leonard_susskind's picture
Felix Bloch Professor in Theoretical Physics, Stanford; Author, The Theoretical Minimum series

Boltzmann's Explanation Of The Second Law Of Thermodynamics

"What is your favorite, deep, elegant, or beautiful explanation?" That's a tough question for a theoretical physicist; theoretical physics is all about deep, elegant, beautiful explanations; and there are just so many to choose from.

Personally my favorites are explanations that that get a lot for a little. In physics that means a simple equation or a very general principle. I have to admit though, that no equation or principle appeals to me more than Darwinian evolution, with the selfish gene mechanism thrown in. To me it has what the best physics explanations have: a kind of mathematical inevitability. But there are many people who can explain evolution much better than I, so I will stick to what I know best.

The guiding star for me, as a physicist, has always been Boltzmann's explanation of second law of thermodynamics: the law that says that entropy never decreases. To the physicists of the late 19th century this was a very serious paradox. Nature is full of irreversible phenomena: things that easily happen but could not possibly happen in reverse order. However, the fundamental laws of physics are completely reversible: any solution of Newton's equations can be run backward and it is still a solution. So if entropy can increase, the laws of physics say it must be able to decrease. But experience says otherwise. For example, if you watch a movie of a nuclear explosion in reverse, you know very well that it's fake. As a rule, things go one way and not the other. Entropy increases.

What Boltzmann realized is that the second law—entropy never decreases—is not a law in the same sense as Newton's law of gravity, or Faraday's law of induction. It's a probabilistic law that has the same status as the following obvious claim; if you flip a coin a million times you will not get a million heads. It simply won't happen. But is it possible? Yes, it is; it violates no law of physics. Is it likely? Not at all. Boltzmann's formulation of the second law was very similar. Instead of saying entropy does not decrease, he said entropy probably doesn't decrease. But if you wait around long enough in a closed environment, you will eventually see entropy decrease: by accident, particles and dust will come together and form a perfectly assembled bomb. How long? According to Boltzmann's principles the answer is the exponential of the entropy created when the bomb explodes. That is a very long time, a lot longer than the time to flip a million heads in a row.

I'll give you a simple example to see how it is possible for things to be more probable one way than the other, despite both being possible. Imagine a high hill that comes to a narrow point—a needle point—at the top. Now imagine a bowling ball balanced at the top of the hill. A tiny breeze comes along. The ball rolls off the hill and you catch it at the bottom. Next, run it in reverse: the ball leaves your hand, rolls up the hill, and with infinite finesse, comes to the top—and stops! Is it possible? It is. Is it likely? It is not. You would have to have almost perfect precision to get the ball to the top, let alone to have it stop dead-balanced. The same is true with the bomb. If you could reverse every atom and particle with sufficient accuracy, you could make the explosion products reassemble themselves. But a tiny inaccuracy in the motion of just one single particle, and all you would get is more junk.

Here's another example: drop a bit of black ink into a tub of water. The ink spreads out and eventually makes the water grey. Will a tub of grey water ever clear up and produce a small drop of ink? Not impossible, but very unlikely.

Boltzmann was the first to understand the statistical foundation for the second law, but he was also the first to understand the inadequacy of his own formulation. Suppose that you came upon a tub that had been filled a zillion years ago, and had not been disturbed since. You notice the odd fact that it contains a somewhat localized cloud of ink. The first thing you might ask is what will happen next. The answer is that the ink will almost certainly spread out more. But by the same token, if you ask what most likely took place a moment before, the answer would be the same: it was probably more spread out a moment ago than it is now. The most likely explanations would be that the ink-blob is just a momentary fluctuation.

Actually I don't think you would come to that conclusion at all. A much more reasonable explanation is that for reasons unknown, the tub started not-so-long-ago with a concentrated drop of ink, which then spread. Understanding why ink and water go one way becomes a problem of "initial conditions". What set up the concentration of ink in the first place?

The water and ink is an analogy for the question of why entropy increases. It increases because it is most likely that it will increase. But the equations say that it is also most likely that it increases toward the past. To understand why we have this sense of direction, one must ask the same question that Boltzmann did: Why was the entropy very small at the beginning? What created the universe in such a special low-entropy way? That's a cosmological question that we are still very uncertain about.

I began telling you what my favorite explanation is, and I ended up telling you what my favorite unsolved problem is. I apologize for not following the instructions. But that's the way of all good explanations. The better they are, the more questions they raise.