Hacker News new | ask | show | jobs
by dynm 426 days ago
It's tricky when you think of a continuous system because the "differential entropy" is different (and more subtle) than the "entropy". Even if a system is time-reversible, the "measure" of a set of states can change.

For example: Say I'm at some distance from you, between 0 and 1 km (all equiprobable). Now I switch to being 10x as far away. This is time-reversible, but because the volume of the set of states changed, the differential entropy changes. This is the kind of thing that happens in time-reversible continuous systems that can't happen in time-reversible discrete systems.
2 comments
im3w1l 426 days ago
Isn't that kind of what we want entropy to capture though? If a particle darts off into the distance then in theory it might be time reversible, but in practice it's not so simple. If the particle escapes the gravitational pull, the only way it can come back is if it bumps into some other object and pushes that object away. So things will inevitably spread out more and more creating an arrow of time.

This can then be related to the big bang, and maybe it could be said that we are all living of the negentropy from that event and the subsequent expansion.

Getting different entropy values based on choice of units is a very nasty property though. It kinda hints that there is one canonical correct unit (plank length?)

link
jdhwosnhw 426 days ago
I have yet to see differential entropy used successfully (beyond its explicitly constructed-for purpose for calculating channel capacity). Similar to your thought experiment is the issue that the differential entropy value depends on your choice of unit system. Fundamentally the issue is that you cant stick a quantity with units into a transcendental function and get meaningful results out
link
dynm 426 days ago
Yeah, it's quite disturbing that the differential entropy (unlike the discrete entropy) depends on the units. Even worse, the differential entropy can be negative!

Interestingly, the differential KL-divergence (differential cross-entropy - differential entropy) doesn't seem to have any of these problems.

link