A Note on Entropy and Time

I just came to a very strange realization:

The amount of time you measure, depends upon your scale of observation.

To understand why this is the case, imagine you spill ink into a single sheet of paper. Now divide the paper into a grid, of equally sized boxes, and count the number of ink molecules in each box over time as the ink stain spreads.

Now imagine the ink stain is your clock.

This is realistic, in that it’ll take time for the stain to spread, and so you could measure time by measuring the distribution of the ink among the boxes over time. Said otherwise, as the ink spreads, the amount of time that passes increases. But we can formalize this by measuring the entropy of the ink stain in the grid. Specifically, the ink stain begins in exactly one box, producing an entropy of zero, since it is the entropy of a distribution over a single event, with a probability of one.

Eventually, the ink should be roughly uniformly distributed among the boxes, producing the maximum possible entropy. As a result, the entropy of the distribution of the ink stain is a measure of time. However, what’s truly bizarre about this, is that the amount of time you measure in this model depends upon the dimensions of the grid.

Just imagine a grid with one box –

That’s a clock that never moves, since the ink is always on the page, and therefore, the entropy is always zero. This is actually profound, and shows that if you want to truly distinguish between moments in time, you need a system that can be in a large number of states. For example, a normal clock repeats every 24 hours, but it’s supplemented by a day, month, and year, to distinguish it from other positions in time. More formally, a calendar is a system that has an infinite number of states. And you can’t truly tell time without such a system, otherwise you’ll end up labelling two different moments with the same name.

That said, calendars are only conceptually infinite, but are practically finite, because human beings live finite lives, and even the human species itself has been around for only some finite amount of time. It follows that unless the Universe itself is capable of being in an infinite number of states, it will eventually repeat the same state. Said otherwise, the state of the Universe as a whole is a clock, and if it’s only capable of being in some finite number of states, then by definition, eventually, if time itself is infinite, the same exact state of the Universe must happen twice.

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