vacuum ENERGY.
Sean clearly makes a good case for the Varying Laws Of Physics, varying of course at either scales, end and the beginning, a general case can be made that the physical laws took a vast amount of time to reach equilibrium (present time constant), and will gradually move to an increased change rate, as the expansion increases, the laws of physics would have to change at a spcific increasing rate in order to maintain “order”.
We note, that as the total particle number in the current Universe reaches its pinnacle, then Gravity rules as far as the eye’s or telescopes can see, but this wont last forever!
As far as the Universe goes, change is always what lays ahead, the past does not and CANNOT change, the Arrow of (present) Time ensure’s the arrow points “one-way”, towards the future.
This is not to say that at the Universe’s end, there would not be moments that become intertwined, or entangled, to the future non existing observers, it would not really matter, what really matters is that the process continue’s.
The only concevable way that the Laws of Physics have not changed, is there to actually be no present time change occuring, for there to be no change occuring now, means no Entropy, and consequently no particle collisions anywhere in the Universe.
Three questions:
1. Which test, experimental or observational, which could distinguish this hypothesis (arrow of time has cosmological origin) from its opposite?
2. Does cosmology also have anything to say about the other problems of time (time direction singled out non-covariantly, Hamiltonian is a constraint, etc.)?
3. According to Pauli’s theorem, time is a c-number rather than an observable in QM (although I expect the opposite to be true in the UV completion of QFT). Do you make any assumption which contradicts this, e.g. that time is measured by clocks?
So, Sean wrote:
whereas when I last thought about this theorem, I claimed otherwise (see Blake Stacey’s comment).
So, is the assumption behind Boltzmann’s H-theorem time-symmetric or not? And what about the theorem’s conclusion: time-symmetric or not? Is the conclusion that entropy increases both in the future and the past… or just towards the future?
I’m pretty darn sure the assumptions and conclusions are time-asymmetric.
Boltzmann actually called his assumption the “Stosszahlansatz”, or “collision number assumption”.
So: what’s the Stosszahlansatz?
It goes like this.
Suppose we have a homogeneous gas of particles and the density of
them with momentum p is f(p). Consider only 2-particle interactions
and let w(p1, p2; p1′, p2′) be the transition rate at which pairs of
particles with momenta p1, p2 bounce off each other and become
pairs with momenta p1′, p2′. To keep things simple let me assume
symmetry of the basic laws under time and space reversal, which
gives:
w(p1, p2; p1′, p2′) = w(p1′, p2′; p1, p2).
In this case the Stosszahlansatz says:
df(p1)/dt =
integral w(p1, p2; p1′, p2′) [f(p1′)f(p2′) – f(p1)f(p2)] dp2 dp1′ dp2′
This is very sensible-looking if you think about it. Using this,
Boltzmann proves the H-theorem. Namely, the derivative of the
following function is less than or equal to zero:
H(t) = integral f(p) ln f(p) dp
This is basically minus the entropy. So, entropy increases, given the Stosszahlansatz!
The proof is an easy calculation, and you can find it in section 3.1
Zeh’s The Physical Basis of the Direction of Time (a good book).
Now: since the output of the H-theorem is time-asymmetric, and all
the inputs are time-symmetric except the Stosszahlansatz, we should
immediately suspect that the Stosszahlansatz is time-asymmetric.
And it is!
“In this case the Stosszahlansatz says:
df(p1)/dt =
integral w(p1, p2; p1′, p2′) [f(p1′)f(p2′) – f(p1)f(p2)] dp2 dp1′ dp2′
”
“Now: since the output of the H-theorem is time-asymmetric, and all
the inputs are time-symmetric except the Stosszahlansatz, we should
immediately suspect that the Stosszahlansatz is time-asymmetric.
And it is!”
The above Stosszahlansatz doesn’t look obviously time-asymmetric to me. In fact, changing t to -t and interchanging primed and unprimed p variables seems to give a symmetrical result. What am I missing?
Just like I tell neodarwinians who try to use the multiverse to lose the implications of the observed universe:
Prove that your multiverse is necessary to the ToE or a complete proven theory of quantum gavity. Otherwise, (and the same goes for inflationary theory)… you’re rambling bla-bla-bla woulda’ coulda’ shoulda’ whatif maybe inferred implied assumed…. crap, that avoids the most obvious solution to the problem.
This one always cracks me up, becuase the people who say this are the same ones who refuse to look for the most apparent resolution to the problem, because, and in spite of the evidence, they still don’t believe that a strong anthropic constraint on the forces is possible. My favorite analogy being the refusal to believe that the guy that’s standing over the dead body while holding a smoking couldn’t have done it, the damning evidence supercedes to the willfully ignorant belief, until proven otherwise, per the scientific method… children.
The problems, entropy, fine-tuning… (you name it!) are very simply resolved without multiverses nor inflationary bandaids in this old post to the moderated research group, and go figure… you don’t need to know anything more than general relativity to understand it, although you can easily see the seeds of a valid theory of quantum gravity contained within the physics.
But the strong anthropic constraint means that I might as well be giving physics to the brick wall across the street. I’ll bet that even hard-core loopers, like Baez, would accept the multiverse before they would admit that the strong anthropic solution already exists.
The Second Law of Thermodynamics says “god” doesn’t throw dice…
Our Darwinian Universe
Continue to willfully ignore the strong anthropic constraint at your own peril…
…conditions there are finely tuned for reasons that remain utterly mysterious
L. O. L.
Kind of missing the point trying to connect the Second Law of Thermodynamics to the arrow of time? After all the Second Law of Thermodynamics is just a restatement of the central limit theorem, with the words “compatible macrostate observable” having the precise mathematical meaning of “a random variable with well defined first and second moments”
The real question is why is there a well ordering on observations, why can’t we make observations in any order we want?
Maybe translations through space-time can’t be represented by an algebra of unitary transformation on a Hilbert Space? But rather by operators with either a non-trivial kernal or non-trivial image sub-space.
Dear Sean —
Thanks for sharing with us your work on this interesting topic.
I would like to dive into the discussion of the main question,
but being only a Bear of Little Brain I have to ask some lower
level questions first.
You start from the basic observation that if entropy is always
increasing then it must have been lower in the past; you then
infer that some early/initial state must therefore have been a
very special, low-entropy state. (Sir Roger goes through this
same logic in The Emperor’s New Mind, and again in his
latest road show.) The question is, do you have to picture that
early low-entropy state as being out of equlibrium?
The standard story of the thermal early Universe is that
(1) essentially all the entropy is in relativistic gases, and
(2) these gases are in thermal equilibrium, and so are already
at maximum entropy. So in what sense is the early thermal
phase a “special” state? I don’t see that it can be.
The state of the Universe once it crosses to being
matter-dominated _is_ a special state since it is very smooth
and once gravity is allowed to act you can pick up a lot of
entropy when matter begins to clump together. Eventually
we reach maximum entropy, and hence equilibrium, again
when all baryons and dark matter are squozen into black holes*,
which is clearly a great increase in entropy over the smoothly
distributed state.
The question then becomes, how are we to understand the
great jump in “specialness” at the radiation-matter transition?
The Universe goes from maximum possible entropy, ie not
special at all, to being welll below maximum entropy, ie
very special. The entropy per co-moving volume didn’t change
at this transition; it’s more like the ceiling on entropy suddenly
moved greatly upward.
To put it differently, it’s like walking along and suddenly having
the floor fall out from under you: your position in the potential
is the same, but somehow you suddenly have a lot more
low-entropy potential energy at your disposal (if you can
figure out, very quickly, how to take advantage of it).
So if what you want to track is “specialness” rather than
entropy per se, you have to somehow decide whether/why
the radiation-matter transition — which seems pretty
non-remarkable microscopically — is such a landmark event.
Thanks for whatever insight
you can offer. Regards,
Paul Stankus
* Even once all matter and dark matter are in black holes
we still haven’t reached equilibrium, since we can always
pick up more entropy by letting the black holes merge,
and merge again, and so on until they can’t collide any
more. So in a Universe which is expanding arbitrarily
slowly there seems to be no upper limit on entropy per
volume and so no ultimate equilibrium.
Historically, the second law concerns ‘the behavior of particles in a ‘gas”.
Information theoretic formulations now focus on probabilities. In the latter context, closed systems tend to move from less probable to more probable configurations.
This contrasts with the classic formulation, that usually talks about moving from ‘ordered’ states towards ‘less ordered states’.
A phase change, on cooling, involves ‘apparent’ increase in order; but is a transition to a more probable state. Likewise for evaporation of back holes, or sublimation of solids, crystalline or amorphous. And in a universe without dark energy, a contracting, heating crunch represents the more likely direction.
So this semantic ‘trick’ solves the problem of the direction of time.
HTH
John’s comment above about the Stosszahlanzatz is basically right, but there is a little bit more behind the story. (And it is all there in Zeh’s book, if you bother to unpack it.)
Boltzmann actually did two innocuous-sounding things that enabled him to miraculously derive a time-asymmetric conclusion from time-symmetric microscopic laws. First, he worked with distribution functions defined in the single-particle phase space. That is, if you have two particles, you can either keep track of them as one point in a two-particle phase space (separate dimensions for the position and momentum of each particle) or as two points in the single-particle phase space. Both are completely equivalent. But if you go to a distribution function in that single-particle phase space, f(q,p), then you have thrown away information — implicitly, you’ve coarse-grained. In particular, you can’t keep track of the correlations between different particle momenta. There’s no way, for example, to encode “the momentum of particle two is opposite to that of particle one” in such a distribution function. (You can keep track of it with a distribution function on the full multi-particle phase space, which is what Gibbs ultimately used. But there entropy is strictly conserved, unless you coarse-grain by hand.)
Because you’ve thrown away information, there is no autonomous dynamics for the distribution function. That is, given f(p,q), you can’t just derive an equation for its time derivative from the Hamiltonian equations of motion. You need to make another assumption, which for Boltzmann was the Stosszahlansatz referred to above. You can justify it by saying that “at the initial moment, there truly are no momentum correlations” (plus some truly innocent technical assumptions, like neglecting collisions with more than two particles). But of course the real Hamiltonian dynamics then instantly creates momentum correlations. So that innocent-sounding assumption is equivalent to “there are no momentum correlations before the collisions (even though there will be afterwards).” Which begins to sound a bit explicitly time-asymmetric.
The way I presented the story in my talk was to strictly impose molecular chaos (no momentum correlations) at one moment in time. That’s really breaking time-translation invariance, not time-reversal. From that you could straightforwardly derive that entropy should increase to the past and the future, given the real Hamiltonian dynamics. What the real Boltzmann equation does is effectively to assume molecular chaos, chug forward one timestep, and then re-assume molecular chaos. It’s equivalent to a dynamical coarse-graning, because the distribution function on the single-particle phase space can’t carry along all the fine-grained information.
The way I was taught (see lectures 8 and 9 in Mehran Kardar’s notes) the irreversibility enters when you approximate the two-particle distribution function f(q1, q2) with the product of the one-particle distribution functions, f(q1)f(q2). Making this approximation to terminate the BBGKY hierarchy yields the Boltzmann equation, which as Kardar proves in lecture 9 is not time-reversal symmetric.
I came across a 1974 paper by Blatt and Opie in which the Boltzmann equation is derived from Liouville’s Theorem by mandating that the approximations involved preserve all one-particle expectation values, while neglecting two-particle and higher. It’s the same idea, I believe: throwing away information (about correlations) means that your distribution function will increase in entropy.
John Baez’s comment (#40) and the responses by Sean and Blake (#s 46 & 47) seem to show that the entropy increase is inextricably linked to information loss. Could it be that the entropy increase is simply caused by information leaking away, perhaps to points beyond a horizon, or lost to interactions with objects outside the domain in question?
To get non-conserved entropy, we must coarse grain, but coarse graining is an intellectual operation, not a physical one. Is information leakage the physical counterpart?
Thanks, Sean! You explained something that I should have remembered from ancient conversations on sci.physics.research: the “assumption of molecular chaos” and the “Stosszahlansatz” are quite different! The former is time-reversal symmetric and (in most situations) unrealistic; since it predicts that entropy increases both in the future and the best, starting from a given moment of low entropy. The latter breaks time-reversal symmetry and is (seemingly often) more realistic, since it predicts that entropy increases only in the future.
Got it.
I’m really fascinated by the arrow of time and its relation to gravity and cosmology. In the future I plan to study this more. In the past, too!
Paul Stankus writes:
Roger Penrose discusses this issue at length in The Emperor’s New Mind. I don’t agree with everything in this book – indeed, some parts are famously controversial, and probably wrong. But, I think his treatment of this issue is quite good.
Briefly: while most of the entropy of the early universe is in radiation and hot gases, and this stuff is close to thermal equilibrium if one neglects gravity, the early universe is very far from equilibrium if one takes gravity into account! Gravitating systems increase their entropy by clumping up and getting hotter! As the universe ages, this is what happens.
So, the early state of the universe is very special – because it’s smeared out and homogeneous, not lumpy the way gravitating systems become as their entropy starts increasing.
If this seems paradoxical, well, it’s because gravity is funny!
Mind you, I don’t think we understand this stuff as well as we should. The whole concept of “thermal equilibrium” becomes rather shaky when one tries to take gravity into account. This is true already at the Newtonian level, but even more so when general relativity is thrown into the mix, and the concept of energy, and thus free energy, becomes much trickier.