Month: December 2008

I wrote up the argument of my last post a bit more carefully and sent it off to the American Journal of Physics as a comment on the original article by Styer.  Here’s a PDF of the submitted version.  I tried to clean up the argument and fill in some gaps, but nothing substantial has changed.

There’s an interesting article by Daniel Styer in the November 2008 issue of the American Journal of Physics.  The full article is behind a subscription wall, so if you’re not a subscriber you can’t see it.  The link should take you to the title and abstract, but in case it doesn’t, here they are:

Daniel Styer, “Entropy and Evolution,” American Journal of Physics, 76, 1031-1033 (November 2008).

Abstract: Quantitative estimates of the entropy involved in biological evolution demonstrate that there is no conflict between evolution and the second law of thermodynamics. The calculations are elementary and could be used to enliven the thermodynamics portion of a high school or introductory college physics course.

The article addresses a claim often made by creationists that the second law of thermodynamics is inconsistent with evolution.  The creationist argument goes like this: The second law of thermodynamics says that entropy always increases, which means that things always progress from order to disorder.  In biological evolution, order is created from disorder, so it is contradicted by the second law.

The creationist argument is wrong.  It rests on misunderstandings about what the second law of thermodynamics actually says.  Debunkings of the argument can be found in a bunch of places.  The point of Styer’s article is to assess quantitatively just how wrong it is.  This is a worthy goal, but Styer makes a mistake in his analysis that seriously distorts the physics. I feel a little bad about even mentioning this. After all, Styer is on the side of the (secular) angels here, and his final conclusion is certainly right: the creationists’ argument is without merit.  Fortunately, as I’ll show, Styer’s mistake can be fixed in a way that demonstrates this unambiguously.

This post is kind of long, so here’s the executive summary:

1. The creationists’ argument that evolution and the second law are incompatible rests on misunderstandings.   (This part is just the conventional view, which has been written about many times before.)
2. Styer’s attempt to make this point more quantitative requires him to guess a value for a certain number.  His guess is a wild underestimate, causing him to underestimate the amount of entropy reduction required for evolution and making his argument unpersuasive.  The reason it’s such a huge underestimate is at the heart of what statistical mechanics is all about, which is why I think it’s worth trying to understand it.
3. We can fix up the argument, estimating things in a way that is guaranteed to overstate the degree of entropy reduction required for evolution.  This approach gives a quantitative and rigorous proof that there’s no problem reconciling evolution and the second law.

The conventional wisdom.  The main reason that the creationist argument is wrong is that the second law applies only to thermodynamically closed systems, that is to say to systems with no heat flow in or out of them.  Heat is constantly flowing into the Earth (from the Sun), and heat is constantly escaping from the Earth into space, so the Earth is not a thermodynamically closed system.  When you’re dealing with a system like that, the way to apply the second law is to think of the system you’re considering as part of a larger system that is (at least approximately) closed.  Then the second law says that the total entropy of the whole big system must increase.  It does not forbid the entropy of one subsystem from decreasing, as long as other parts increase still more.  In fact, when heat flows from a very hot thing (such as the Sun) to a colder thing (such as the Earth), that heat flow produces a huge increase in entropy, and as a result it’s very easy for decreases in entropy to occur in the rest of the system without violating the second law.

Imagine raking the leaves in your yard into a pile.  The entropy of a localized pile of leaves is less than the entropy of the leaves when they were evenly spread over your yard.  You managed to decrease that entropy by expending some energy, which was dissipated in the form of heat into the environment.  That energy dissipation increased the total entropy by more (much, much more, as it turns out) than the decrease in the entropy of the leaves.

There is a second reason the creationist argument is wrong, which is that entropy is not exactly the same thing as disorder.  Often, entropy can be thought of as disorder, but that correspondence is imperfect.  It seems clear that in some sense a world with lots of organism in it is more “organized” than a world with just a primordial soup, but the exact translation of that vague idea into a statement about entropy is tricky.  The first reason (Earth is not a closed system) is much more important, though.

The main purpose of Styer’s article is to quantify these points. In particular, he quantifies the amount of entropy supplied by sunlight on the Earth and the amount of entropy decrease supposedly required for biological evolution.  The point is to show that the first number is much greater than the second, and so there’s no problem reconciling the second law and evolution.

What’s wrong with Styer’s argument.  The problem with his argument comes in Section III, in which he tries to estimate the change in entropy of living things due to evolution:

Suppose that, due to evolution, each individual organism is 1000 times “more improbable” than the corresponding individual was 100 years ago.  In other words, if Ω_i is the number of microstates consistent with the specification of an organism 100 years ago, and Ω_f is the number of microstates consistent with the specification of today’s “improved and less probable” organism, then Ω_f =  10^-3Ω_i. I regard this as a very generous rate of evolution, but you may make your own assumption.

Thanks!  I don’t mind if I do.

1000 is a terrible, horrible, ridiculous underestimate of this “improbability ratio.”  The reason gets at the heart of what statistical mechanics is all about. One of the most important ideas of statistical mechanics is that probability ratios like this, for macroscopic systems, are almost always exponentially large quantities.  It’s very hard to imagine anything you could do to a system as large as, say, a cell, that would change the number of available microstates (generally known as the “multiplicity”) by a mere factor of 1000.

If a single chemical bond is broken, for instance, then energy of about E=1 eV is absorbed from the system, reducing the multiplicity by a factor of about e^E/kT, or about 10¹⁷ at typical biological temperatures.  That’s not the sort of thing Styer is talking about, of course: he’s talking about the degree to which evolution makes things “more organized.”  But changes of that sort always result in  reductions in multiplicity that are at least as drastic.  Let me illustrate with an example. [The fainthearted can feel free to skip the next paragraph — just look at the large numbers in the last couple of sentences.]

Suppose that the difference between an organism now and the same organism 100 years ago is that one additional protein molecule has formed somewhere inside the organism.  (Not one kind of protein — one single molecule.)  Suppose that that protein contains 20 amino acids, and that those amino acids were already present in the cell in the old organism, so that all that happened was that they got assembled in the right order.  That results in an enormous reduction of multiplicity: before the protein formed, the individual amino acids could have been anywhere in the cell, but afterwards, they have to be in a specific arrangement. A crude estimate of the multiplicity reduction is just the degree to which the multiplicity of a dilute solution of amino acids in water goes down when 19 amino acids are taken out of it (since those 19 have to be put in a certain place relative to the 20th).  The answer to that question is e^-19μ/kT, where μ is the chemical potential of an amino acid.  Armed with a statistical mechanics textbook [e.g., equation (3.63) of the one by Schroeder that I’m currently teaching from], you can estimate the chemical potential.  Making the most pessimistic assumptions I can, I can’t get -μ/kT below about 10, which means that producing that one protein reduces the multiplicity of the organism (that is, makes it more “improbable”) by a factor of at least 10⁸⁰ or so (that’s e^-190).  And that’s to produce one protein.  If instead we imagine that a single gene gets turned on and produces a bunch of copies of this protein, then you have to raise this factor to the power of the number of protein molecules created.

That calculation is based on a bunch of crude assumptions, but nothing you do to refine them is going to change the fact that multiplicity changes are always by exponentially large factors in a system like this.  Generically, anything you do to a single molecule results in multiplicity changes given by  e^-μ/kT, μ is always at least of order eV, and kT is only about 0.025 eV.  So the ante to enter this game is something like e⁴⁰, or 10¹⁷.

Fixing the problem.  It’s been a few paragraphs since I said this, so let me say it again: Despite this error, the overall conclusion that there is no problem with evolution and the second law of thermodynamics is still correct.  Here’s a way to see why.

Let’s make an obviously ridiculous overestimate of the total entropy reduction required to create life in its present form on Earth.  Even this overestimate is still far less than what we get from the Sun.

Suppose that we started with a planet exactly like Earth, but with no life on it.  I’ll call this planet “Dead-Earth.” The difference between Dead-Earth and actual Earth is that every atom in the Earth’s biomass is in Dead-Earth’s atmosphere in its simplest possible form.  Now imagine that an omnipotent being (a Maxwell’s demon, if you like) turned Dead-Earth into an exact replica of Earth, with every single molecule in the position that it’s in right now, by plucking atoms from the atmosphere and putting them by hand into the right place.  Let’s estimate how much that would reduce the entropy of Dead-Earth.  Clearly the entropy reduction required to produce an exact copy of life on Earth is much less than the entropy reduction required to produce life in general, so this is an overestimate of how much entropy reduction the Sun has to provide us.

Earth’s total biomass has been estimated at 10¹⁵ kg or so, which works out to about 10⁴¹ atoms.  The reduction in multiplicity when you pluck one atom from the atmosphere and put it into a single, definite location is once again  e^-μ/kT . This factor works out to at most about e¹⁰.  [To get it, use the same equation as before, (3.63) in Schroeder’s book, but this time use values appropriate for the simple molecules like N₂ in air, not for amino acids in an aqueous solution.]  That means that putting each atom in place costs an amount of entropy ΔS = 10k where k is Boltzmann’s constant.  To put all 10⁴¹ atoms in place, therefore, requires an entropy reduction of 10⁴²k.

Styer calculates (correctly) the amount of entropy increase caused by the throughput of solar energy on Earth, finding it to be of order  10³⁷k every second.  So the Sun supplies us with enough entropy every 100,000 seconds (about a day) to produce all of life on Earth.

The conclusion: Even if we allow ourselves a ridiculous overestimate of the amount of entropy reduction associated with biological evolution, the Sun supplies us with enough entropy to evolve all of life on Earth in just a couple of days (six days, if you like).