Today was the first week in months that the main trail in Chadwick Lake (Newbugh, NY) wasn't caked in snow, which gave me (and mom) a chance to look at trees instead of our feet. It was also windy, and trees were visibly un-rooted, which led me to consider two things: (1) my safety and (2) the chances of a tree falling by natural causes.

So when I got home I asked ChatGPT to explain the likelihood of a tree falling like Richard Feynman would explain it to his grown but ordinary child. It eventually offered to explain the percolation threhold and how it's used in model clusters of trees falling in a forest.

Apparently trees fall in forests randomly until clusters form, and then eventually the clusters become "connected" once it passes the percolation threshold. I'm further told that when a tree falls (and nobody's there to hear it), neighboring trees are more likely to fall due to reduced canopy (i.e. leaf coverage) and increased localized wind velocities cranking at Treebeard's roots.

I don't fully get it and I'm okay with that, but the main take-away for me was that it sounded similar to the phase transfers I learned about in intro physics (e.g. from liquid to gas). I asked "dad" if that was a valid comparison and that's when I learned about phase changes of different order. Something that changes from a liquid to gas is an example of a first order phase transition, while percolation threshold relates to phenomena described as a second order phase transition. The order refers to which derivative of some representative function is discontinuous, and i guess more importantly what that derivative represents physically. So if you know a function that relates to the energy transfer involved in the liquid to gas phase change, and you observe in real life that there's discontinuous jumps in the physical measurement that's represented by its first derivative (e.g. volume) with respect to energy (e.g. heat), then this phenomena can be labeled a first order phase transition. So while liquid-to-gas transitions involve discrete jumps in measurements like entropy, percolation involves a continuous change in entropy, again with respect to energy.

Cool makes sense-ish but the real take away, for real this time, was that nature tends towards states of high entropy. That's what the gpt chats said and that didn't make sense, since entropy from intro physics if I recall was analogous to "disorder." I asked it to explain and I got a more mature understanding of entropy, which is that entropy can be described, roughly, as the number of distinct micro-states that can represent a single macro-state. The molecules of a liquid in a static container generally hold the same positions within that container, so the the number of unique micro-states is lower, ergo low entropy.

So the rationale is that nature tends toward states of high entropy because high entropy states are more numerous, and thus more probable. From this perspective nature doesn't actually tend toward things, rather what we observe (which are phenomena with higher probabilities) is what we happen to consider "natural." I've always heard math and physics are intertwined but this might be the first time I see how the abstractions of math can connect with natural observations.

Michael Scott described Fall as being the most contemplative of seasons. That's indubitably due to the foliage; trees are majestic and spark the curiosites.