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Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

Second Law, "Hard" and "Soft" Irreversibility, and NKS

In our NKS Online guestbook we got a reasonable question, which I thought others might be interested in, along with my response. It concerns the section at the start of chapter 9 that discusses the second law of thermodynamics. Here is the way the question was posed -

"I'm not sure I agree that the second law of thermodynamics is disproved or somehow not supported by any of these examples. The second law states the total entropy of the universe either increases or stays the same as a result of a physical process. None of Dr. Wolfram's examples show evidence of a spontaneous decrease in entropy which would be required to deny the universality of the Second Law. I would look forward to a discussion of any evidence to the contrary that I may have missed".

The relevant section of the book starts on page 435 and continues to the end of page 457. In the notes, all of them are relevant from page 1017 to the bottom of the first column on page 1022.

To start with, NKS does not say that the second law is disproven, and indeed the note on page 1020 titled "My explanation of the Second Law" explicitly states that what NKS says about the second law is not incompatible with past understandings of it. It claims only to have clarified some points about it, and especially about the puzzle created by the contrast between the apparent reversibility of all known laws of physics in the small, and the definite direction of evolution provided by the second law in the large (frequently called "the arrow of time"). Since the behavior described by the second is supposed to be an emergent consequence of the (apparently reversible) micro-laws, there is a puzzle to explain.

NKS discusses the issue in the context of reversibility. One needs to understand at the outset that there are two distinct ways something like the second law might be taken, which I will call hard irreversibility and soft irreversibility. By hard irreversible, I mean any rule or transformation that takes multiple prior states to the same subsequent state. Such a mapping is a contraction, in a state-space sense. Rule 254 is the classic example - it takes any initial to all black in a few steps. It is easy to see how an emergent measure of rule 254 would show irreversibility, a constant increase in black cells, etc. If the micro-laws of physics were irreversible in a hard sense, then the second law or things like it would be expected.

But the micro-laws we actually see, in great detail e.g. in particle physics, appear to be completely reversible in every respect. They are not contractions. From a local state (including perhaps some local rates of change to be sure, but that is just a local state in a phase-space sense), the past can be projected as readily as the future, according to the same laws. (In QM some of these projections are probabilistic, but that is again true in both directions. In QM, a given current state could have come from a blurred variety of priors).

So there is no obvious reason from the micro-laws themselves, to expect irreversibility up at a coarse-grained level. Coarse graining might be lossy - it might lump many micro-configurations into the same "bin" (of temperature e.g.). But the micro-states themselves need not evolve from more to less ordered, just considering the micro-laws.

Wolfram deals with a few of the classic attempts to explain this in passing, in the note on page 1020 titled "Current thinking on the Second Law". Ergodic explanations, for example, suggest that systems visit all of their states, and that their typical average behavior can be found just by portioning out time among the possibilities. (See also the previous note, discussing Gibb's contributions to this subject). As Wolfram points out, however, only some systems are ergodic to begin with, and the number of possible (real, physical) states for systems with large numbers of components is so astronomical they cannot possibly actually visit them all. More components does not help in that respect, because the number of configurations always grows much more rapidly than components.

So we have systems visiting only tiny portions of their state space according to reversible laws, not populating all of them, nor (it appears) evolving in a contractive way. Yet the observed macro behavior is still irreversible. And the question is why.

Wolfram notices that for any reversible rule, one can find definite micro configurations that would evolve toward increasing order, rather than the opposite. To see this, take such a system (which in addition produces complexity, not always simple behavior) from some ordered state, call it B, and evolve it backward in time. You will reach a state, call it A, that has high entropy measure in all conventional senses. The same as if you evolved it forward to C - a necessary consequence of reversibility of the micro rule. Entropy measured at A or C is high, and at B is low.

Now, imagine starting the system from A and running forward, only to B. You have a reversible deterministic system capable of generating complex behavior, that shows a decrease in entropy measure over the course of its evolution - that portion of it anyway. There is no "hard" increase in entropy going on. Nor is it constant - it is in fact decreasing. But from a highly special initial condition, though one with a high conventional entropy measure, as well. The system state at A is a disordered jumble, that happens to be so configured, that it evolves to an ordered state at B. There is no hard impossibility of this.

But it depends on a rare initial state, A. Of all possible configurations of the systems elements at the time of A, only some tiny portion will evolve to ordered configurations at B, or other nearby times B' or B''. Most possible configurations at A will go from disorder to disorder. A decrease in entropy is then possible, but improbable, if we select initial conditions at time A in some arbitrary manner, instead of constructing them by evolving backward from a known ordered state at B.

This understanding of the second law, therefore, predicts there will be local and temporary deviations from it, on some scale damping out as the special conditions around A that are ancestor states of an ordered B, die out in a sea of disordered states at time A that are ancestors of disordered states at B. The second law is a statistical law about ordered areas of configuration space, not a hardwired requirement from step to step.

The analog of the Maxwell's Demon question then becomes, can one find ancestors of ordered Bs, by some method more efficient than actually evolving the system to see what it does? And there Wolfram invokes computational irreducibility, to explain why it will in general be as hard to know what the ancestors of ordered Bs are, as it is to overcome the entropy of the system in any other way. The inability to short-cut the backward calculation, "what microstates will lead to order at B?", sets a limit on the sort of initials that can be considered. Otherwise put, a large information (negative entropy) "investment" would be needed to pick out exactly the prior configurations A - themselves apparently disordered - that happen to evolve to order at B.

For any system that intrinsically generates randomness, the above is the whole story. It shows the kind of thing the second law is about, and how and why it should be expected to hold. If such systems are the most common forms of complexity, formally and in the real world, again we expect the second law to hold wherever we look, with only local and temporary, small scale deviations from it, as regularly covered in basic statistics e.g. in the theory of sampling, when there are small sample sizes.

But Wolfram looked for and found (purely formally) some reversible rules that do not seem to always produce increasing disorder in this sense, from typical initial conditions. Without being simple in their behavior all of the time. Andrew Wuensche noticed a similar phenomenon, when he looked for a classifier to distinguish class 4 from class 3 complexity, by looking at a 2D graph of entropy vs. change in entropy. The 3s rapidly get to maximum entropy and stay there. Entropy high, delta entropy low. But 4s show fluctuations in both. 4s have local particles on simple or periodic backgrounds, which are in effect quite ordered states compared to seething class 3 randomness. But they can also produce regions of disorder that look essentially like class 3. As the portions of a system in one or the other "phase", changes, an entropy measure will move, and in either direction.

What Wolfram found in his rule 37R example was essentially this phenomenon in a reversible rule (as opposed to general rules, where it also happens with 4s, most of which are not reversible). And the idea is that selecting a state at random back at A for such a system, there is no reason to expect, even statistically let alone by any step-by-step necessity, that the entropy measure of the system at B will be higher than it was at A.

The second law is correct for lots of systems (formal as well as empirical), because intrinsic generation of randomness is everywhere. It can be expected in a hard way if the underlying rules are contractions (irreversible themselves). It can be expected in a statistical way if the underlying rule is reversible but class 3 randomizing. But if the underlying rule of a system is both reversible and class 4, local-particle-supporting, there is no good reason to expect the second law to apply to that system. Overall it is certainly true - that is confirmed by experiment, and to be expected, purely formally, from the rariety of special initials, the preponderance of 3s over 4s, etc. But from formal considerations alone, we should expect there might be (limited measure, rare, etc) exceptions.

I hope this helps.

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Old Post 11-10-2005 04:41 PM
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Tony Smith
Meme Media
Melbourne, Australia

Registered: Oct 2003
Posts: 168

Re: "the apparent reversibility of all known laws of physics in the small"

I am gaining more and more confidence that the reversibility we observe "in the small" exists only because of an inescapable requirement for a predominance of things which persist in order to produce a world in which pattern recognition is possible.

By hard irreversible, I mean any rule or transformation that takes multiple prior states to the same subsequent state. Such a mapping is a contraction, in a state-space sense.
As suggested in my first post here, such an irreversible rule run for a sufficient time can evolve to overwhelmingly cyclic local states. Given that the tools with which we macroscale creatures have to directly observe the microscale would be cyclic local states themselves (photons et al), it is unsurprising that such states provide a persuasive illusion of simple persistence and thus reversibility and ubiquitous emergent conservation "laws". Thinking about quantum indeterminacy and the first femtoseconds of the Big Bang from that perspective starts to make a lot more sensible natural history.

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Old Post 11-12-2005 08:26 AM
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Gershom Zajicek M.D.
Hebrew University of Jerusalem
Jerusalem, Israel

Registered: Feb 2004
Posts: 152

A reversible CA

Gentlemen I fully agree with the anonymous note in your guestbook: "I'm not sure I agree that the second law of thermodynamics is disproved or somehow not supported by any of these examples. . . .”

The question is whether the CA[rule=214] (p.437) might be regarded as an appropriate model for investigating the nature of entropy. The answer is No! Since this CA has a memory in which it stores the color of the cell two steps back, while entropy lacks a memory. Actually CA[rule=214] is a neat example of Maxwell’s demon (p.1021)

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Old Post 11-12-2005 12:59 PM
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Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

Actually one look-back is often used to mimic wave phenomena; it readily mimics second difference effects. E.g. what Rudy Rucker calls "the wave rule" is a totalistic rule with one lookback over the center cell. Real systems have positions and momenta, not just positions. Peirce already pointed this out, discussing the essential spannedness of sufficient knowledge even in a Laplacean picture of the world.

If you want, you can also encode the effect in a CA with a few more colors (and in some cases of lookback, longer range) encoding their current value and their previous value in the immediately preceding step, effectively. Wolfram notes that possibility in passing in a caption on page 437, stating four colors are sufficient in this case.

As for the idea that the fundamental laws might be time assymmetric (contractions), but have symmetric consequences on an emergent level much of the time, it is certainly possible, formally. One might also suspect we'd find symmetric laws more readily, if the math we use to try to model a system is much more easily solved with extra symmetries that effectively reduce the degrees of freedom we have to contend with.

But it is speculative at this point. We can't point to a particular particle physics experiment that shows detectable, fundamental time assymmetry in the small. Weak force phenomena are the closest phenomenally, and one might ask whether they've been shoehorned into a successful QED-like theory (in the standard model I mean) when some other model might be more natural for them.

Another purely speculative idea is to notice that multiway systems are contractions "turned upside down". So you'd expect apparent non-determinism (multivalued transitions) backtracking a contractive system. One poster suggested our emergent sense of time is simply flipped over from an underlying assymmetric, contractive system.

A slight difficulty with this last idea is the apparent reversibility in the small of working QM calculations. On a sufficiently small scale, the past doesn't seem to be more readily predicted than the local future. But one might imagine workarounds that make that prediction difficulty epistemological (about what we can know) rather than objective.

FWIW.

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Old Post 11-13-2005 11:42 AM
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Tony Smith
Meme Media
Melbourne, Australia

Registered: Oct 2003
Posts: 168

Momenta easy, conservation of hard

Equally hard is near continuous variation of momentum, at least until we stop presuming that standard model fundamental physics will have any detectable relationship to Planck scale processes.

If the only knowledge at your disposal was quantum mechanics you would be hard put to predict any of the everyday macroscale physics we take for granted. Go down as far again into the microstructure and you aren't likely to find anything which would be useful in the absence of an understanding of how the world works at higher levels.

CA provide a neat conceptual demonstration of the importance of moving entities in Class 4 processes. It would be really interesting if somebody could identify simple or directed graph-theoretic network evolution rules that provide a persuasive analogy of conservation of linear and angular momentum, but I suspect it is more likely that nature has found an easier way to get from the elemental network to a liquid-like state from which apparent continuity and conservation could much more easily emerge.

There are problems with overrating "equivalence". We actually need a natural history story which does not demand leaps of faith.

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Old Post 11-14-2005 05:16 AM
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Alexandre Ismail

NYC

Registered: Nov 2005
Posts: 4

entropy: a common misunderstanding

Jason Cawley wrote:

So we have systems visiting only tiny portions of their state space according to reversible laws, not populating all of them, nor (it appears) evolving in a contractive way. Yet the observed macro behavior is still irreversible. And the question is why.

and

Coarse graining might be lossy - it might lump many micro-configurations into the same "bin" (of temperature e.g.).

I'm pretty sure our perception of "entropy" at a macro-scale comes from the fact that almost all of our human perceptual processes involve coarse graining and incomplete sampling of the underlying system of reality.

I think that's all there between the micro-scale of _demonstrated_ and _computed_ reversibility and the macro-scale of _observed_ (_or sampled_) irreversibility. Its just the inability of humans to distinguish microstates, followed by an insistence on writing equations about them - while not looking at them.

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Old Post 07-23-2006 04:12 AM
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