1  Setting the Alarm Clock – A Story about Symmetry and Information

Once upon a time, there was no such thing as GPS and no such thing as the internet. There existed purely mechanical clocks and watches, which were not particularly accurate, so they needed to be set fairly often. Starting in the 1930s, in many places there was a phone number you could call to obtain the current time.

Back when I was in first or second grade, my father got a fancy new clock-radio for his birthday. He gave me his old mechanical wind-up alarm clock. He went into great detail explaining how to use it:

• Wind this to energize the time-keeping mechanism.
• Wind this to energize the ringer for the alarm.
• Turn knob only one way, to set the time.
• Turn knob only the other way, to set the alarm.

That evening I set the alarm-hand for the appointed time and went off to sleep. I woke up the next morning before the appointed time, but just for fun I waited in bed, waiting for the alarm to go off....

It didn’t.

At dinner the next evening I mentioned that the clock hadn’t done what I’d expected. My father asked

"Did you set the clock to the right time?"

I said

"No... I didn’t want to look at the clock, I just wanted the alarm to go off on time."

My father was about to say something, but before he even got started I solved the mystery and blurted out the answer:

"Oh... I guess you have to set the face-hands; otherwise it won’t know what time it is. If the alarm worked all by itself, you could just wait for the alarm to go off and set the face-hands from that, and you would never need to call time-and-temperature."

Figure 1: Wind-Up Alarm Clock

At the time, I really liked this line of argument. I thought it was powerful and beautiful. I still like it. The most amazing part is that it is independent of mechanism. The clock could have been made of springs and gears, or made of hydraulics, or whatever – the mechanism didn’t matter. There was no way the clock could, by itself, figure out what time it was.

In grown-up terms, I can explain it in more sophisticated terms, but the idea is the same:

• As long as the clock is an isolated system, it will exhibit time-shift invariance. It’s a fundamental symmetry of the universe.

  This time-shift invariance is related to conservation of energy, which is one of the most profound and useful principles of physics. Indeed, every continuous symmetry corresponds to a conservation law. And vice versa. See reference 1.

  
  Figure 2: Emmy Noether
• Information, which at first seems about as abstract as anything can be, has a hard core of reality. In an isolated system, if you don’t have information about what time it is, you simply cannot manufacture that information.

  This limitation on the flow of information is related to the second law of thermodynamics, since entropy is basically just the opposite of information. This is another very profound and useful law of physics.

Both of these laws (conservation of energy and paraconservation of entropy) are independent of mechanism.

Sometimes you can use a mechanism-based argument to obtain a thermodynamic result, but the result is usually more general than the argument used to obtain it. For example, you might use an argument about rates to discover something about chemical equilibrium. However, the result is more general than the argument used to obtain it, because catalysis will change the rates by orders of magnitude, without changing the equilibrium point at all. As another example, Einstein’s analysis of the Brownian motion of a pollen grain led him to an equation that relates drift to diffusion. The equation applies to a wide range of situations – including the electrons in transistors – not just pollen grains in water.

This stuff is like magic, only better, because it’s real.

2  References

1.

Wikipedia article, “Noether’s Theorem”
https://en.wikipedia.org/wiki/Noether’s_theorem