@bolbteppa I was able to derive it, but thx for your reply, regardless

tho I have one question

No worries

Assuming we do this type of transformation $$\Sigma_{\vec p} \rightarrow V\int \frac{d^3p}{(2\pi\hbar)^3}\rightarrow V\int \nu(E)dE$$

And we have $\nu(E)=\int \frac{\delta(\epsilon-\epsilon_p)}{(2\pi\hbar)^3}d^3p$, density of states

I see. the following ambiguity here

if you consider both integrals

in order to eliminate the $\int dE$ you have a specific, finite value of the energy

while for $\int d^3 p$ , since we have in the denominator $2 \pi '\hbar$

the boundaries for p are $-\infty$ and $+\infty$

isn't this odd?

While $\espilon_{\vec p}$ has a finite value and it's related to $\vec p $ as $\espilon_{\vec p} = \frac {\vec p^2}{2m}$

p can have values that range from - to + infinity

or am I misunderstanding something here?

but you didn't had to remove it TT

The denominator is irrelevant to what you said, and it's not clear what the problem is

I remember that when I was considering a single particle in 3D, and wanted to find the canonical partition function of it

I had the same denominator, in the same power

If I had more particles, then I had $\int d^3p_1 d^3p_2...d^3p_{3N}$

for N,3D particles

that's why I assumed, this is what we were considering

is there a physical interpretation of the unitarity bound in CFTs, i.e. why the operators always have positive anomalous dimension?

@NiharKarve it's just "causality" - the unitarity bounds are equivalent to Euclidean reflection positivity, and reflection positivity is the OS axiom corresponding to Lorentzian causality

another viewpoint is just to say it means your theory is unitary and has positive energies, cf. physics.stackexchange.com/a/478127/50583

sorry, I meant in particular an interpretation of the anomalous dimensions being positive

I understand the derivation based on reflection positivity

I'm not sure what the "physical interpretation" of an anomalous dimension is to begin with, no matter any bounds on it :P

A physical system is said to be ergodic if any representative point of the system eventually comes to visit the entire volume of the system. The representative point of the system is an initial microstate. Keeping in mind that the change of the system is governed by the hamiltonian mechanics, so along the trajectory the energy is the same. Doesn't this imply that all across the phase space, the energy is the same, which in general is not true?

@imbAF stafusa already answered that when you asked this question on the main site 2 days ago

why do you just ask the same question here without any indication that you already got an answer that you apparently didn't understand?

if you want people to help you, you need to be specific in what you actually want, not ask generic questions when you actually have something much more specific in mind

implicit/explicit

always confuses me

implicit, he means, that it's assumed that we consider the accessible volume right?

but we say, entire volume for some reason?

If I understand things correctly, I think you can fix the entire metric structure of the spacetime by just considering the 2-frame G-structure

That will fix the metric components within the appropriate coordinates

The Polyakov soldering

@imbAF sure, the entire accessible volume (i.e. states with the same energy in this case) as opposed to the system just moving in a tiny corner of the space of states with the same energy, or even just sitting still in a single stationary state

Ok

there is apparently some equivalence between the metric components/choice of coordinates (up to 2-germs)/BRST operators/2-frame G structure

one more thing @ACuriousMind

How, I don't know how to formulate this question correctly, how do the boundaries of the volume, came to be?

This volume, is it a result of the trajectories?

what volume? what boundary?

entire accessible volume

that "volume" is just the region of the space of states with the same energies

it should have boundaries?

Or it doesn't make sense to speak about that?

I don't understand what you mean by it "having" boundaries

it's a subset of a topological space, so we certainly can speak of its boundary in that sense, but I don't understand what the question about that is

two examples, and you can tell me if they represent the same thing

1 example is the MCE, in which, if I am not being wrong, the hyper-surface is the region, where $\rho$ is non zero and constant. This is a region/volume with dimensions in phase space (Again I might be badly expressing myself). 2. case of the classic harmonic oscillator, in phase space we have an elliptic trajectory, which I assume is something similar to example 1, but I am not sure

I gate these two examples, regarding our discussion about this "volume" , "boundary"

those are two examples of the volumes we're talking about, yes

I don't see any question about boundaries :P

well

it's a volume

as a result of the fact that $\rho$ is non zero there

But in the case of the 1D harmonic oscillator

$\rho$ hasn't got anything to do with it "being a volume"

But if you ask yourself, where $\rho$ is non zero

doesn't that indirectly define a volume in phase space?

@imbAF why are you asking yourself that?

ergodicity isn't about any $\rho$s

it's just a statement about where a single point in phase space with energy $E$ will move given enough time

in the entire region, which contains all the microstates with roughly the same energy

if the system is ergodic, then given infinite time it will visit every other point with energy $E$ with equal probability

that's all

there are no $\rho$s, no MCEs or whatever here

aha

so we don't need to specify whether we are in a MCE,CE,GCE

ok I didn't know that

And it wasn't pointed out anywhere

I find it a bit strange that you have issues with the "implicit" assumption of the volume having to be accessible and then you implicitly assume we're talking about ensembles when nothing in the definition of ergodicity mentions ensembles :P

so I thought

because it's statistics and phase space and all that

the type of ensemble should be accounted for

One other thing, when a system is not in equilibrium, is the state of non-equilibrium chararterized by a set of microstates?

or it's stupid to think about this?

I don't know what you mean by " chararterized by a set of microstates"

a non-equilibrium state can just be any $\rho$ you want

I am returning in 15, so I will explain myself

But thanks for what you explained until now

I was going to ask you about the microstates and stuff for the 1d harmonic oscillator and it's elliptic trajectory in phase space, but from what you said I was able to deduce this particular case

@ACuriousMind I want to say beforehand, that in our lecture the only case we considered when we were introduced to phase space,was that of MCE. Which why I always have this as my starting point and then try to generalize it and that is hard sometimes. That's why for me things should be explained in general terms, because then you can trickle down a specific case.

With this in mind, when I said " is a non-eq. macrostate characterized by a set of microstates", in equivalence to the MCE case, where the macrostate of eq. is chracterized by a certain multiplicity, by a set of microstates with the same energy.

we've already been over this some time ago - a generic non-equilibrium macrostate is just any probability density $\rho$, not more, not less

Yes

it doesn't have to be constant, it doesn't have to care about energies, there's literally no restriction on this function other than its integral over all of phase space being 1

but my questions is not about this

ok

The physical process of a system going from a non eq. state to an eq. state, can it be showcased in phase space?

that would have to be some sort of non-equilibrium dynamics, which you really shouldn't think about if you're already struggling to comprehend the equilibrium case

I understand the eq. case

but I was thinking of this non-eq dynamic case

But if you say that I shouldn't care, ok

the problem is that how you reach equilibrium is very specific to the actual system under consideration

Ofc,it's a case by case scenario

e.g. an ideal gas cannot start in non-equilibrium and then reach equilibrium - it doesn't have any interactions that could mediate that, but a real world gas clearly can do that

Yes

how the real world gas does that depends on how you model it, different from the ideal gas

and the mechanism by which e.g. a strip of metal that's heated at one end reaches thermal equilibrium is completely different, and understanding one of these cases doesn't give you lot for the other

Basically

I have to be more specific about the case

at hand

in other words

and how these systems reached equilibrium is completely irrelevant to their behaviour in equilibrium, which is what you're currently studying with canonical ensembles etc.

In case of CE in equilibrium

the microstates which belong here, do not have the same energy

right?

I don't understand the question

@ACuriousMind You said this. And you said states with the same energy

Oh, I don't know how to tag an existing comment xD

that was when we were discussing ergodicity and the motion of a single particle/point

I thought we had made clear ensembles played no role there

so I don't see how it is relevant at all to any question that starts with "In case of CE"

IDK what to say other then statistical mechanics is stupid

We never considered, or talked about the phase space for CE

only for MCE

so me trying to adjust something specific, to something more general

while having caveman understanding of it

it's not that easy

this, when you ask me, what do you mean with this

is this chat only for asking physics related questions?

@UmeshKonduru no, it's a general chat for physics.SE - you can talk about whatever you like, but naturally the topics here tend to be physics

I wanted to talk to physics students about their experience at college

I'm in high school right now, I'll be going to college this year

ask away - but keep in mind that experiences will vary wildly with where people go or went to college

sure, I'll keep that in mind

I'd like to hear about what y'all think about the differences between high school and undergrad physics

And anyone here involved in research? I'd love to talk about that too - I'm really curious about how it goes

"undergrad physics" can include a lot - I took a lot of math and theoretical physics and the mathematical approach is often very different from the way physics is taught in school, but I think generally the difference is less pronounced than the difference between math in school and actual mathematics

would it be possible for an engineering graduate to take up physics later?

how easy depends on your university, but sure - we've had several people who started with engineering here over the years

You can see the terminator between daylight and night is faster than the shockwaves.

Where does this relation comes from $\Omega= - PV$ where $\Omega$ is the grand canonical potential