The Second Law of Thermodynamics is one of the few scientific laws that has attained a status in wider culture, even featuring in rock tracks by Muse. Famously, C.P. Snow cited an understanding of the 2nd Law as something that every educated person should have.
The 2nd Law is often stated in technical language that makes its meaning hard to understand, but the basic principles are actually readily grasped. I was recently challenged to explain the 2nd Law at the level of a bright 13-year-old, and so here is my attempt.
The world is pretty complicated, so let’s make a simple “toy” version of it. This will still tell us the essential point about the 2nd law of thermodynamics. Take a box containing 100 coins. The coins are either heads-up or tails-up and there is enough room in the box to shake them so that they turn over.
So shake the box. About half the coins will be heads and half tails. There are lots and lots of possible states that have about half heads and about half tails. Shake the box again and you get another such state. And again; you get another one.
It doesn’t matter which particular coins are heads, and which particular coins are tails, which means that there are lots of possible ways of arranging each individual coin such that roughly half are heads and roughly half tails. There is nothing special about such an outcome, and so we call the overall state a disordered state.
Now consider the state where every coin is a heads. This is a special state because it requires every individual coin to be in a particular state — namely heads. There is only one possible way of arranging the coins such that all are heads. Thus we call this a highly ordered state.
Suppose we start in the all-heads state and we shake the box hard. What’s going to happen? Well, since there there is only one state with all heads (or, indeed, all tails) the chance of it landing back in that state are very low.
But, since there are lots and lots of states in which roughly half the coins are heads and half tails, it is overwhelmingly probable that you’ll end up with a nearly half-and-half state. In other words the system has moved from an ordered state to a disordered one. The more lop-sided an outcome is (heads outnumbering tails or vice versa) the fewer ways there are of producing that outcome, and so the less likely it is.
What happens if you’re in a disordered state to start? Well, the chances of moving to a highly ordered state are very low, just because there are few such states. Chances are you’ll land back in a disordered state, because there are lots of configurations that are disordered.
And that’s the second law of thermodynamics. It says that it is easy and likely for an ordered state to move naturally to a disordered state; but it’s highly unlikely for a disordered state to move spontaneously to an ordered one (that would take an intervention from outside, such as a kid coming along and turning over each coin as appropriate).
That’s the end of the explanation, and so far I’ve used no maths and no jargon. Depending on the kid, though, one could then introduce some jargon.
The “macrostate” is the overall ratio of heads to tails. The “microstate” is the particular configuration of every coin (imagine them numbered 1 to 100, and then the microstate is the list HTTHHTH … et cetera).
“Entropy” is then just a fancy word for how ordered the state is. Formally, it is a count of the number of microstates that would give the same macrostate (thus for the all-heads macrostate there is only one possible microstate, whereas for the 50:50 macrostate there are lots of microstates). A high-entropy macrostate has lots of possible microstates; a low-entropy macrostate has few possible microstates.
Saying “entropy always increases” is then just saying what I said above, namely that, since ordered macrostates have very few microstates, whereas disordered macrostates have lots, there is a natural tendency for systems to move from ordered states to disordered states, and it is very unlikely for a system to move spontaneously from a disordered state to an ordered state.
Note that, in the above reasoning, we have made the implicit assumption that each microstate is equally probable. This is usually a pretty good assumption.
The mathematics of counting microstates is accessible to a maths-able teenager. Start with four coins. There is one microstate that gives all-heads, four than give three heads and one tail, and six that give two heads and two tails (these are: HHTT, HTHT, HTTH, THTH, THHT & TTHH). The number of microstates for each of the macrostates (0, 1, 2, 3 and 4 heads) is thus 1, 4, 6, 4, 1 respectively. Thus two-of-each is six times more probable than all-heads.
Note that the 2nd law is probabilistic, not absolute — there is a tiny chance that a shake of the coins would indeed give all-heads, but that chance decreases as the number of coins increases. With four coins it would be one-in-24, and with 100 coins it would be one-in-2100.
In the real world we’re talking about molecules, which are tiny, and so there are about 1023 molecules (Avogadro’s number) in each teaspoon of stuff. Thus the chances of significant large-scale violations of the 2nd law effectively vanish to zero.
Nothing above says that a system cannot move from a disordered to an ordered state. There are lots of possible ways in which it can (the above possibility of a boy coming along and turning the coins over to attain a desired result is an example), but all of those possibilities require an input of energy into the system.
Thus they can occur in one part of a system, if that part receives energy from elsewhere in the system, but they cannot occur in the system overall if it is isolated (that is, not gaining energy). To quote Muse: “In an isolated system, entropy can only increase”.
(And by the way, if you’re wondering about the tiresome creationist claim that the evolution of complex and highly ordered life on Earth violates the 2nd law, no it doesn’t, since the Earth is not isolated, it gets a large energy input from the Sun.)