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One deleted. Let's remain respectful about each other's abilities and intelligence.

But that's what tides do! They stretch things! HEYOOOO!

I don't know that I have ever visited Physics Forums before. I don't see any ChatGPT stuff there, but I haven't looked very hard.

@ACuriousMind The one on Quora is definitely a "confident idiot" kind of response, saying some things that are correct and then using circular reasoning to get at the heart of the subject.

So I clicked the Quora link in this question and learned that Quora automatically generates a ChatGPT interaction prompt next to some or all questions. The ChatGPT response is currently useless, even though there is a correct human-written answer right next to it.

@Relativisticcucumber Well, that's no fun. I guess I should spend more time in here being friendly.

@user16217248-OnStrike That's actually very helpful. Thank you!

@user16217248-OnStrike It depends on the request and on some circumstances which may or may not be public. But raising the issue in a public forum, like chat, invites other users to participate in your drama. The general goal of site moderation is to have fewer site users involved in drama.

@user16217248-OnStrike One removed. Please use flags for this sort of thing.

But posts whose copy-paste includes the "regenerate response" prompt, and similar, are taken as an "admission" of chatbot usage.

@Slereah I don't know how much of the private moderator guidance I can share publicly.

If they had asked the moderator corps for more data before (or instead of) issuing orders to their volunteers, the whole shitstorm might have been avoided.

@ZeroTheHero The claim was a that (a) software GPT detectors have a pretty high error rate, that (b) the errors discriminate against users who aren't native writers of English, and that (c) the geography of GPT-related suspensions suggests those biases apply on the SE network. I think those are legitimate concerns. But the company essentially forbade most GPT-related removals and suspensions, based on their internal analysis, and doubled down when the moderators were unconvinced.

The removed featured might be a test of the "strike," since restoring that tag is a moderation action, which the moderators have said they won't engage in.

@JohnRennie I see smaller type next to shorter messages.

@JackRod I'm not terribly active in chat, but the people who are active more regularly also know about radio reflectors. I'll watch my notifications and try to reply eventually.

The answer is that they're in different parts of the phase space because they have different positions and momenta.

So I guess I'm still a little confused about your initial question:

Liouville's theorem basically says that the total probability is conserved during this process.

@imbAF I seem to have been begging the question. I think of a "macrostate" as a region of phase space whose shape and probability density don't evolve over time. Your linked video shows that, if you were to identify the microstate of such a system as a small pointlike region of the phase space, later on that probability would have smeared out over the region corresponding to the macrostate.

@imbAF I'll leave it on while I'm making dinner, and come back later.

@imbAF Perhaps you should send a link to the video you watched — we seem to be missing some context between us.

For a one-dimensional simple harmonic oscillator, the macrostate "has energy E" corresponds to a circle in (position, momentum) space. Macroscopically, all of the oscillators with energy E are indistinguishable, whether they happen to instantaneously be at their left turning point, their right turning point, or somewhere in the middle..

A thermodynamical macrostate has a well-defined thermodynamical observable, like the total energy. But if you look at the bottom of the first animation, you see that the potential energy surface is for a single particle, and the different parts of the phase space map to different total energies.

But in those animations, the initial regions are all square blobs which don't correspond to thermodynamical macrostates.

@imbAF You mentioned Liouville's theorem. That Wikipedia page has some animations of how regions of phase space evolve in some single-particle systems.

Like I said a few minutes ago: for an ideal gas, the particles' positions are changing continuously. The momenta in an ideal gas only change when two particles' positions get too close to each other — a collision.

if you make "nice" assumptions about units so I don't have to write p^2/2m

The momentum surface is p_1^2 + p_2^2 + p_3^2 + ... = E

I would say this macrostate corresponds to a surface in momentum space, and the entire volume of position space.

Suppose the macrostate is "this gas has total energy E"

I think those are different things.

I don’t understand “the phase space probability is a macro state.” Let me think out loud a moment.

That’s what I meant, too.

Okay. Poof! In a gas, every particle has a position and a momentum, which we can label as a point in phase space. But the positions evolve continuously, while the momenta only change when two gas atoms collide.

Let’s pick one model. Gases need a different toolset from solids. Should we switch?

How is an oscillator state a “point” in that phase space? You have to transform to some (amplitude, phase offset) coordinates for an oscillation to be a “point”?

What are the dimensions in your phase space?

Maybe? “Phase space” means different things under different circumstances.

Oh, you asked a question after I wrote “sure”

Sure. Consider that the balls-in-buckets model is isomorphic to the Einstein solid. If all of the balls are in the first ten buckets, then that side of the solid contains more heat than the other side. Probabilistically, you expect the random fluctuations to end up in one of the innumerable more states where the heat is evenly distributed.

The “fundamental assumption of thermodynamics” is that, in equilibrium, all of the indistinguishable microstates are equally likely.

Maybe you have a anisotropic crystal, so the heat spreads out more rapidly along some directions than along others.

You can handwave “this atom jiggles its neighbors,” so maybe the excitation energy is mechanically unlikely to stay put.

@JohnDoty That seems like an answer, rather than a comment.

Some minor comments (v10). (1) The intermix of ALL CAPS, bold, italic, (parentheses), and [brackets], combined with unconventional capitalization and unconventional spacing around punctuation, make this question unnecessarily hard to read. As a rule, it's better to edit text until it is readable with a minimum of decoration. (2) If you have found a possibly-relevant related question, please link to it in addition to describing it. If the explanation at a related question is confusing, elaborate about which parts make sense and which don't, so that we can focus our answers.

Search term: geoid. The surface of a body of water makes a better "level" than a rolling ball.

I've removed a bunch of comments, some of which bordered on being inappropriate. @Mauricio, please consider converting your comment to an answer. Further comments here should serve to improve of clarify the question.