The apology published in the November issue of Applied Mathematics Letters, for withdrawing my accepted article “A Second Look at the Second Law” at the last minute, says it was withdrawn “not because of any errors or technical problems found by the reviewers or editors but because the Editor-in-Chief subsequently concluded that the content was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.” This might have been a valid reason for rejecting my article, since the level of mathematics is lower than in typical AML articles, although the journal web page says “potential contributions include any work involving a novel application or utilization of mathematics.” But it was certainly a violation of the publisher’s own standards to withdraw the article for such reasons after it was accepted. See ENV’s coverage here; a video version of the article is here.
Yet articles that are more philosophical than logical are routinely published in scientific journals. To find an example, we need look no farther than the article that was cited as a “concise refutation” of my article by the blogger whose letter to the editor apparently triggered the withdrawal. The article, by Daniel Styer, was in fact the only evidence cited by this blogger to support his claim that my arguments on this topic were “often refuted nonsense.”
In that 2008 American Journal of Physics article¹, 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 k_Blog(1000), multiplied this by a generous overestimate for the number of organisms on Earth, and divided by the number of seconds in a century.
In a 2009 American Journal of Physics article², Emory Bunn 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 10^15000000 times more improbable than those at the beginning of the Cambrian (10^125000000 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 and Bunn, 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.
Thus I would like to extend Styer and Bunn’s results to the game of poker. The Boltzmann formula allows us to define the entropy of a poker hand as S = k_Blog(W) where k_B = 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 = k_Blog(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 redeal, and end up with a full house, the resulting entropy change is S2 – S1 = k_Blog (3744) – k_Blog (54912) = k_Blog (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 S_p = k_plog (W) and have a nice formula for entropy which increases when probability increases, why should the constant k_p used be equal to the Boltzmann constant k_B? 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 k_p can be chosen arbitrarily. If we do arbitrarily set k_p=k_B, 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.
Some readers will by now have realized that there is something terribly wrong with the whole concept of “compensation,” as used by Styer and Bunn and many others [3 (p366),4,5 (p160)]. There are different kinds of entropy, and poker entropy has very little to do with thermal entropy. If you want to show that evolution does not violate the second law, you cannot simply say, sure, evolution is astronomically improbable, but the Earth is an open system, so there is no problem as long as something (anything!) is happening outside the Earth that, if reversed, would be even more improbable.
In my withdrawn Applied Mathematics Letters article I defined “X-entropy” to be the entropy associated with any diffusing component X (if X is heat, X-entropy is identical to thermal entropy), and, since entropy measures disorder, I defined “X-order” to be the negative of X-entropy, and showed that the equations for entropy change not only say that X-order cannot increase in an isolated system, they also say that in a non-isolated system the X-order cannot increase faster than it is imported through the boundary. (I had published this analysis previously in a Mathematical Intelligencer piece⁶ and in appendix D of my 2005 John Wiley book⁷].) Thus the equations for entropy change do not support the illogical “compensation” idea; instead, they illustrate the tautology that “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 (or leaving) which makes it not extremely improbable.”
Of course no one on either side believes my article was withdrawn because it was “more philosophical than mathematical,” it was withdrawn because it supported the wrong philosophy. The article was perceived as being supportive of intelligent design (ID), and it was discovered that I am a known ID supporter (see In the Beginning…⁸). But in fact, this article did not mention ID or make any appeal 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 it has not, you have to argue that, thanks to the influx of solar energy, it is not really extremely improbable (in the sense of footnote 4 there) that the four forces of physics would rearrange the basic particles of physics into spaceships, nuclear power plants, computers and the Internet. But you cannot hide behind the absurd compensation argument made by Styer, Bunn, Asimov and many others.
On a topic of less philosophical importance, arguments with such grotesque errors of logic as those made by Styer and Bunn could never be published in a physics journal, yet these errors go almost unchallenged to this day. Articles that are more philosophical than scientific are welcomed by many science journals, as long as they support the right philosophy.
References
(1) Styer D (2008) “Entropy and Evolution,” American Journal of Physics 76, issue 11, 1031-1033.
(2) Bunn E (2009) “Evolution and the Second Law of Thermodynamics,” American Journal of Physics 77, issue 10, 922-925.
(3) Urone PP (2001) College Physics, Brooks/Cole, Pacific Grove, CA.
(4) Asimov I (1970) “In the Game of Energy and Thermodynamics, You Can’t Even Break Even,” Smithsonian 1, August 1970, p4.
(5) Angrist S, Hepler L (1967) Order and Chaos, Basic Books, New York.
(6) Sewell G (2001) “Can ANYTHING Happen in an Open System?,” The Mathematical Intelligencer 23, number 4, 8-10.
(7) Sewell G (2005) The Numerical Solution of Ordinary and Partial Differential Equations, 2nd Ed., John Wiley and Sons, Hoboken, NJ.
(8) Sewell G (2010) In the Beginning and Other Essays on Intelligent Design, Discovery Institute Press, Seattle, Washington.