@ACuriousMind Sorry I had to go earlier when we were discussing about the conditions under which the Liouville Theorem holds true or is relevant

Is it correct to say that, the theorem is relevant only for isolated systems? Which are the only systems with conserved energy?

Liouville's theorem is very symple(ctic)

@imbAF I think that certainly: it only holds for closed systems where the total energy is conserved.

Ok so

in other words

we concern ourselves with the Microcanonical ensemble

No idea what that is :)

Me and statistical mechanics aren't friends lol

Can we say for an eigenvector of the hamilton operator H or of the angular momentum $\vec J$, that it is an eigenstate of the system?

Eigenvectors of the Hamiltonian are energy eigenstates

Angular momentum eigenstates are defined similarly...

yes

I know

but can you say

Eigenstate of a system?

the eigenstate is something that characterizes the corresponding operator

Eigenstate is always related to at least one specific operator (can be more than one)

But I don't think it is a property of the entire system, perhaps you can say something else if you want to give such a state a more global characterization

I'm having deja vu

But you do say

@ACuriousMind Ok I forgot

@SillyGoose Yes. Their combined action is the action of an "actual" rotation

But we do say, the system occupies a state $\Psi$

what do you mean you forgot, we've had this conversation at least twice already

I'm not having it a third time for you to forget it again :P

It's actually hard man, to articulate

specially when it's said in class

and I am dubious

@ACuriousMind One more question regarding the Gibbs paradox. Is it correct to say that by considering the indistinguishability of the particles, entropy becomes an extensive quantity? (because we add that 1/N! factor)

The paradox happens, from what I understand, because Gibbs entropy (or whatever he was using to measure it) wasn't an extensive quantity

I think that every eigenstate is a pure state, but not the other way around, correct me anyone if I'm wrong

I don't really feel qualified to discuss the Gibbs paradox

classical statistical mechanics is something I somewhat understand but am not really comfortable with

Yes that's what fixes things

like, Hamiltonian mechanics and phase space densities and all that make sense to me but I don't think e.g. my understanding of entropy matches up with how it's usually taught

@Amit every eigenstate of what

any eigenstate is a pure state because the definition of "eigenstate" is just that it's some vector with special properties w.r.t. some observable

and a "pure state" is just a vector

mixed states are density matrices that don't correspond to single vectors

@imbAF everything in stat mech is basically a closed isolated subsystem or a(n open/closed) subsystem of this, including Liouville

@ACuriousMind Oh, okay, so probably even the other direction is correct to some extent, e.g. there exists some observable for which a pure state is an eigenstate... but never mind really, I only wanted to supply imbAF with a "global" characterization of an eigenstate (since pure does relate to the system, as opposed to an eigenstate which relates to observable/s).

"..or a(n open/closed) subsystem of this" of which?

(there's another technical issue here where you should probably think of Hilbert spaces at different temperatures as being different Hilbert spaces in the $C^\ast$-algebra formalism but let's not get into that as I don't really understand that either)

of this closed system*

@Amit every vector $\lvert \psi\rangle$ is trivially the eigenstate of the projector onto it, $\lvert \psi\rangle\langle \psi\rvert$

hah, right :)

It's just that I myself got confused about this pure vs. eigen business once, I was only trying to reconstruct the resolution of the confusion ('cause I suspect that's what makes people tie eigenstates to "system" wrongly...)

a pure state can be expressed as a linear combination of eigenstates of some observable

And the pure state, is the state of the system

No stat mech system is in a pure state, it's all density matrices, but density matrices are constructed in terms of 'pure states', so where the hell are the pure states coming from, e.g. where are the pure states in the Boltzmann distribution (and how does this relate to the Gibbs paradox)

@bolbteppa than, that would mean, that the Liouville theorem is always valid, since energy is conserved, and the hamiltian mechanics are vaild

valid*

Yes, in classical stat mech, Hamiltonian mechanics is always valid, but so what, we can't solve $\approx 10^{23}$ equations and impose the initial conditions, so we need an alternative which is stat mech

I don't really get why you'd worry about Liouville's theorem being valid

it's just a straightforward consequence of Hamiltonian mechanics

But see this is the problem

we never

got the liouville theorem as a consequence of the Hamiltonian mechanics

never

yuh I think that's right. This stuff only matters for terminology :) It's all self apparent once you get working with the math machinery

we took a thermodynamical system, considered it's phase space, and said what we said, which is the liouvill theorem

that's it

But as I told you

Liouville's theorem is just $\partial_t \rho = \{ H, \rho\}$

that's essentially just how all functions on phase space evolve in Hamiltonian mechanics

I understand that many courses don't teach it that way because for inscrutable reasons they never actually bother teaching proper Hamiltonian mechanics before using it, but that's just how it is

Oddly enough, we also didn't cover it in Analytical mechanics, first time I saw it was in a Susskind lecture :) and he is terrifically rigorous as we all know lol. But it was still great, totally understandable (probably even rigorously it's quite simple)

everything we did, in statistical mechanics, revolved around an isolated system, to which we associated the MCE. And since the Liouville theorem came up, when we were discussing isolated systems to which we associate the MCE, I thought that the Liouville theorem is ONLY valid or relevant for systems with conserved energy, and wouldn't be a thing for system where energy is not conserved

oddly enough, a non-quantum example for "actually understanding Hamiltonian mechanics makes the rest much easier"

"...all functions on phase space evolve in Hamiltonian mechanics"

which functions evolve in phase space?

All i know is that, trajectories in phase space

represent the time evolution of microstates in reality

Enough with the oddly? :) lol

Liouville is basically the basis for statistical mechanics, it says the distribution function for an isolated system only depends on conserved quantities like energy, momentum, particle number etc... you can then study subsystems of this which interact with one another and frame things in terms of the distribution function (i.e. apply it to open subsystems of a larger closed system)

Yes I understand this

If you ignore the larger closed subsystem, you have to get into axiomatics and hypotheses etc...

observation scale change

I talk about Liouville's theorem and why I think it's not really a theorem if you build Hamiltonian mechanics properly in this answer and the general notion of evolving functions on phase space along flow in this answer

@ACuriousMind thanks for these links

I hope you understand where I am coming from

Hamiltonian and Liouville are of no relation to me

because it's not how I learned about it

Always remain with a pure state of heart, no matter what observable you bump into

I do understand that the trajectory of a point in phase space, satisfices the Hamiltonian mechanics but that's about it

If Liouville is too confusing, just frame it in terms of the Schrodinger equation for a density matrix i.e. the von Neumann equation

My custom game server has voxelized regions within my terrain. When inside a region, my characters are getting stuck on things often. So I wanted to implement something like what Unity calls "skin width" where your collider can partially penetrate the obstruction. From their description, I can't figure out exactly how to implement it. (FYI: the reason the server is doing this is my game is extremely server authoritative, simulating what the client is doing)

The Unity client will slide along edges with ease with its character controller. I want my server's simulation to do the same, only given 0.1m x 0.1m voxels that say if a location is passable or not.

that true rotations are a composition of "spatial" and "spin" rotations clears everything up for me :D @ACuriousMind

If I have a hilbert space $\mathcal{H} \cong \mathcal{C}^{\otimes n}$, then every operator over this space is of the form $O^{\otimes n}$?

or i guess a sum of operators of that form?