A few months ago I was reading Physics Today on the weekend, and in a historical article about Tatiana Ehrenfest-Afanassjewa, a passing reference was made to Paul Ehrenhfest’s “Dog-flea” model, which illustrates entropy by proposing two dogs lying side-by-side; one initially covered in very many fleas, and one without. Even if the fleas are as likely to jump from the first to the second dog, as from the second to the first, they will eventually end up covered with the same number of fleas. Well, almost certainly, almost exactly the same the number of fleas.
But this works even if both dogs start covered with fleas. Imagine numbering the fleas — the first dog with even numbers 2, 4, 6, 8 and so on; the second with odd numbers 1, 3, 5, 7, 9, etc. After some suitably large multiple of flea-hopping timescales has elapsed, both dogs would have an unremarkable and comparable mixture of even and odd numbered fleas. Even though to someone unaware of the tiny flea numbers nothing has changed, entropy has still increased.
Of course, any other initial numbering scheme would work equally well. The dog-flea arrangement starts off in some specific state, and gradually — by the mechanism of random flea-hopping — evolves into something with fewer and fewer traces of the starting point. Disorder has increased, even if a simple flea count suggests that nothing has changed.
Knocking over a glass, and turning the ordered arrangement of its specific shape into a pile of scattered broken pieces is another common example. But imagine those “disordered” shards swept into a box, and then imagine that you carefully document the sizes and shapes of the shards and their positions and orientations. You now have a new and very specific initial state, and were you to claim — with your “modern artist” hat on —- that this box of broken glass was a significant new artwork, one could claim that it is highly ordered. In fact, your new artwork is — in its own context — now the ordered state, and any substantial alteration, even a reconstruction into the original intact glass — however achieved — would, I argue, have to be counted as part of the disordered collection (because it is not the chosen “ordered” artwork state) … although there are many other much more likely “disordered” states.
As it happens, you may have to fight your cultural biases to see things this way. But we can choose the states we want to call “ordered”, and that choice does not have to conform to notions of good housekeeping.
Next week: something about maps