## Friday, December 07, 2007

### Myths about the arrow of time

Originally published on December 3rd. At the end of this text, I added a textbook example of a postdiction, showing that the whole controversy hides in the Bayesian priors once again.
Certain physicists keep on promoting the arrow of time - a scientific question that was settled between 1850 and 1872 and that became standard material of college physics (and maybe high-school physics) - as an area of active research in 2007.

Because most of the 21st century debates about the arrow of time are profoundly irrational, I will try to reveal and clarify several inherently religious (more precisely, anti-religious but equally metaphysical) myths underlying this debate.

Boltzmann's brains
Predicting the past
Is cosmology behind the second law?
The arrow of time is the arrow pointing from the past to the future. This concept is a symbol of the macroscopic asymmetry between the future and the past. Eggs break but they never unbreak, chicks get older but they never get younger, the heat always flows from the hotter object to the cooler one, the friction force slows things down. The reverse processes don't seem to exist. The origin of this asymmetry might be confusing for beginners but it has been well understood for a long time.

After each myth written in italics, I will describe the reality. The first myth will be about the history but I will jump to the essence of the problem right after that.

Arrow of time according to beer physics.

Myth: The arrow of time became very important or confusing in recent years.

The arrow of time has been understood in terms of physical quantities for more than 150 years: Rudolf Clausius formulated the second law of thermodynamics in 1850 even though the idea goes back to Sadi Carnot's paper from 1824. Lord Kelvin, James Clerk Maxwell, Ludwig Boltzmann, Josiah Willard Gibbs, and others have shown that the origin of these phenomena is microscopic (not cosmological!) in nature. Boltzmann proved his H-theorem, a quantitative and rigorous form of the second law, in 1872. Gibbs introduced the key notion of an "ensemble" of microstates (or "points in phase space", using his old classical setup): we will talk about "indistinguishable microstates" later. There have always been people who misunderstood statistical physics but there are no recent scientific results that would justify a qualitative revision of Boltzmann's, Gibbs's and related answers.

Discussions about the arrow of time are not the core of the research of well-known scientists and no papers in which the arrow of time plays a crucial role are famous. There are no experiments and no non-trivial calculations in recent literature that would justify a new kind investigation of this concept and if you are a sponsor of science and a scientist tells you something else in order to get funding, you are being had.

The arrow of time belongs to the basic material of physics that is not affected by any known modern results in particle physics or general relativity or string theory or cosmology and the well-known particle physicists and string theorists have nothing to say beyond what good physicists in broader groups, including those outside high-energy and gravitational physics, should know anyway.

The term "arrow of time" was coined by Arthur Eddington in 1927. Although I have criticized his numerology from the 1930s, of course that he knew basic physics - at least in the 1920s - and correctly said that the arrow of time "can be determined by a study of organizations of atoms, molecules, and bodies" and in physics, "it is a property of entropy alone": see Wikipedia.

Myth: The time asymmetry of macroscopic processes contradicts the time-reversal symmetry (or, more precisely, CPT symmetry) of the underlying microscopic dynamical laws.

This myth - also referred to as the "irreversibility paradox" - is the main driver behind the irrational debate about the arrow of time. Most people who present the arrow of time as a "hot" topic believe this myth although they never state it clearly. They only give you "hints" about a "subtle tension" and they hope that this fog will make you think that there is a contradiction. Of course, they can't replace this fog by a particular well-defined contradiction because no such contradiction exists. Whether they feel some tension is a problem for their physician, not a physicist. And there exists no verified principle that would imply that this kind of "tension" should be rationally viewed as a problem by a physicist.

Why is there no contradiction? Well, the microscopic laws are time-reversal-symmetric - or at least approximately time-reversal symmetric and exactly CPT-symmetric. But as we will discuss in detail, the microscopic laws are not "everything" one needs to make physical predictions. While the microscopic laws or "dynamics" contain pretty much all the non-trivial information about the physical system and all the difficult maths, one also needs another part of the structure that I will generally call the "interpretation of physics" in order to connect the formulae with reality. This interpretation includes basic logic, inference, and it gives meaning to words used for various macroscopic concepts, objects, and phenomena.

We have known for 80 years that the world obeys the laws of quantum mechanics and the "interpretation of physics" therefore mostly means "interpretation of quantum mechanics". However, even classical physics needs some "interpretation" even though it is less subtle. The arrow of time always hides in the "interpretation of physics" component of a physical theory, especially in the definition of words from the "macroscopic vocabulary" and in the basic logical rules to work with them, and whether the theory is classical or quantum is pretty much irrelevant for this qualitative question.

There is of course no contradiction between our "dynamical microscopic laws" and the "interpretation of physics". Only the latter structure breaks the symmetry between the future and the past but it is of course enough for macroscopic physics to be time-reversal-asymmetric. We will see more details later.

Myth: The arrow of time is a consequence of quantum mechanics.

It is quite easy to see why this sentence is incorrect: even when the world was thought to be classical, people knew that eggs were breaking but not unbreaking. Quantum mechanics changed many qualitative facts about the real world but the arrow of time is not one of them.

Classical statistical physics (with its Maxwell-Boltzmann distribution and other insights) was replaced by quantum statistical physics (with its Bose-Einstein and Fermi-Dirac distributions and similar wisdom). But both of these disciplines imply pretty much the same large N limit called thermodynamics. In the quantum case, the large N limit also includes the classical limit. The qualitative relation between thermodynamics on one side and statistical physics on the other side is pretty much identical in the classical and quantum world.

Another simple way to debunk the myth is to notice that the strength of all quantum phenomena is controlled by Planck's constant which is very tiny in "everyday units" while the manifestations of the arrow of time - such as friction - are clearly macroscopically large. So they can't be proportional to Planck's constant: their essence can't depend on quantum mechanics.

Myth: The arrow of time is a consequence of gravity.

This myth was promoted by people such as Roger Penrose (who has also combined this myth with misinterpreted collapses of wave functions and psychology). Roger Penrose is an original thinker, a hero of mathematical physics, but his belief in his myth represented a profound breakdown of very basic scientific intuition and rational reasoning in general.

Why is the sentence a myth? Well, some astronauts can tell you that even in spaceships with no gravity, eggs break but never unbreak. In fact, they could offer as many examples as you find necessary. All effects of gravity are controlled by Newton's gravitational constant. Can the arrow of time be one of them?

Well, one of the nice, quantitative examples of the arrow of time is friction: it slows objects down but it never speeds them up. It is trivial to see that the arrow of time associated with friction (or diffusion or many other phenomena that add first time derivatives into classical equations of motion) is identical with the arrows associated with eggs and chicks. If someone says that the arrow of time is due to gravity, he also says that friction is due to gravity. I think that such a statement would be embarrassing even for many smart pupils from elementary schools.

Friction is something very different than gravity.

The magnitude of friction is clearly controlled by atomic physics and electromagnetism; gravity has obviously nothing whatsoever to do with it. Gravity is irrelevant. The friction force is not proportional to the tiny Newton's constant. A good enough student should be able to make an order-of-magnitude estimate of the size of friction from the first principles. Collide atoms and calculate the elastic and inelastic cross sections. The inelastic scattering will control dissipation, friction, and consequently the strength of the arrow of time.

Myth: Cosmology is the source of the arrow of time.

Cosmology is a discipline that studies certain solutions to general relativity, a theory of gravity, relevant for the Universe at the longest time scales and distance scales or the Universe near its beginning. If gravity is not behind the arrow of time, it actually follows that cosmology can't be behind it either. Thinking that the friction force or the aging processes are due to the phase transition of a young Universe or due to the Sun or something else in the vast Cosmos is obviously a form of astrology.

Let me say a few more words anyway. When it was born, the Universe probably had a very low entropy. We should really expect it was zero; for example, the Hartle-Hawking wave function is a unique state and the entropy of a unique state is "ln(1)=0". The vanishing entropy of a young Universe is consistent with the second law of thermodynamics that requires the entropy to increase as the Universe gets older. But the low initial entropy is not equivalent to the second law. Why?

Myth: A small or vanishing entropy of a newborn Universe is a sufficient condition that explains all effects of the arrow of time.

But the vanishing entropy of a newborn Universe is surely not a sufficient condition that is enough to derive the applications of the second law of thermodynamics that we normally care about. If the total entropy were zero at the beginning, it would be natural for it to be increasing afterwards, at least for some time, because the entropy can't be negative.

But it could still be true that the total entropy of the Universe increases but in 40% of the Universe, eggs are unbreaking. The remaining 60% of the Universe where the eggs are breaking would overcompensate this anomaly. Believe me or not but the eggs are breaking in the whole observable Universe and regions with opposite arrows of time simply can't co-exist. I will try to discuss this fact later.

The "strength" of the processes that break the time-reversal symmetry, such as friction, is controlled by dissipation and the calculations of inelastic processes we mentioned previously. These calculations are completely local and they apply to every small piece of spacetime. The friction seems to be universal across the Cosmos; you could never derive the universality of the friction force across the Universe from a global or cosmological assumption. In fact, some cosmological details about the Universe a few billion years earlier are absolutely irrelevant for the size of a friction force. Every student who has actually understood where the friction force comes from - and it is the electromagnetic interaction between many elementary particles that makes it work, not gravity - should feel certain about this simple fact.

Myth: A low entropy of the early Universe is a necessary condition for the second law of thermodynamics to hold today.

An "opposite" statement, namely that a vanishing total entropy of the newborn Universe is a necessary condition for the second law of thermodynamics, is also incorrect. The entropy could have been nonzero and the second law would still hold. In fact, many people believe that the total entropy was always nonzero. And Boltzmann himself even believed that the entropy in the distant past could have been very high, higher than today - exactly the opposite than what Sean Carroll considers necessary - and we were born from an unlikely low-entropy fluctuation (recall the Boltzmann brain). The assumptions about the distant past have nothing whatsoever to do with the assumptions we need to make to scientifically derive things like the Boltzmann equation and other macroscopic violations of the T symmetry.

The correct assumptions we need are local ones and they are effectively equivalent to the molecular chaos, namely the absence of correlations between positions and velocities of individual molecules of the gas in the initial state. The assumption of strictly vanishing correlations simplified Boltzmann's calculations considerably. Now, these quantities are not really 100% uncorrelated but what is important is that it is insanely exponentially unlikely that their correlation is exactly the right one to change anything about the result of the calculation of diffusion and other time-asymmetric processes, for example they are insanely unlikely to make the entropy decrease for a macroscopic amount of time.

Myth: The arrow of time is a manifestation of quantum gravity.

This statement is as sexy as the previous myths about quantum mechanics and gravity combined. Unfortunately, it is also as stupid and manifestly wrong as these previous two myths combined. Equivalently, we may formulate the myth as follows: "There would be no arrow of time if the Universe were not quantum or if it didn't include the gravitational force." Check that this is equivalent.

This is doubly incorrect because the arrow of time (e.g. a friction force) clearly exists in a classical world (there is also inelastic scattering in classical physics) and it also exists in a non-gravitational world (ask the astronauts).

It is plausible that in a hypothetical future description of quantum gravity that may unify dynamics and interpretation of physics in a very surprising way, it will be possible to "derive" the arrow of time by a novel procedure. But it would still be true that the arrow of time would follow from those ingredients of the new theory that would supersede the "interpretation of quantum mechanics" as we know it today. The arrow of time does not and cannot inherently depend on any insights of quantum gravity because if it did, it couldn't hold in theories simpler than quantum gravity.

And it cannot have global, cosmological reasons. Boltzmann, Gibbs, and their colleagues have derived nothing else than that the origin of the irreversible processes is microscopic and local. Whoever says that it is cosmological and global should have been failed in high school physics.

I think that to associate concepts of cutting-edge research such as "quantum gravity" with mundane notions of 19th century physics such as the "arrow of time" is a form of intellectual porn that builds purely on human ignorance and rudimentary misunderstandings of the real world.

Myth: The arrow of time is a consequence of CP-symmetry violation.

The weak nuclear interactions violate the CP symmetry which is equivalent to saying that they violate the T symmetry. Is it the reason why eggs don't unbreak? Of course not. There are two basic ways to see why. First, the weak interactions much like all other interactions preserve the CPT symmetry - there is extensive theoretical as well as experimental evidence supporting this assertion. And the CPT symmetry would be enough to show that eggs break as often as unbreak. More precisely, eggs break as often as mirror anti-eggs unbreak. ;-)

That's not the case. However, there is another argument that doesn't require you to find any mirror anti-eggs. If the CP violation were responsible for the time asymmetry of eggs and of the friction force, friction would have to be proportional to the small numbers that encode the CP violation by the weak interactions (a small angle from the CKM matrix, if you care). Again, this is clearly not the case because the friction force is much stronger and it is controlled by electromagnetic collisions - collisions caused by a force whose microscopic description is time-reversal-symmetric.

Myth: The time-reversal symmetry of microscopic dynamical laws implies that the arrow of time shouldn't exist.

For the sake of clarity, let us represent the arrow of time by the friction force again. The incorrect statement above would imply that the friction force has to vanish because zero is the only number that is equal to minus itself. Obviously, the friction force doesn't vanish. We can observe this fact empirically but a good enough theoretical physicist should also be able to show theoretically - from the first principles - why the friction force doesn't vanish. And to roughly estimate it.

How is it possible that the time-reversal-invariant microscopic laws allow us to calculate a non-zero friction force? Well, as we have already mentioned, it is because the microscopic laws are not everything we need to make physical predictions. We also need an "interpretation of physics" to relate the mathematical symbols to our observations.

So how do we make predictions? We always have to approximately know the state of the physical system (or the Universe) at one moment "P" and to use the evolution encoded in the microscopic dynamical laws to calculate some (not all) pretty much macroscopic features of the physical system at a different moment "F". Predictions in science always follow this template. In English, the moment "P" is referred to as the past while the moment "F" is called the future. "F" must come after "P". More precisely, we define the word "after" (and the logical arrow of time) in such a way that "F" comes after "P".

Note that the role of "P" and "F" in the previous paragraph is asymmetric. It is important that in the past, we know the state of the physical system approximately. And it is equally important that in most cases, we are only interested in some questions about the system in the future: we are not interested in some "environmental" degrees of freedom or some very convoluted microscopic correlations at the moment "F".

This is the whole source of asymmetry that drives the friction force as well as decoherence (these two arrows are guaranteed to agree; all the older arrows except for the decoherence arrow were proven to be equivalent by Enrico Fermi in his 1936 book "Thermodynamics" although he was certainly not the first guy to understand the reason).

If you have systems where you either know the configuration at "P" exactly or you are able to study all degrees of freedom at "F", the arrow of time indeed disappears: microscopic or structureless objects are the best examples. But it doesn't disappear for eggs, chicks, and friction. In the case of complex or macroscopic objects, it is very important that our incomplete knowledge of the physical system and the environmental degrees of freedom - that are associated with "P" and "F" asymmetrically - introduce an asymmetry to the whole calculation that allows the entropy to increase.

Even in particle physics, we know about these asymmetries. For example, we often calculate inclusive cross sections where we sum over many different final states if we don't care about their differences. On the other hand, we don't sum over initial states. The closest thing we do is to average over N initial states when we don't know the initial state accurately. (Think about summing or averaging over projections of spins.) But the additional factor of "1/N" from the averaging is a key example of a time asymmetry in our "interpretation of physics". Why do we add the "1/N" for the initial states but not the final states? Think about it - it is about pure logic. You don't need to understand any particle physics to know why. And this logic doesn't respect any symmetry between the past and the future.

And by the way, caring and knowing are two different things, too. Sometimes we don't know the exact initial state even if we would care about it. On the other hand, we often don't care about the details of the final state even though it would be possible to know it. ;-) "Not knowing" is associated with averaging and the past but "not caring" is associated with summing and the future. There is no symmetry here.

Myth: Postdicting the past from the present follows the same formulae as predicting the future from the present, with "t" replaced by "-t".

I am convinced that most talented 12-years-old kids know that this belief - a belief of Sean Carroll and lots of other deeply confused people - is not true once we talk about macroscopic objects, concepts, and phenomena. Five minutes ago, the friction force was slowing objects down just like it is doing now. The procedure sketched above would imply that five minutes ago, the friction force was speeding objects up. I think that you have a damn serious problem with your physical theory or a physical approach if it implies that five minutes ago, the friction force was speeding things up!

It is much more serious a problem than some subtle cosmological deviations that can be measured after billions of years or by analyzing memories from the Big Bang. It is a problem that completely destroys the agreement of your theory with pretty much every local event that has ever occurred anywhere in the Universe.

Because it seems that there is a lot of experimental evidence that the friction force was slowing things down just like it is doing now and much like in the future, I dare to say that their theory or their approach is falsified, it is dead, and we should no longer try to revive it or organize silly conferences about such an approach. And be sure that "my" old-fashioned method to predict the sign of the friction force implies that the friction forces were always slowing things down. ;-)

It is incredible but this is really what the debate is all about. Some people just think that the laws of physics imply that the friction force was speeding things up five minutes ago and only a profound discovery in quantum gravity could bring the world back to normal. :-)

Well, I beg to differ. In fact, "my" approach to physics respects the time-translational symmetry (up to tiny corrections implied by cosmology). That means that if the friction will be doing something in 5 minutes, it was inevitably doing pretty much the same thing 5 minutes ago. When a physicist calculates the probability of the configuration A at time T1 evolving to another configuration B at time T2, it doesn't matter what the "current time" is, whether it is before T1, T1, after T1, before T2, T2, or after T2. Physics doesn't and cannot depend on "current time" because it is a subjective notion. Physical laws and methods are constant and eternal. On the other hand, it is important to know whether T1 is before T2 or the other way around.

What's wrong with the approach of the people who believe this myth is their "interpretation of physics". They don't understand the difference between the assumptions and the assertions, between the future and the past, between the cause and the effect. It's probably because they never deal with any complex systems - only with time-reversible orbiting planets or elastic scattering.

Even in mathematical logic and probability theory, the role of assumptions and assertions is highly asymmetric. To see it, let me describe the Bayes' theorem. It is a prescription to correct your idea about the probability of a hypothesis "H" after you see evidence "E". The corrected probability, the so-called posterior probability which is really the conditional probability of the hypothesis "H" given the evidence "E", is calculated as
P(H/E) = P(E/H) P(H) / P(E)
On the right-hand side, P(H) is the prior probability of the hypothesis "H" that you knew before you saw evidence "E": the "primordial" priors before you collect your first evidence are of course the main sources of controversies in the Bayesian approach but this subtlety has nothing to do with the topic we discuss in this article. "P(E)" is the so-called marginal probability of evidence "E" in all contradicting hypotheses, equal to the sum of "P(E/H_i) P(H_i)" over "i".

Finally, "P(E/H)" is the conditional probability of seeing evidence "E" assuming the hypothesis "H". The main point of this discussion is to emphasize that the conditional probabilities "P(E/H)" and "P(H/E)" are two very different animals. There is no symmetry between assumptions and assertions in conditional probability calculus. The additional factor of "P(H) / P(E)" that distinguishes them is a more general version of the "1/N" factor in the averaging over the initial states that we discussed previously, in the comments about particle physics.
At the end of this article, I added a textbook example of a Bayesian postdiction in which it is very clear that the controversy hides in the "prior", as expected.
The very same factor must also be added when you are postdicting the past, as we discussed in the article dedicated to this subtlety and linked above. One can use physical theories to say something about the past but the formulae for the probabilities are different from the formulae controlling the predictions for the future. If you didn't include the extra factor, you would incorrectly postdict that the entropy in the past was probably higher than today and the friction were negative (accelerating). The additional factor of "exp(-entropy)" that prefers low-entropy postdictions for the past configurations has the very same origin as the "P(H) / P(E)" factor in the Bayes' formula.

The people who believe the myth think that all these factors are equal to one. Logicians have a part of their brain occupied by the Bayes' formula; ordinary people store their common sense in the same piece of their brain. The believers in this myth have a hole in that region instead. A related myth caused by this hole is the belief that the high-entropy configurations are the natural, primordial state of all objects. Quite on the contrary: high, maximized entropy is the feature that physical systems only acquire in the future, once they have enough time to reach the equilibrium. Completely different (and, in fact, opposite) observation applies to the past: it is not only acceptable but really natural (or necessary) for objects to have a low(er) entropy in the past.

These differences between the past and the future are not facts related to cosmology or quantum gravity or string theory but rather very universal facts associated with elementary logic. The people who say that the arrow of time is due to cosmology or quantum gravity or global warming misunderstand elementary logic. They will tell you that the arrow of time is due to cosmology or even their more or less bizarre version of it. They will tell you "trust me". But even if you don't get the explanation above, trust me, they are just being incredibly silly. ;-) They are not saviors who will protect you and science from seemingly absurd predictions. The reason is that science, when used properly, was never making any absurd predictions of this kind.

Even many regular people who don't know the Bayes' formula and similar things realize that there is a difference between our thinking about the past and the future and this difference is a fundamental ingredient of any rational thinking about the real world that other laws of physics must be compatible with. And they are consistent, of course. The other laws of physics such as Maxwell's equations are just being added to the basic logical framework - the "interpretation of physics" or "interpretation of science" in general - that deals with knowledge, memories, and predictions.

Let us try to localize the problem again for the case of a breaking egg. For the microstates of atoms of an egg, the dynamical laws are time-reversal-symmetric and reversible. But this fact doesn't allow us to deduce that "an egg was unbreaking in the last 5 minutes". The reason is that the very sentence "an egg will break" or "an egg will unbreak" implicitly includes the averaging over the initial microscopic states of the macroscopic object called "an egg" and summing over the final microscopic states of "an egg" and its environment. This is because of the very definition of "an egg" and of "breaking". An egg as a concept is not a particular microstate: it is a macroscopic state that can be represented by each of many microstates. And this union is and must be treated asymmetrically with respect to the time reversal. The summing vs. averaging above is the reason why the entropy is (almost) always higher for the final states of the macrosopic objects. It is the reason why we may derive that an egg - like other macroscopic objects - breaks but (exponentially) almost never unbreaks.
Slogan: All statements, questions, and probabilities of processes involving macroscopic objects include averaging over indistinguishable initial states in the past and summing over indistinguishable final states in the future. The previous sentence is manifestly past/future asymmetric and this is where the whole asymmetry comes from. The macroscopic systems thus prefer to evolve from macrostates with a few indistinguishable microstates in the past (because they're being averaged over, adding the "1/N" factor) to macrostates with many more indistinguishable microstates in the future (because the probability is summed over all of them). Therefore, the entropy increases and the second law holds. No extra explanations are needed and no extra explanations exist.
Some people talk about "preparing" initial and final states. But those who have heard about the causal arrow of time realize that controlling the future, or causing something to happen, creates correlations between the doer and the effect, and these can only be created as we move forwards in time, not backwards. After all, similar correlations between the effect and its environment that increase as we move forwards in time are also the reason behind decoherence and its arrow of time.

If you want a book that explains that the "irreversibility paradox" has been solved a very long time ago and the solution is fully based on a careful logical analysis of the differences between the microscopic and macroscopic descriptions of a physical system, see e.g. Michael Bushev 1994 (at amazon.com: Synergetics) or Jack Hokikian 2002 (The Science of Disorder, also includes quotations of Richard Feynman). See also a book by Peter Harman (1982), Cyril Domb (1996), and Derek York (1997). York is the only person who includes some cosmological speculations but he still presents Boltzmann's conventional picture. Susan Friedlander and Denis Serre (2002) argue, in Handbook of Mathematical Fluid Dynamics, that the rigorous definition of the conditions in which the Boltzmann equation holds was given by Grad in 1949.

OK, so I actually think that most people have enough common sense to know that the past is fundamentally different from the future and there exists no meaningful logical, predictive framework in which this difference between assumptions and predictions, and between cause and effect, is absent. In this sense, the time asymmetry can't be due to a "spontaneous symmetry breaking" because there exists no logical framework in which the symmetry between assumptions and assertions holds. So why do some people defend the undefendable equivalence between the future and the past?

Shockingly enough, the answer is political correctness. Sean Carroll, one of the staunchest champions of this totalitarian ideology, writes very seriously: "We human beings are terrible temporal chauvinists." So it is really politically incorrect to point out that his opinions about the origin of the arrow of time are silly - because you would become a chauvinist! ;-)

Well, I am not only a "temporal chauvinist" but I happen to think that those who can't admit a fundamental, logical asymmetry between the past and the future in any world qualitatively resembling ours lack basic knowledge of logic as well as common sense. Denying the difference between the past and the future is even worse than not seeing the cognitive differences between the two sexes. It is insane.

Myth: The cosmological arrow of time must always coincide with the thermodynamic arrow of time.

The cosmological arrow of time is the arrow from the moment when the Universe is small to the moment when the Universe is large. Because the Universe is expanding right now, this arrow agrees with the thermodynamic arrow of time. But it is always the case? Stephen Hawking used to argue that the answer was Yes. He used an incorrect continuation to the imaginary time to derive this hypothesis.

The Yes answer is clearly wrong: if you considered a Universe that ends with a Big Crunch, the eggs can't start to unbreak once the Universe reaches the maximum volume and begins to shrink. One can't even calculate the exact moment at each point of the Universe when the Universe is just beginning to shrink. But the eggs and chicks would need to know the moment very accurately. These two arrows obviously don't have anything to do with each other - eggs are controlled by accurate and fast local physics while the expansion of the Universe is about some global, slow properties of the Universe - and their directions may, in principle, be independent. Hawking has agreed that his previous statement was wrong.

Myth: Inflation is a sufficient condition to explain the arrow of time.

During an inflationary era, the cosmological arrow of time is "almost always" coincident with the logical arrow of time - because the Universe is "almost never" found in the fine-tuned state that would lead to a long, exponential shrinking or "deflation". But other kinds of shrinking of the Universe don't depend on any such "fine-tuning" and they can therefore occur in both ways, as explained in the previous paragraph.

Moreover, you can't say that inflation "explains" the arrow of time. Inflation during which the Universe accelerates its expansion is just another example of the arrow of time in which the rules of the game "look" kind of opposite than those controlling the friction force. But as we have already argued, inflation or any other cosmological evolution is insufficient to explain why invidividual eggs break but don't unbreak. In fact, the Universe after inflation (more precisely, after re-heating) already has a pretty high entropy. If a low-entropy beginning was your strategy to "derive" the second law, inflation will make your task harder, not easier.

Long after inflation, matter starts to clump. Gravity differs from non-gravitational gases in one respect: non-uniform configurations - such as those with black holes - are actually the highest-entropy ones. In fact, black holes have the highest entropy that you can squeeze into a given volume. This fact allows the seeds of galaxies to emerge without violating the second law. Gases are normally getting increasingly more uniform but gravity can change this conclusion.

But this is just a technicality about the geometric shape of generic and special configurations. If you only care about the second law - about the increasing entropy - there is no qualitative difference between inflation, post-inflationary cosmology, breaking eggs, or other processes in the Universe. The entropy always increases and it increases because of a fundamental logical arrow that must exist before one learns additional scientific laws.

Conclusions

Let me summarize: there exists some knowledge about the arrow of time and the second law of thermodynamics that is not being taught sufficiently clearly at schools but those physicists who actually understand how the real world is connected with the formulae of physics have understood for more than a century while those physicists who try to connect the arrow of time with some esoteric details of their cosmological models are the slower ones, if I have to be completely polite.

And that's the memo.
Update...
Bonus: a textbook example of a postdiction

Imagine that your system has N microstates. We will study the evolution from time T1 (earlier) to T2 (later). At T1, we will distinguish two macroscopic states A1, A2. A1, A2 include M1, M2 microstates, respectively. Analogously, we will distinguish macroscopic states B1, B2 at time T2 whose corresponding number of microstates is N1, N2, respectively. We have M1+M2 = N1+N2 = N.

What about the dynamics? N11 microstates from A1 evolve into states in the set B1. N12 microstates in A1 evolve into states in B2. N21 microstates in A2 evolve into something in B1. N22 microstates in the A2 set evolve into states in B2. We have N11+N12 = M1, N21+N22 = M2, N11+N21 = N1, N12+N22 = N2. Among these four relations, three are independent because M1+M2 = N1+N2.

Imagine we know that at T1, we have measured the macroscopic state to be A1. What is the probability of getting B1 at T2? Among the M1 microstates representing A1, N11 and N12 give us B1 and B2, respectively. So the probabilities to get B1 and B2 (from A1) are N11/M1, N12/M1, respectively. These two probabilities add up to one. Everyone agrees with this paragraph.

But now imagine that we know that we have the final macroscopic state B1 at T2 and we want to postdict the macroscopic state at the earlier time, T1. The unbreaking-egg people will just interchange the role of T1 and T2. They will tell you that the probabilities that you had A1, A2 at T1 were N11/N1, N21/N1, respectively.

But this time-reversal-symmetric result is not the correct answer because this would imply a high entropy in the past. The correct answers have nothing like N1, N2 (associated with B1, B2 and time T2) in the denominator. Instead, it is always M1, M2 (associated with A1, A2, and T1) that appear in the denominator: recall that we should always average over the initial microstates. Because postdictions are a special case of inference, a proper formula may be obtained from the Bayes' formula blockquoted above if we identify the "hypotheses" with the "initial conditions" A1=H1, A2=H2, and the "evidence" with our assumed "final state", E=B1. We therefore have
P(A1/B1) = P(B1/A1) P(A1) / P(B1)
The conditional probability P(B1/A1) is N11/M1, as explained three paragraphs ago. As always in the Bayesian reasoning, the "prior" probability P(A1) is the main quantity that encodes the whole controversy while the "marginal" probability P(B1) in the denominator is a normalization factor that guarantees that the probabilities P(A1/B1) and P(A2/B1) add up to one.

Prior, a standard Czechoslovak shopping center from the age of socialism. "Prior" stands for "Přijdeš Rychle I Odejdeš Rychle" which means "You arrive quickly and you leave quickly." :-)

OK, so what's the result?

The key player is the "prior" probability P(A1) of a macroscopic state A1. If you chose it to reflect the number of microstates, as the unbreaking-egg people always do, i.e. P(A1) = M1/N, you would end up with probabilities N11/N1, N21/N1 for having A1, A2 at T1, respectively (assuming B1 at T2). However, it is not correct to punish initial states for having a low entropy in this brutal way. There is no a priori reason to assume that the initial state should have a high entropy.

The correct method to postdict the initial state dictates that you treat all your different hypotheses about the macroscopic form of the initial state as a priori equally likely. In our case it really means that P(A1) = P(A2) = 1/2. Consequently, your postdicted probability of A1 at T1 is
(N11/M1) / (N11/M1+N21/M2) = ...
... = N11 M2 / (N11 M2 + N21 M1)
By construction, the probabilities add up to one. Anyway, that's quite a different result than N11/N1, isn't it? For example, when M1/N goes to zero while M2/N, N11/N1, N21/N1 are kept finite somewhere between 0 and 1, the probability above goes to one: a low-entropy initial state A1 is heavily preferred in this case which would surely not be the case of the wrong N11/N1 formula. This Bayesian method of postdiction can be easily generalized to arbitrary physical systems, both in classical and quantum physics.

Note that I didn't have to add an additional assumption about a low entropy of the early Universe or something like that: it wouldn't be useful to quantitatively calculate the probabilities anyway. I only needed to drop the wrong assumption that the initial state had a high entropy. The contrarians who probably think that my prior is "obviously" wrong or "unnatural" should realize that they are choosing a prior, too. If there are two competing theories in physics, we should make observations to decide. Observations unanimously show that my prior is pretty much correct and the contrarians' prior is hugely wrong.

If you are irritated that the result depends on how finely we clump microstates into macrostates - i.e. on a somewhat arbitrary choice of the prior - you are correct: they do depend on it. For example, if we were suddenly able to distinguish two macroscopic states A2, A3 that used to be grouped under the same (old) A2 umbrella, the natural prior would be P(A1)=1/3, P(A2+A3)=2/3. We could have changed the priors even without this justification. However, there exists no "canonical" or "more rigorous" or "unique" way to postdict in physics that would be independent of any priors. In a sense, the well-defined predictions are irreversible - something that becomes intuitively clear when we deal with irreversible processes that make it difficult to determine the past because the information is getting lost. If you know that both 1 ton of banknotes and 1 ton of books burns into the same pile of ash that you see, there is no canonical rule to tell you whether the initial state contained banknotes or books. You need to use other wisdom to make a qualified guess. I think that the (inconvenient) books are more likely than the (convenient) banknotes but the reason has nothing to do with the entropy of books vs banknotes. :-) Common sense and the context is crucial; physics of the phenomenon itself is not enough.

This discussion is another example of discussions we have had in the context of the stringy landscape. Does a much larger number of vacua in a set (KKLT vacua, for example) make the corresponding scenario much more likely? Because the details of the stringy vacuum may be viewed as a feature of our past (the early Universe), the discussion about the likelihood of classes of vacua is really a special example of the considerations from this article.

My answer has always been that the different classes of vacua or different ideas (a special case of the macroscopic states) must be treated as a priori comparably likely, regardless of gigantic differences between the numbers of vacua (or microstates) in the set, and the general case we discuss now is no different. At the same moment, there is no "canonical" way to assign the priors which is why we cannot settle the question about the initial state of the Universe, including the preferred scenario in string theory, by a well-defined and trustworthy calculation of probabilities. Priors always play a role in any postdiction or inference.

One of the qualitative consequences of "my" resulting formulae is that all the postdictions of a high entropy in the past disappear. We have really inserted this outcome to the Bayesian formulae as an assumption, as the "prior". But it is a correct assumption because there is no rational reason to think that the high-entropy states in the past should be preferred, as observations clearly demonstrate. It would be completely absurd to punish the states in the past for having a low entropy.

Imagine that you have two hypotheses: for example, man evolved from a monkey vs. man evolved from a tulip. The initial state of the Earth with tulips preparing to evolve into human beings would have a higher entropy, by "10^{30}" or so, which is very plausible. Does it mean that you should a priori assume that the tulip theory of evolution is "exp(10^{30})" times more likely? It's completely ludicrous. Warm enough tulips with a high enough entropy could then beat the dynamical difficulty of getting one human DNA from one tulip DNA as a mutation and you could end up claiming that the tulip theory of human evolution (including a massive violation of the second law of thermodynamics) is more likely. Even though it was explained above that there are no correct God-given priors (1/2 or 1/3 for P(A1) is comparably reasonable), we see that Sean Carroll's priors are wrong by a factor of "exp(10^{30})" or so which is a pretty large number.

Physics as understood by your humble correspondent should agree with observations and the observations show that eggs were not unbreaking five minutes ago and high-entropy tulips didn't evolve into humans; the unbreaking-egg people disagree because the primary player of their physics is a stubborn (albeit nonsensical) metaphysical dogma that everything we say about physics must be time-reversal-symmetric.

It's not.

1. Nice snowflakes Lubos, but they have the wrong symmetry for our universe.

About the H-Theorem. You should read your own references: Boltzman was only able to prove his H-Theorem by assuming an equivalent (and equally time assymetric postulate - molecular chaos). Neither is consistent with time symmetric physical laws. This was also pointed out more than 100 years ago. You can't get the behavior you want without some kind of coarse graining - either that or rejecting time symmetric physical laws.

2. Dear Pig, it's been explained many times that the assumption is first of all effectively correct, that any conceivable modification of it (incl. random microscopic correlations) is inconsequential, and whether it is equivalent to another correct assumption is not a problem in any way.

Of course that if someone prevents us from using any time-asymmetric step in our reasoning or assumption, we will never derive anything asymmetric and our cognitive abilities would be stuck at the level of those who don't allow high-tech methods such as logical inference. Because I am not your mirror, it is not my job to present examples of such people.

But time-asymmetric reasoning is absolutely essential to get any of the macroscopic conclusions right. It is an inseparable part of any kind of inference based on incomplete information and whoever doesn't use time-asymmetric reasoning is guaranteed to have problems to think rationally.

The correct reasoning about macroscopic systems is time-asymmetric - posterior probabilities are calculated after priors are decided, for example - while your reasoning is incorrect. I happen to think that being right vs wrong is more important than being convenient for someone who likes to spread metaphysical dogmas rather than to explain reality.

We are surrounded by mammals who never want to study whether things are right or wrong and they never want to do any useful quantitative calculation. They never want to listen to rational arguments. They never want to question the validity and value of what they are writing. They just want to harrass people who are more reasonable than they are.

Also, my article fully reflects the cutting-edge explanation of all these things that indeed does involve coarse-graining - developed by Gibbs and Boltzmann as a result of the irrelevant criticism you mention - but it doesn't change anything about Boltzmann's correct derivation of his correct equations.

3. Unifying twistor strings with spinor loops?

I have found a dynamic graphic simulation on Wiki which is consistent with David Hestenes concept of Zitterbewegung.

Nonsymmetric velocity time dilation by Cleonis [28 January 2006, cc-by-sa-2.5] appears to be consistent with an electron [red] traveling about a proton [blue].

This “gif” might also be interpreted as a single planet revolving about a moving star.
The star could be a galactic core with only one star shown revolving and with a little thought experiment, one may be able to imagine at least one planet revolving about this star.
If the “red” were a system of 8 planets about a star or a galaxy of stars about the galactic core then one might even imagine a virtual torus.

If one looks at
a - figure 11, page 120 Kapustin, Witten, ‘EM Duality And The Geometric Langlands‘ there is a resemblance to the gif;
b - then at figure 2, page 13 of Maloney, Witten ‘Partition Functions’ the gif forms a virtual cylinder resembling AdS3.

Tokamak basics from Pitts, Buttery and Pinches, ’Fusion: the way ahead’ demonstrates the relation of the helix and torus to electromagnetism in plasma physics.

There are helical equations of Fourier Transforms from Hamish Meikle.

The 3D helix can be decomposed into a 2D sine, 2D cosine and 1D loop.
The 1D loop resembles the nearly circular, elliptical planetary equations of Newton.

4. Lubos, good article, but I think it was highly misleading. You seem to be equating the arrow of time with the second law of thermodynamics far too closely. As you know, the second law says entropy will tend to increase, and that is symmetric in both the forward and reverse time directions. So there's no particular time direction introduced by the second law. The arrow of time gives a **particular** time direction to processes, introduces consistent time-asymmetry to processes. Where does that come from?

As I said, the arrow of time is not the same as the second law. The second law is responsible for time-directed change, but as to why the we see change in a PARTICULAR direction - that's the arrow of time. They're different things! For example, the reason we see the entropy of the universe increasing in the forward time direction is because of the combination of the second law and the low entropy state of the universe in the past. The combination of the two forms an arrow of time, time-directed change in one consistent direction. Surely you can't disagree with that?

"It is incredible but this is really what the debate is all about. Some people just think that the laws of physics imply that the friction force was speeding things up five minutes ago" Well, yes, absolutely. I'm one of them. And I would imagine the vast majority of physicists would agree with me! Surely it's pretty straightforward: if you think friction is based on Newtonian mechanics then you could take a video of an object's atoms crashing into another objects atoms (basically, a frictional encounter) and time reverse that video and it would look correct. But, of course, the entropy of that closed system would be increasing in the reverse direction so we don't see it in practice. But we don't need quantum gravity to explain why friction would speed things up in the reverse direction - just Newtonian mechanics.

I've got my own web page on this: The Arrow of Time. Feel free to have a go at me!

5. Oops, should read "the entropy of that closed system would be DECREASING in the reverse direction so we don't see it in practice"

6. Dear Andrew, thanks, but you are completely wrong. The second law of thermodynamics works in one direction only and its whole point is that it introduces an asymmetry between the two directions of time.

This preferred direction of time (to the future) as dictated by the increase of entropy i.e. the second law is the most important arrow of time, the thermodynamic arrow of time, and it can be shown to coincide with the logical arrow of time, the decoherence arrow of time, and related arrows of time.

The existence of a universal (thermodynamic) arrow of time and the second law of thermodynamics is the very same thing.

If you think that anything goes and the arrow of time can be arbitrarily switched, then you are simply confused about the basic point of this whole discussion.

I don't think that any physicist who has received an A from statistical physics and thermodynamics in a decent enough course would agree with your confused answer about your "anything-goes arrow of time".

7. Hi Lubos. Firstly, I would like to say that I was in favour of your earlier posting which stated that the second law could never be a product of the initial conditions of the universe. Very right you are. However, the second law and the arrow of time are not the same thing. You say: "The second law of thermodynamics works in one direction only and its whole point is that it introduces an asymmetry between the two directions of time." Well, we'll have to disagree there. The second law introduces no time asymmetry. Here's what Brian Greene says in The Fabric of the Cosmos (page 160): "All of the reasoning we have used to argue that systems will evolve from lower to higher entropy toward the future works equally well when applied toward the past". And again: "The laws offer no temporal orientation ... both toward what we call the future and toward what we call the past - the statistical/probabilistic reasoning behind the second law of thermodynamics applies equally well in both temporal directions." Also see the example from Richard Gott's book, the ice cube example which Brian Greene also uses in his book, and which I describe on my site.

You say: "This preferred direction of time (to the future) as dictated by the increase of entropy i.e. the second law is the most important arrow of time, the thermodynamic arrow of time, and it can be shown to coincide with the logical arrow of time, the decoherence arrow of time, and related arrows of time." I agree absolutely with this. But the laws of increasing entropy - the second law - is time symmetrical, as just discussed. The reason for time asymmetry is a combination of time-symmetrical laws and the initial conditions of the universe. Honestly!

Your arrow of time seems to be defined by your egg-breaking example, but then you also seem to deny any role for the initial conditions of the universe. I would say, if change of entropy is time-symmetric, then the initial conditions of the universe are clearly the only reason why entropy is increasing in the universe. But I would say the initial conditions even define the arrow of time for your "egg breaking" example. Consider the following three events, and decide in which sequence they could possibly occur:

1) The birth of the low-entropy universe.
2) A perfect egg in existence.
3) The egg breaks.

Or alternatively:

1) The birth of the low-entropy universe.
2) The egg breaks.
3) The perfect egg in existence.

According to you, the initial conditions of the universe are irrelevant, so both of the sequences are possible. I would say only the first sequence is possible. The initial conditions of the universe are responsible for your arrow of time - but in quite an indirect way. The production of the egg follows the birth of the universe via the formation of stars, formation of planets, and evolution of life. And then the egg breaks. It simply could not work the other way round. Sequence two could not happen. Your arrow of time is a result of the initial conditions of the universe, together with time-symmetric laws.

8. Dear Andrew, I realize that this whole chapter of Brian's second book is wrong because we have spent some time arguing about it before the book was released.

The second law would be completely vacuous and meaningless if the direction of the increasing entropy could depend on the context - on space or time.

The second law is not vacuous. It says that the increasing entropy has the same universal direction for all macroscopic (or high-enough-entropy) systems in the whole Universe. This direction can be seen to be the same direction as the direction in which the decoherence diagonalizes the density matrices; in which the microscopic correlations grow; in which we think about evolution of the Universe.

The right terminology is to call the early times in this time-asymmetric setup "the past" and the opposite end "the future". There is absolutely no freedom here in exchanging the past or the future.

Your examples are completely silly. I am, on the contrary, saying that the egg can never unbreak or become perfect as time increases - as long as the time coordinate is chosen to agree with any of the major arrows of time (logical, thermodynamic). So the second triple of events you listed clearly can't occur. It is obvious that an egg can't be "perfect" after it broke. This depends on basic laws of logic only and has nothing to do with anything else, certainly not with cosmology.

Your second triple of events can't occur in either order

A) low entropy small universe; broken egg; perfect egg

B) perfect egg; broken egg; low entropy small universe

because neither of them agrees with the second law. Neither of them has a universal arrow of time. In the flawed approaches, one of the two histories or both could occur.

I don't have more time for these exchanges that don't lead anywhere - if someone can't understand that the entropy increases with time and this fact implies an universal arrow of time, it is just far too hopeless to make any debate - and I will be erasing comments that look as stupid to me as yours.

9. I think that gravity could be used to help create Time's Arrow. Friction need not enter into it.

It could be something as simple as:
10 bowling balls floating in deep space, all in a cubic kilometer. Gravity will bring them closer together.
Due to gravity, they are closer now as opposed to then.

Yes, this does not apply to the quantum world. This could be considered a classical version of Time's Arrow, not a quantum one.

Yes, not everything is a perfect example of this. The moon is receeding, but the moon's creation itself could be used for time's arrow. Gravity made the rocks/material that made up the moon closer to all other bits of rock/mateiral now as oppposed to them.

10. Motion and the Arrow of Time

Much ado…The arrow of time is implicit in the Lorentz-Fitzgerald-Einstein (LFE) contractions, and it‘s curious no one seems to have noticed. Time and motion are complements. Any change of position in space corresponds to a change of position in time; the temporal change always in one direction only, from present to subsequent present, regardless of the direction of movement in space, its variables subject only to the degree of motion (i.e., speed). The relationship between motion and time is given by the complementary ratios

(v/c)^2 + (t_0/t)^2 = 1

derived from the LFE time dilation equation

t= t_0/√(1-(v/c)^2 )

Definitions

A relative inertial rest frame is implicit in any reference herein to speed, velocity or motion.

If a traveler departs and returns at a high average relativistic speed, he returns at a time x years in the future relative to his starting frame of reference at a cost of less than x years subjective time. This variable subjective interval can be reasonably described as travel forward in time or artificially accelerated movement through time. Because this subjective cost is variable and dependent upon spatial velocity, we will use the term “speed through time” for the sake of descriptive simplicity. Also, for simplicity, we will use the term “orientation in time” (i.e., space-time) to describe the source of this temporal speed, the variability of which is consequent to change in relative spatial velocity.

All motion can be described in terms of v/c. Every value for v/c has a corresponding specific orientation in time, and a calculable speed through time (in terms of subjective interval).

The range of v/c from an asymptotic approach toward hypothetical absolute zero motion to the asymptotic approach to c encompasses all possible motion. Each incremental value for v has an associated temporal orientation, and each results in a specific positive (present toward future) rate of motion through time. The ratio v/c cannot have a negative value, so there is no possible negative value for t--no possible negative interval. The arrow of time, therefore, is implicit, and founded in the fundamental geometry of space-time. This is more easily intuitive if we note that the same is true for length and mass: in the corresponding LFE transformations, the requisite positive value for v/c limits the dimensions for mass or length to positive values (which we already accept intuitively). Hence one could say that the arrow of time is implicit in the length of the telephone pole outside your window or the heft of the keys in your pocket.

Duration

H.G. Wells’ time machine, while obviously not a real world possibility, offers a number of useful analogous insights into relativistic effects, in particular the nature of duration. More accurately, the shortcomings of Wells’ and others’ conceptions suggest a definition for persistence. Wells viewed past and future as pre-existing and apparently constant locations subject to access by a vehicle with the right properties. He noted that the traveler looking out from inside his machine would see motion and change in the outside world accelerated as he accelerated forward through time. He failed to note the opposite. Wells’ machine disappeared from the present as it moved into the future--it “skipped” to a future time. Had he noted (as he did for the internal observer) that in order to move from point A to point C in space it is necessary to reside in all points B along a specified path between, he might have recognized the same as being a logical probability for time. In order to move from the present to a point in the future, it would be necessary for his machine to reside in all points along a given path between, so it would not have disappeared, but remained visible in its spatial location. An external observer would note, however, a slowing of motion within the confines of the machine as it accelerated forward in time as the occupant’s relative orientation in space-time changed. (All of this intentionally overlooks purely speculative concepts like wormholes and dimensional short-cuts.)

Science fiction has long offered one way of looking at duration in terms of the temporal “worm,” a single continuous meandering 4-dimensional body that represents a person, for example, from roughly the time of conception to roughly the time of demise (or dissolution); a meandering knot encompassing all the movement of a lifetime as a single convoluted 4-dimensional entity (even Barbour does essentially the same thing, though he breaks his entity into what amount to motion picture frames). The problem with this idea is that such a structure requires a quality of duration in its own right. Any structure must persist to exist, whether in 1 dimension or 20 (or in configuration space), so the need to add dimensions solely for the sake of duration goes on ad infinitum. This is where the various “time doesn’t exist” theories fall apart. Barbour, for example, postulates the immensely complex structure he calls “Platonia” but fails to acknowledge that it needs to persist to exist.

Because motion and time are inseparably connected, and because all frames of reference are in motion in relation to some other frames, and all motion creates a calculable positive value for t, then it is simpler and more reasonable to view duration as natural motion through time, as a complement to spatial motion. Three-dimensional objects are just that, but they “endure” because they move with the observer through space and time. They don’t “have” a fourth dimension; they move through the fourth dimension.

This has interesting implications. There is no past, and there is no pre-existing future, though the future can be treated as a destination, whereas the past, as a corporeal structure at least, does not exist at all. The Universe exists in a constantly moving local “present,“ an infinitesimally thin 3-dimensional surface moving with a 4th dimensional vector. All record of the past exists as the effect of prior causative motion and interaction. The 4-dimensional Universe has an edge, and it is “now”. Entropy is a marker, not a cause. Block time is a myth.

Time and Gravity

Here’s a more radical thought: Motion establishes a moving object’s orientation in time. All motion dilates, or warps time, only notably at relativistic speeds, but calculably at any speed, per the LFE time dilation equation above.

The time-motion complementarity equation,

(v/c)^2 + (t0/t)^2 = 1

which is a simple algebraic transformation of the time dilation equation, links motion and time as an ontological whole, so if motion warps time, then the converse should be true: any warping of time independent of motion should result in a change in motion. The only sources of time dilation are velocity and mass. Because velocity and time are complementary, any time warp created by the presence of mass necessarily results in motion or a tendency to motion of any object in the vicinity of the mass (the attraction is mutual, but described here in terms of very small mass in the vicinity of a very large mass for the sake of simplicity). Hence any object entering the range of influence of a mass will tend to move toward the center of that mass and accelerate as the degree of time warp increases in inverse proportion to the distance.

Michael Cleveland

11. Sorry, second iteration of the complementarity equation should have read (v/c)^2 + (t_0/t)^2 = 1.

12. I'm not worth a damn early in the morning. Disregard reference in previous to negative v/c. It should state that negative interval would be possible only if v/c > 1.

13. Lubos, is there any professional philosopher of science who agrees with the view you present here about time asymmetry?

14. Dear Robin, I don't know: I am trying to avoid philosophers because most of them are stupid, when it comes to science.

Do you realize that the negation of your question is the question whether all philosophers are ignorant about all basics of science?

What I am writing is surely not controversial among those who should be allowed to complete their science degree in a college.

15. Lubos, you are essentially correct, but a little too hasty in dismissing remaining issues with our understanding of the arrow of time.

Firstly, while the dynamical equations in both quantum and classical mechanics are time [or CPT] symmetric, particular solutions need not be. This is an additional reason for the "non contradiction". In general, solutions of lesser symmetry form families whose average bears the original symmetry.

This leads me to my dissatisfaction with the ensemble approach of Gibbs. The ensemble average has the same symmetry as the original equations. This means that the Gibbs entropy is an invariant of the Liouville equation - so no arrow of time there.

The H-theorem is achieved by sleight of hand. If the Liouville equation is written as a BBGKY hierarchy, the H-theorem can be seen as a way to truncate it at the first correlation level.

The phenomenal success of the Gibbs ensemble for homogeneous systems like perfect gases has blinded us to the fact that real experiments are performed on individual systems, for which entropy and the 2nd law are still good.

Nor does the ergodic hypothesis save the ensemble approach to entropy - real experiments are made in far less time than a Poincare recurrence!

In a sense, the scheme is retrieved by "course graining" but that is ad hoc and arbitrary. Another approach is needed which restores some objectivity to the theory. And who knows, perhaps pursuit of the answer will be as far-reaching as Bohrs attempt to reconcile "orbiting" electrons with non-collapsing atoms.

Just as no real family has the 2.6 children of the average, so any approach to defining entropy microscopically must identify what makes a sub-system identifiable as an entity and over what time-scale. It means looking for new kinds of symmetries and approximate symmetries.

The arrow of time is not solved in all its details, and those details could be the seeds of new ways of thinking.

16. Dear Fresh-eye, I agree that symmetries of the laws of physics don't have to be satisfied - and usually are not satisfied - by the actual configurations and I am absolutely certain I have never contradicted this fact.

Under the time evolution, ensembles of N microstates evolve to ensembles of N microstates. But it is not true that if the final ones are macroscopically indistinguishable, then the initial N states have to be macroscopically indistinguishable, too.

If you define the entropy as log(N) where N is the number of macroscopically indistinguishable microstates from your microstate - or any microstate in your ensemble - then the entropy always increases.

I don't understand your complaints about the H-theorem. The only extra thing I can agree with you is that indeed, Poincare recurrence is far too long a time to matter for anything here. It's the time after which it may happen that the entropy brutally increases at least once. However, the entropy of a large enough system grows all the time, after each Planck time.

The relation of the H-theorem to 2.6 children or Bohr atoms' (non)collapse is not clear to me.

Indeed, you may say that the statistical treatment of the states brings - even classically - some kind of non-objectivity or non-realism to the laws of physics. That's how the things work. Physics is a gadget to deduce correct statements about the objects and their evolution. So yes, one should manipulate with assertions and their probabilities and he shouldn't imagine that there's always an objective classical "model" behind the calculations.

The arrow of time has been understood since the time of Boltzmann, and even the quantum version of the H-theorem etc. was just a straightforward rewriting of the insights that have been known for more than a century.

Your instincts and explanations are correct - especially Bayesian post-diction. One time I was discussing the Arrow of Time on the radio. The host asked me if the arrow would reverse in the event of a cosmic collapse. I said no, for essentially the reasons you give.

H-THEOREM: the Boltzmann equation on which it is based truncates the BBGKY hierarchy by assuming that pair correlations are negligible. This is good for a near perfect gas not too far from equilibrium. But if the initial state has significant high order correlation, the evolution of the Boltzmann equation cannot tell you why it dissipates.

There is a whole hierarchy of N entropies which can be defined for 1- up to N-particle distribution functions, where N is the total number of particles in the system. The H-Theorem speaks to the 1-particle distribution. For most initial conditions, H approaches the equilibrium value monotonically. The interesting thing is what happens to the higher order entropies when you start far from equilibrium [a recently stirred cup of coffee, for example]. I am recalling a talk by the late Illya Prigogine which I heard over 20 years ago - sorry no reference. He displayed a graph showing one-particle H_1 going monotonically, as expected. The 2-particle H_2 peaked before tailing off monotonically. H_3 peaked later and had a longer tail and so on. Of course, H_N would be invariant.

Bottom line is: the H-Theorem is neither exact nor universally applicable, but it does embody an essentially correct insight.

ENSEMBLES: My main discomfort is with using ensembles as any more than calculation devices because they smear away much of the interesting behaviour of real systems. Consider a box containing a liquid in equilibrium with its vapour - the whole system in free fall. There is a whole family of solutions to the equations of motion representing different locations of the liquid phase within the box. The ensemble average homogenises the gas and vapour [like the family with 2.6 children] so tells us nothing about how any one example behaves - unless we put it in by hand.

As you know, this unreality is similar to the Quantum Measurement Problem, and the solution is similar too. My views on this resemble those of Deutch and Gellman. When we measure anything, we are gathering evidence about our location in phase-space [or Hilbert space] which we try to fit to a map [i.e. theory]. Everything you wrote about Bayesian inference comes into play. Wave function "collapse" is a calculation device by which we declare that we know where we are at that point. There is no mystery. It is not that there is an objective classical model "behind" quantum mechanics - rather the entire wave function is the objective model. Gellman brought to my attention a paper by Mott [Proc. Ry. Soc. A, 126, pp. 79-84 (1929)] which antedates path integrals by almost 20 years but is in the same spirit and shows clearly why we perceive the world as "classical". It surprises me is that it is never cited in elementary text books on QM.

I think that our preoccupation with "averaging" needs to be replaced by a theory of "sampling" the distribution. In a sense, QM procedures already do this when we look for eigenvalues and assert that our measurement picks out one. Particle physicists identify fleeting entities by energy resonances - something that exists identifiably for short period. I think the same sort of thing can be done for thermodynamic states - finding new kinds of complex symmetries and variational principles which cause them to "drop out" of the dynamics rather have to be inserted "by hand" a priori.

P.S. My reference to Bohr was only to say that asking a niggling question can sometimes lead to revolutionary consequences - QM in his case.

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20. To continue my remarks to Lubos:

ENSEMBLES: My main discomfort is with using ensembles as any more than calculation devices because they smear away much of the interesting behaviour of real systems. Consider a box containing a liquid in equilibrium with its vapour - the whole system in free fall. There is a whole family of solutions to the equations of motion representing different locations of the liquid phase within the box. The ensemble average homogenises the gas and vapour [like the family with 2.6 children] so tells us nothing about how any one example behaves - unless we put it in by hand.

As you know, this unreality is similar to the Quantum Measurement Problem, and the solution is similar too. My views on this resemble those of Deutch and Gellman. When we measure anything, we are gathering evidence about our location in phase-space [or Hilbert space] which we try to fit to a map [i.e. theory]. Everything you wrote about Bayesian inference comes into play. Wave function "collapse" is a calculation device by which we declare that we know where we are at that point. There is no mystery. It is not that there is an objective classical model "behind" quantum mechanics - rather the entire wave function is the objective model. Gellman brought to my attention a paper by Mott [Proc. Ry. Soc. A, 126, pp. 79-84 (1929)] which antedates path integrals by almost 20 years but is in the same spirit and shows clearly why we perceive the world as "classical". It surprises me is that it is never cited in elementary text books on QM.

I think that our preoccupation with "averaging" needs to be replaced by a theory of "sampling" the distribution. In a sense, QM procedures already do this when we look for eigenvalues and assert that our measurement picks out one. Particle physicists identify fleeting entities by energy resonances - something that exists identifiably for short period. I think the same sort of thing can be done for thermodynamic states - finding new kinds of complex symmetries and variational principles which cause them to "drop out" of the dynamics rather have to be inserted "by hand" a priori.

P.S. My reference to Bohr was only to say that asking a niggling question can sometimes lead to revolutionary consequences - QM in his case.

21. To continue my remarks to Lubos:

ENSEMBLES: My main discomfort is with using ensembles as any more than calculation devices because they smear away much of the interesting behaviour of real systems. Consider a box containing a liquid in equilibrium with its vapour - the whole system in free fall. There is a whole family of solutions to the equations of motion representing different locations of the liquid phase within the box. The ensemble average homogenises the gas and vapour [like the family with 2.6 children] so tells us nothing about how any one example behaves - unless we put it in by hand.

As you know, this unreality is similar to the Quantum Measurement Problem, and the solution is similar too. My views on this resemble those of Deutch and Gellman. When we measure anything, we are gathering evidence about our location in phase-space [or Hilbert space] which we try to fit to a map [i.e. theory]. Everything you wrote about Bayesian inference comes into play. Wave function "collapse" is a calculation device by which we declare that we know where we are at that point. There is no mystery. It is not that there is an objective classical model "behind" quantum mechanics - rather the entire wave function is the objective model. Gellman brought to my attention a paper by Mott [Proc. Ry. Soc. A, 126, pp. 79-84 (1929)] which antedates path integrals by almost 20 years but is in the same spirit and shows clearly why we perceive the world as "classical". It surprises me is that it is never cited in elementary text books on QM.

22. Remarks to Lubos part 3:

I think that our preoccupation with "averaging" needs to be replaced by a theory of "sampling" the distribution. In a sense, QM procedures already do this when we look for eigenvalues and assert that our measurement picks out one. Particle physicists identify fleeting entities by energy resonances - something that exists identifiably for short period. I think the same sort of thing can be done for thermodynamic states - finding new kinds of complex symmetries and variational principles which cause them to "drop out" of the dynamics rather have to be inserted "by hand" a priori.

P.S. My reference to Bohr was only to say that asking a niggling question can sometimes lead to revolutionary consequences - QM in his case.

23. Dear fresh-eye,

the assumption about no correlations in the initial state is just a technicality to simplify the calculation and make it doable. It doesn't affect the very conclusion that the entropy is increasing.

In particular, there is an expo-exponentially tiny probability that the correlations in the initial state are such that they will allow the state to reduce its entropy macroscopically as it evolves.

The H-theorem is more than some vague philosophical cliche; it is actually the first step in the calculation of the rate of increase of the entropy in a particular physical system.

There is no real process that could be called "collapse of the wave function" because the wave function is not a "real" object.

I completely agree that even in thinking about ensembles, one uses some kind of "unrealism" which is compatible with the unrealism of quantum mechanics. That's an issue that Boltzmann himself has understood very well. While the classical statistical physics doesn't interfere, the treatment of probabilities of different microstates is pretty much inherited by quantum mechanics. When one tries to derive macroscopic conclusions - e.g. the "theorem" that a soup will get cooler when it stands on the table - one has to approach ensembles with the same kind of uncertainty about the exact microstate as one would do in quantum mechanics.

After all, the only conceptual difference between quantum statistical physics and the classical statistical physics is that there is no interference in the latter. However, both pictures are probabilistic, and in both pictures, one may obtain some "objective" numbers - a classical limit - by averaging many building blocks.

It is just a fundamentally flawed approach to think about a particular microstate when one is actually trying to derive a statement about the behavior of a macroscopic system such as soup - i.e. about the ensembles.

Best wishes
Lubos

24. Lubos, Happy New Year.

MICROSTATES:

I prefer to think of it as micro-canonical versus canonical ensemble. Every real system we measure is micro-canonical [albeit not necessarily isolated]. Yet the 2nd law still applies and arises from the size of the system.

Many years ago, I wanted to extract equilibrium entropies from Metropolis Monte-Carlo simulations [of water] which produce snapshots of a system biassed by the Boltzmann distribution. To find the canonical ensemble average for almost any variable one could perform a simple average over all the snapshots. But not for entropy.

Entropy seemed to require an estimate of the total size of the phase-space. Formally, this could be obtained by averaging the inverse Boltzmann factor. The paradox was that this factor is largest where the snapshots are most sparsely sampled. Yet intuition tells me that practical entropy ought to be calculable from the near neighbourhood of a "typical" point in phase-space. By "typical" I mean what one would likely encounter in a real experiment.

I found pointers to how this might be done in the work of Zeev Alexandrowicz and his student, Haggai Meirovitch:
See Z. Alexandrowicz, J. Chem. Phys. 55, 2765(1971)
and H. Meirovitch,  Chem. Phys. Lett. 45, 389(1977).

Alexandrowicz' paper introduces a variational method for optimizing model distributions analogous to what we do well known method for quantum problems.

25. H-THEOREM

Is the H-Theorem right? Yes! But its range of rigorous validity is limited to that of the Boltzmann equation itself. That is: not too far from equilibrium, when higher order correlations can be neglected.

However, the insights gained seem to have a validity which goes beyond the formal validity of the equation itself. Indeed, this would not be the first time that a method was unreasonably effective beyond its range of formal validity. A well-known example is the use of the Born approximation in quantum scattering theory.

It is also possible to have a theory which seems to produce the "right" answers, but for the wrong reasons. Sooner or later that theory will mislead us, although it may remain a useful calculation tool for engineers. Think of Ptolemy's geocentric epicycle theory versus Kepler's Solar focal ellipses. It is much easier to predict an earth-based observation with the former, but would be completely hopeless if you wished to calculate what we would observe from anywhere else.

What we still lack is a usable theory to link a path from extreme disequilibrium to equilibrium. Hence, arrow of time research is not dead. In teasing it out, we may yet uncover useful structures.