As ENV readers may know, I wrote an article, “A Second Look at the Second Law,” that was reviewed and accepted by Applied Mathematics Letters (AML) in 2011, then withdrawn at the last minute because “our editors simply found that it does not consist of the kind of content that we are interested in publishing.” The circumstances surrounding the withdrawal were so embarrassing to the publisher that it ended up paying $10,000 in damages and publishing an apology in the journal. But they still refused to publish the article itself.
Now the Mathematical Intelligencer (MI) has published a rebuttal to my (unpublished, remember!) article. So I submitted a letter to the editor (which, I was told, are normally published “as received,” if they are published) responding to the MI article, but my letter was sent to a referee anyway, and rejected, so I will not be allowed to defend myself, even in a letter to the editor. One of the referee’s comments is especially interesting:
He quotes [Bob] Lloyd criticizing him for being too casual in equating entropy and disorder. He replies that he should not be faulted, since physics textbooks often discuss the second law in scenarios in which precise quantification is difficult. Surely, though, there is a big difference between the level of rigor called for when communicating the flavor of the second law to students, and the level to which one should be held when arguing that a major branch of science must be discarded as conflicting with the second law. Hand-waving arguments about films running backward are not adequate for drawing the momentous conclusions Sewell wants to draw. That was Lloyd’s point.
In other words, it’s OK for physics texts to apply the second law to applications beyond thermodynamics, such as books burning or wine glasses breaking, but not to computers arising on a rocky planet, because that might threaten “a major branch of science.” Even publishing one letter to the editor that draws such a “momentous” conclusion must not be permitted!
Since I will not be allowed to respond in the journal, below is my response, so at least ENV readers can see it. In his MI Viewpoint, Bob Lloyd begins by successfully linking me to Discovery Institute (after which, further rebuttal is hardly necessary), and the rest of the paper is almost entirely dedicated to showing that my “X-entropies” are not always independent of each other, that under certain circumstances they can influence each other.
In fact, in the AML paper I acknowledged that while these different entropies are independent of each other in “our simple models, where it is assumed that only heat conduction or diffusion is going on, naturally, in more complex situations, the laws of probability do not make such simple predictions.” Lloyd says that universal independence of these X-entropies “is central to all the version of his arguments.” Except that I never claimed or believed they were always independent — see point one in my response for more detail on his primary criticism, which entirely misses the main point of the AML paper. In 11 exhausting years of writing on this topic, I have noticed that criticisms are invariably directed toward some minor, peripheral point, while completely ignoring the main argument, which is extraordinarily simple, and made clearly in the last paragraph of my rejected response…and even more clearly in the video below.
So the AML article was not worthy of publication, even after it was accepted, an article slamming the unpublished article is worth publishing, but not any response to that. Well, now you have an illustration of how the scientific “consensus” on certain controversial issues is maintained. And if you watch the video you will understand why, on this issue at least, suppression of all opposing viewpoints is so necessary to maintain the consensus.
A Response to Bob Lloyd
By Granville Sewell
What follows is my response to Bob Lloyd’s Viewpoint piece “Is there any Conflict between Evolution and the Second Law of Thermodynamics?”6, published in the last issue of The Mathematical Intelligencer. But first a little background.
The Applied Mathematics Letters article referred to in Lloyd’s piece, and in my response, was actually never published in AML. I received a letter from the editor about a week before it was to appear telling me it had been withdrawn, because “our editors simply found that it does not consist of the kind of content that we are interested in publishing.” The journal later published a formal apology5, acknowledging that it was withdrawn, “not because of any errors or technical problems found by the reviewers or editors, but because the Editor-in-Chief subsequently concluded that the content was more philosophical than mathematical.” Hence the article can only be viewed on my personal web site, here, but it is important that readers do see this article, otherwise they will not really understand either Lloyd’s criticism, or my response.
It is also important to note that, while I am an intelligent design advocate4, the AML article did not mention intelligent design, did not include any appeals to the supernatural, and did not even conclude that the second law has definitely been violated by what has happened on Earth. It only concluded that if you want to believe that the spontaneous rearrangement of atoms on this planet into intelligent brains and machines capable of mathematical computation and long distance air travel did not violate it, you cannot hide behind the widely accepted “compensation” idea, you have to believe that, thanks to the influx of solar energy, and natural selection, and whatever, what happened here was not really extremely improbable — or more correctly, macroscopically describable and extremely improbable from the microscopic point of view. (In my AML paper I tried to state what is meant by “extremely improbable” a bit more precisely: “If we repeat an experiment 2k times, and define an event to be ‘simply describable’ (macroscopically describable) if it can be described in m or fewer bits (so that there are 2m or fewer such events), and ‘extremely improbable’ when it has probability 1/2n or less, then the probability that any extremely improbable, simply describable event will ever occur is less than 2(k+m) /2n. Thus we just have to make sure to choose n to be much larger than k+m.”)
Lloyd cites my example, given in my letter to the editor1 in a 2001 Mathematical Intelligencer issue, of carbon and heat diffusing independently of each other in a solid, and claims that:
This is a very clear statement that he believes that the different “X-entropies” in the later AML formulation, behave independently. This point is central to all the versions of his argument…Sewell’s argument is general and should apply equally to solutes in liquid solutions. Here also, thermal gradients should have no effect on the concentrations of dissolved species if the independent treatment of entropies is valid.
Then he proceeds to show that these “entropies” are not independent of each other in certain experiments in liquids. This seems to be his primary criticism of my writings on this topic.
I may have left the impression in my 2001 letter that I believed these different “X-entropies” were always independent of each other, but in the more recent Applied Mathematics Letters paper, I wrote:
The “compensation” counter-argument was produced by people who generalized the model equation for closed systems, but forgot to generalize the equation for open systems. Both equations are only valid for our simple models, where it is assumed that only heat conduction or diffusion is going on; naturally, in more complex situations, the laws of probability do not make such simple predictions. Nevertheless, in “Can ANYTHING Happen in an Open System?” I generalized the equations for open systems to the following tautology, which is valid in all situations: “if an increase in order is extremely improbable when a system is isolated, it is still extremely improbable when the system is open, unless something is entering which makes it NOT extremely improbable.”
The examples of “X-entropy” (e.g., “carbon-entropy”) were given only to illustrate this tautology, as was clearly stated in my AML abstract. And in a solid, where the simplest models for diffusion and heat conduction are assumed, that is, without convection or sources/sinks, these “entropies” are independent of each other and each X-entropy cannot decrease faster than it is exported through the boundary (i.e., each X-order cannot increase faster than it is imported), so in this case, the equations of entropy change do illustrate the tautology nicely, though I’m not sure a tautology needs illustrating. And there is really nothing special, except for historical considerations, about the case X=heat (thermal entropy), the entropies associated with any diffusing component X are all defined by the same equations, and are all equally quantifiable. In other situations, as I clearly stated, things are not that simple, the universal “independence” of X-entropies was never claimed, and is certainly not a critical point.
Lloyd did not of course provide any counterexample for my tautology. Contrary to what seems to be common belief, when thermal entropy decreases in an open system, there is not anything macroscopically describable happening which is extremely improbable from the microscopic point of view, something is just entering the open system which makes the decrease not extremely improbable. The tautology was clearly the central point of my argument in each paper.
It is difficult to know where this mistaken idea, that entropy can be separated into independent components, has come from. One possibility is that this comes from assuming a precise equivalence between entropy, to which the formalisms of thermodynamics apply, and disorder, which is too ill-defined for thermodynamics to be applied.
Then Lloyd ridicules me for talking about the order associated with airplanes or TV sets, as though I were the first to try to apply the second law to less quantifiable applications such as these. But most any college physics text that mentions the second law cites things like wine glasses breaking or books burning, as examples of entropy increases. All I have done is extend this in an obvious way: if decreases in entropy such as atoms rearranging themselves into computers and jet airplanes would be forbidden by the second law on an isolated planet, because they are macroscopically describable things that are extremely improbable from the microscopic point of view, then they are still forbidden by the same law (or at least the same natural principle) if the only thing entering the system is solar energy, for the same reason: they are still extremely improbable. At least it seems clear to me and most other people that such an increase in order would still be extremely improbable, though I acknowledged in the Conclusions of the AML paper that one can still argue that it is not extremely improbable, and thus avoid the conclusion that the second law has been violated.
Suppose we take a video of a tornado sweeping through a town, and run the video backward. Would we argue that although tornados turning rubble into houses and cars represents a decrease in entropy, tornados derive their energy from the sun, and the increase in entropy outside the Earth more than compensates the decrease seen in the video, so there is no conflict with the second law? Would we argue that what we were seeing was too difficult to quantify, and “too ill-defined for thermodynamics to be applied,” and conclude that we cannot tell if there is a problem with the second law or not? If so, the second law should never have been applied to any applications outside thermodynamics; but I am certainly not the first to do this. Some things are obvious even if they are difficult to quantify. And tornados turning rubble into houses and cars is nothing compared to what has happened on Earth in the last 3-4 billion years, in my opinion at least.
The qualitative point associated with the solar input to Earth, which was dismissed so casually in the abstract of the AML paper, and the quantitative formulations of this by Styer and Bunn, stand, and are unchallenged by Sewell’s work.
The American Journal of Physics papers by Styer2 and Bunn3 illustrate beautifully the type of logic my writings are criticizing, so let’s look at these papers.
Styer estimated the rate of decrease in entropy associated with biological evolution as less than 302 Joules/degree Kelvin/second, noted that this rate is very small, and concluded “Presumably the entropy of the Earth’s biosphere is indeed decreasing by a tiny amount due to evolution and the entropy of the cosmic microwave background is increasing by an even greater amount to compensate for that decrease.” To arrive at this estimate, Styer assumed that “each individual organism is 1000 times more improbable than the corresponding individual was 100 years ago” (a “very generous” assumption), used the Boltzmann formula to calculate that a 1000-fold decrease in probability corresponds to an entropy decrease of kBlog(1000), multiplied this by a generous overestimate for the number of organisms on Earth, and divided by the number of seconds in a century. ?
Bunn later concluded that Styer’s factor of 1000 was not really generous, that in fact organisms should be considered to be, on average, about 1025 times more improbable each century, but shows that, still, “the second law of thermodynamics is safe.” ?
Since about 5 million centuries have passed since the beginning of the Cambrian era, if organisms are, on average, 1000 times more improbable every century, that would mean that today’s organisms are, on average, about 1015000000 times more improbable than those at the beginning of the Cambrian (10125000000 times more improbable, if you use Bunn’s estimate). And since nothing can have probability more than 1, this would presumably mean today’s organisms have a probability of less than 10-15000000 (or 10-125000000) of having arisen. But, Styer and Bunn argue, there is no conflict with the second law because the Earth is an open system, so any extremely improbable events here can be compensated by events elsewhere in the universe. ?
According to Styer, the Boltzmann formula, which relates the thermal entropy of an ideal gas state to the number of possible microstates, and thus to the probability of the state, can be used to compute the change in thermal entropy associated with any change in probability: not just the probability of an ideal gas state, but the probability of anything. This is very much like finding a Texas State Lottery sheet that lists the probabilities of winning each monetary award and saying, aha, now we know how to convert the probability of anything into its dollar equivalent.?
To see how absurd this logic is, let’s extend Styer’s calculations to the game of poker. The Boltzmann formula allows us to define the entropy of a poker hand as S = kBlog(W) where kB = 1.38*10-23 Joules/degree Kelvin is the Boltzmann constant and W is the number of possible hands of a given type (number of “microstates”, W = p*C(52,5), so S = kBlog (p) + Constant, where p is the probability of the hand). For example, there are 54912 possible “three of a kind” poker hands and 3744 hands that would represent a “full house,” so if I am dealt a three of a kind hand, return some cards, reshuffle and re-deal, and end up with a full house, the resulting entropy change is S2 – S1 = kBlog (3744) – kBlog (54912) = kBlog (1/14.666) = -3.7*10-23 Joules/degree. Of course, a decrease in probability by a factor of only 15 leads to a very small decrease in entropy, which is very easily compensated by the entropy increase in the cosmic microwave background, so there is certainly no conflict with the second law here. ?
There are some problems, however. While one can certainly define a “poker entropy” as Sp = kplog (W) and have a nice formula for entropy which increases when probability increases, why should the constant kp used be equal to the Boltzmann constant kB? In fact, it is not clear why poker entropy should have units of Joules/degree Kelvin. In the case of thermal entropy, the constant is chosen so that the statistical definition of thermal entropy agrees with the standard macroscopic definition. But there is no standard definition for poker entropy to match, so the constant kp can be chosen arbitrarily. If we do arbitrarily set kp = kB, so that the units match, it still does not make any sense to add poker entropy and thermal entropy changes to see if the result is positive or not. It is not clear how the fact that thermal entropy is increasing in the rest of the universe makes it easier to get a highly improbable poker hand. Of course, all these problems also exist with respect to Styer and Bunn’s analyses of the entropy associated with evolution; at least with poker entropy we don’t have to take wild guesses at the probabilities involved. ???
If you want to show that the spontaneous rearrangement of atoms into machines capable of mathematical computation and interplanetary travel does not violate the fundamental natural principle behind the second law, you cannot simply say, as Styer and Bunn and so many others do, sure, evolution is astronomically improbable, but the Earth is an open system, so there is no problem as long as something (anything, apparently!) is happening outside the Earth which, if reversed, would be even more improbable. You have to argue that what has happened on Earth is not really astronomically improbable, given what has entered (and exited) our open system. Why is such a simple and obvious point so controversial?
(1) Sewell G (2001) “Can ANYTHING Happen in an Open System?,” The Mathematical Intelligencer 23, issue 4, pp. 8-10.
(2) Styer D (2008) “Entropy and Evolution,” American Journal of Physics 76, issue 11, 1031-1033.
(3) Bunn E (2009) “Evolution and the Second Law of Thermodynamics,” American Journal of Physics 77, issue 10, pp. 922-925.
(4) Sewell G (2010) In the Beginning and Other Essays on Intelligent Design, Discovery Institute Press.
(5) — (2011) “Editors note: A Second Look at the Second Law,” Applied Mathematics Letters 24, issue 11, p. 1968.
(6) Lloyd B (2012) “Is there any Conflict between Evolution and the Second Law of Thermodynamics?” The Mathematical Intelligencer 34, issue 1, pp. 29-33.