Trivia about time, to be precise. Prior to my current job, my knowledge of timekeeping was pretty much knowing how to read and adjust a clock, and because I’m a physicist, Einstein synchronization (basically accounting for propagation delay of light) and the effects of general and special relativity. All of the physics-related exercises with time conjure up a perfect clock, so you don’t have to worry about all the little details that arise when dealing with real-world hardware. Now, I don’t actually do time measurements, I “just” work on building clocks, but there are some things I’ve picked up.

A clock will have an oscillator in it, and some way of counting the oscillations. Time is the phase of these oscillations — one “tick” represents one cycle or some number of cycles. The derivative is the frequency, and if you take another derivative you get the rate of change of the frequency, which is the drift. Which sounds just like kinematics — the basic equation that describes all of this looks just like basic kinematics, as long as the rate of change of your frequency is a constant. And that brings up a point commonly fumbled by the popular press: leap seconds are often described as being added because the earth’s rotation is slowing. And while it’s true that over long times, the rate is slowing, that term could be zero and you’d still have to add leap seconds. The frequency represented by an earth that has slowed (but is no longer slowing) is different than that of atomic time, and so one will accumulate a phase difference (i.e. one will run slow compared to the other). That the rotation rate is slowing means that we will add leap seconds more often, assuming other effects on the rotation rate don’t mask this.

The above assumes “perfect” clocks. However, in all real processes that we measure, there is noise. Different kinds of noise, too, depending on the systems being measured. The best you can hope for is random, (i.e. white) noise, which gets averaged down as you take more measurements, and varies with the inverse square root of the number of data points (in this case, time). There are noise processes that average down faster, but eventually white frequency noise will dominate, and then the best case is that there are no other noise processes that dominate at longer times (like flicker or drift).

You integrate white frequency noise to get the effect on the phase, or time. The integral of this white noise gives you a random walk. That is, for any two real clocks, with exactly the same frequency, the best you can do is have them random-walk with respect to each other. They will never stay synchronized.