@Bml Those values are exactly what you get if you divide the interval $[T_i,T_f]$ into $N$ equally large subintervals.

@Bml Look, that is just many applications of $y=mx+c$ in disguise. Can you not see that this thermal chain is just doing linear interpolation of the temperatures at equal steps in temperature?

that's what nI means by "linear interpolation"

Oh, come on, ACM! You ignored my trailer of you :P

@HerrFeinmann Extremely bad timing on your end to try to be funny in that way, sorry :P

We can recycle it next time, I guess

Anyway, ACM, this is on you: I've repeated told you that your hands-off moderation is going to cause problems, and now when you actually have to stomp out a bad discussion, you get sidetracked into a mess because you have left that problem to fester.

Well, actually, I recorded that and after coming back I found out hell had broken loose

@HerrFeinmann yet more reasons to not like voice messages

Although, to be fair, it is one of the better uses for voice messages

There are so many things bad with voice messages, but playing with flair is indeed one small perk.

@naturallyInconsistent voice messages are justice

@HerrFeinmann how can that be?

@ACuriousMind I remember something like equispaced decomposition and Newton Interpolation, but I cannot recall a formal proof of this. Can you please recommend one?

@naturallyInconsistent ??? what are u talking about

saying "ignore this user" does not remove their messages from the permalink

r u just trying to call me a liar?

@Relativisticcucumber you said you blocked. Blocking means that their messages will not appear.

is there an actual block? i thought the only option is to ignore the user?

@Relativisticcucumber nope, for us it is incredible strange, I'd say. (in the sense that here you would never do such a thing. at least until now).

folks, can't we get all along? :/

@naturallyInconsistent It only removes them from the chat room interface, not the transcript (also you could just log out to see the messages again). Do we really have to fight even over the basic workings of the site now?

Let's calm down, people

Remember that this is a place where many cat pictures have been shared

wrong link

im literally so sick of people telling me to calm down. im not going to be called a liar or made to feel like im in the wrong for getting mad that someone is calling me a liar

I said "people" :(

ok but why isnt it just "hey nI dont call people a liar without fact-checking basic info"

@SineoftheTime stai seminando discordia bro (ahahah jk)

@ACuriousMind Well, I'd have trouble seeing how the lorentzian vegetable would be inclined to log out or open the transcript for a recent conversation enough to take that screenshot if the user interface already removed those messages from the chat interface.

instead of "do we really have to fight" and "calm down" and all of this BS

I just wanted everyone to avoid escalation, I wasn't aiming at you :(

@naturallyInconsistent ...they would do that in order to take that screenshot, obviously. Please stop being needlessly contrarian over an insignificant detail

@HerrFeinmann colpa mia, adesso vado :(

@Bml I think you're overthinking this: If you have an interval $[a,b]$, then if you divide it into $N$ equal parts you get $[a,a+\frac{b-a}{2}], [a+\frac{b-a}{2}, a+2\frac{b-a}{2}],\dots$

@SineoftheTime jk=just kidding, in case you don't know :P

Yeah yeah :)

"Life is too short to be arguing over some little things." Sun Tzu, The Art of War

"Life is too short" -Me, my room

@ACuriousMind and that's the part im being suspicious of. The transcript up till the point of mister nobody making the meta post and all the heated discussions, if removing Allie's posts, do not show any reason that I can see that would have warranted such an annoying move at all. I dont see why the lorentzian vegetable would have taken such an awkward move as opposed to just taking a screenshot directly of the chat interface.

As in, I dont see this as an insignificant detail. You cannot just insist that the problem go away.

@Relativisticcucumber in fact, yes to this. I'd agree that you would have the right to be angry.

You're entitled to your opinion, strange as I find it. Like everyone else, please let's drop the topic now for good. Everyone's had plenty of time to say their piece, nothing productive is going to come out of going in circles. Take it up later or on meta if you find you still have something to say about it.

$\uparrow$ this

hiya tobias

hi Tobias

But you keep doing this. We'll bring up that you aren't doing enough moderation, and then you'll let the topic go around in circles, and then everybody gets back to the slightly toxic usual hbar. Some nice people might leave for good, making the hbar even more toxic. Surely people can air that they wish that hbar gets better moderation overall?

hello

I don't know if we need better moderation or so... I don't think so, actually. We should try to a) not act in bad faith and b) calm down, think a minute before sending a message

Are you sure that it is appropriate to come into a heated discussion that other people had already taken offense on "calm down" and give yet more of that?

...and keep in mind that we all are humans, have our own flaws, have bad days and so on. also, some messages might sound harsher than they were meant (e.g. due to written text, and/or language barrier), which brings be back to point a)

@naturallyInconsistent I am addressing this to everyone, and not only this particular discussion

@TobiasFünke and I'm specifically pointing out that when people have a righteous reason to be angry, telling them to calm down is precisely the fuel that will make the fire burn brighter

i think that heated discussions are rare on the chat. this was a case of false accusations and ACuriousMind already apologised and mister nobody accepted the apology. it is settled now

@naturallyInconsistent I am not telling anyone to calm down (or do anything). I said "we should try".

this was the only heated discussions I have ever come across here

@TobiasFünke Do you think that will be a meaningful difference from the perspective of the person who is already offended?

bruh

@Allie I hope I did not confuse you too much this morning :s

no you didnt!

ok, glad :d I hope you saw my comment that I had a mistake (I meant $N(N-1)$ instead of $N-2$, IIRC) :d I don't know what I was trying to calculate there hehe

...and I did not know that the textbook is that much expensive. holy moly

ikr!

I found it on ebay for cheaper but the price is still steep

Dark Allie is back

is it still in press?

hehe

i dont know

i want it though

im really liking it so far!

@TobiasFünke books in general tend to get more expensive once you climb up the academia hierarchy.

mhmh makes sense

does it? :P

how are you doing? :) long time no see

user!!!!

bestie user

Just note that unless those books are worldwide famous, the authors earn a very marginal royalty.

@Allie hey @Allie. Happy weekend.

exactly :(

@User1865345 weekend doesn't exist, im on break

although I go back to school this week... sorta...

My profile has a monograph in CRC. One of the best in his field. The book is overly expensive though.

@User1865345 book are at a constant price of $0 for me (unless they're very old and lost to time...)

@Allie I see.

I dont really understand how 6 copies are up on ebay for >$200, like how many people on ebay are actually going to buy this specific DFT book

Since I got my first physical books I way prefer them to PDF

i have the PDF but i dont like looking at a screen

When I asked what's the point of fixing such a price, he said these are actually meant for institutional copies. Not for personal use, although if one wishes, they could buy. But the sale over the year is pretty low.

@TobiasFünke that's a good question.

canonical ensemble

science terms sound so much more complicated than the actually are sometimes

@Allie I know 🥺

this solid state book tho is only $10 on ebay! I have it rented from the library and if i like the rest of the chapters i might buy it

@Allie once you get admitted to a phd degree, you will be given access to departmental resources.

@Allie very good.

i like how thats phrased as if i am surely getting admittd :P ill roll with it

@TobiasFünke did you ask me Or I am replying to a question intended for others? 😅

@Allie you will. Have faith.

@Allie said the chemist!!!

@User1865345 I asked you :)

I'm barely a chemist :P i just happen to know organic chemistry

@Allie that counts! hehe

In my experience chemists invent the strangest terminology for no reason (hehe)

and work in bases all the time

sorry for the rant

i did synthesis for a year and it was cool but it hurt my back. and it was frustrating. and i am a clumsy person who should not be around dangerous chemicals

haha yeah I also better stay away from any experiment

@ACuriousMind Yes, I understood, but I was looking for formal proof of that :-)

@TobiasFünke do you have any examples that come to your head?

@Allie not right now lol it was just in discussions with some chemists...

@Bml I mean the proof is just that each of those intervals has size $\frac{b-a}{N}$, as you can just directly compute

I don't really understand what you think is missing here

so N-representable just means derivable from an antisymmetric N-electron wave function?

in regards to reduced density matrices

@TobiasFünke sorry. I got busy in Ten Fold.

ah np, just reply whenever you have time and want to

@Allie in essence, yes

sometimes there is a condition on the kinetic energy added (when dealing with continuous system, as opposed to lattices, for example) for "practical" purposes.

I am bit busy, of late. Reading some good materials. Enjoying like @Allie enjoying her dft book.

reading is awesome!

nice :) that sounds like a good way of being busy

@Allie (and in general you might ask this question for indistinguishable bosons, too). the $N$-rep problem is a specific instance of the so-called "quantum marginal problem"

hmmm

my OFDFT project has to do with treating a system of N bosons occupying one orbital

and then including a term to account for the pauli exclusion principle

for electronic structure calculations

@Allie yes. Of course!!

interesting

@TobiasFünke exactly 😌

I did not manage to read a lot in the last weeks, sadly. But hopefully soon again

currently busy with stupid work :d

@TobiasFünke no problem. 👍🏻

oh geez

entropy was just mentioned in the book

NOOOO

@Allie you fear entropy?!

how does one take the natural log of an operator? is this analogous to taking the exponential of an operator where its just a taylor series expansion?

What I appreciate more that rhe current reading has been inspired by some questions at CV. I always like when I have to explore new sources for a potential answer to that question.

I just love learning :3

@Allie this looks tame, I guess.

What is $\hat \Gamma$?

@TobiasFünke you're cheating on us

yeah I was exaggerating, less fear and more an awareness of my lack of knowledge

@User1865345 the ensemble density operator

@HerrFeinmann it was not my free choice

@Allie heavy name. It seems they have just taken a trace of a matrix.

@Allie At an operational level, yes, but the Taylor series version is not actually a mathematically rigorous definition for all operators. I.e. you can think of it that way, but the rigorous version of functions of operators is given by Borel functional calculus

okay, thanks!

There you go.

@User1865345 it's actually a bit more involved since the operators here are typically linear operators on infinite-dimensional spaces, i.e. not necessarily matrices

Sure @ACuriousMind. Appreciate the clarification. 👍🏻

We deal with entropy in information geometry frequently.

@Allie first you define the operator exponential via the exponential series of an operator (which is formally the same as the usual exponential series), then you define the log as its inverse, in the sense that $\log\exp A=A$. There are some caveats about convergence but this is the idea

I see

At least in the matrix case this works well

to add what other's have said: For normal (hermitian, unitary...) operators you can always resort to the spectral decomposition. If $A=\sum\limits_{k} a_k P_k$, then $f(A):=\sum\limits_k f(a_k) P_k$ if $f$ is defined on the spectrum of $A$. For matrices this is totally fine, in the infinite-dimensional case you need some more effort, but the idea is the very same

In one answer of his ACM called it "functional calculus"

For your example, you must also define that for $f(x):=x\log(x)$ you set $f(0)=0$.

@HerrFeinmann I already linked to Borel functional calculus above :P

@ACuriousMind I got it, thank you. I was just overthinking this concept, the solution was trivial. Sorry for wasting your time.

I have so much to study

I proved your consistency!

@ACuriousMind lol

So it is a sub-branch of FA, I see

@Bml no worries

HI all

I'm going to get to linear algebra one day but for now I just have too much on my plate that I just want to focus on

@Allie studying is like onion. You need to go layers after layers. :-)

Hi

How can the energy scales of the temperature and scalar mass can be compared?

my glases have been broken for a week and ive procrastinated calling the optometrist

I am getting pretty confused.

@Allie by no means should you necessarily worry about learning the details of what I said, this is the kind of formal mathematics even many physicists never learn :P

@Allie you have not fixed it yet? gasps...

@LittleBlue who's comparing them and in what context?

@ACuriousMind Of course, your answer was more than satisfactory for the question i had

@HerrFeinmann now the question is can you prove existence and uniqueness :P

But even just in QM in general, I would say I have a good basis (pun intended) for linear algebra but there's a lot of detail and rigor I would like to eventually understand

How much linear algebra is needed for QM?

the idea where we use eigenvalues extends to functions that we don't have a series expansion of. e.g. one can take the floor function of a matrix

so i think it is a more general idea

I'm a bit lost now, what goes wrong in the Taylor series in the general (non matrix) case? The metric structure? I'm thinking about how in the context of Lie groups the Taylor series is only good for matrices because Lie Algebras don't have a product (an associative one, I mean), but this looks like a totally different can of worms. Operators form a linear space even in the finite dimensional case, so my guess now is that it's a metric problem, right?

@User1865345 I'm just a baby in QM but from what I've seen, it's almost all linear algebra

Woww!!

i think its very pretty actually

@HerrFeinmann Proving the convergence/domain of definition of the series definition can be tricky if you plug in unbounded (and hence only densely defined) operators

ive always thought linear algebra was elegant but the way its used in QM just tickles my brain the right way

I see.

i plan to start studying again now

@User1865345 to do it mathematically rigorously you need all of linear algebra and the part of functional analysis that deals with operators on Hilbert spaces

@ACuriousMind Oh, ok, it's about that. The series is formally there

Okay.

(you do not need the horrible Sobolev space stuff that I got when I took a "functional analysis" class :P)

@HerrFeinmann as long as you have a suitable metric, it works exactly analogous to the "usual" case

@ACuriousMind I hated it

@RyderRude you could also use that definition but beware! It requires that your matrix can be diagonalized. IIRC, the general definition takes care also of non-diagonalizable ones

@ACuriousMind that PDE course really hurt you, buddy

@ACuriousMind we have studied functional analysis. Mostly confined to spectrum and few other stuffs needed for forecasting results.

I think you also mentioned Sobolev spaces in that context

@HerrFeinmann That was the same course, yes :P

Like, everytime you want to mention a horrible memory, Sobolev spaces return

Is it related to pde?

They make diffeomorphism invariance look like child's play

@HerrFeinmann ooh. it seems like neither definition is more general than the other then. the series definition doesn't work for non series functions and the eigenvalue definition doesn't work for non diagonal matrices

@User1865345 well, a PDE features differential operators :P

I went in there thinking we'd do some beautiful algebra and then we spent months doing weird analytical inequalities to prove some n,k,p,t Sobolev space embeds into some r,k,l Schwartz space or whatever :P

@ACuriousMind ok, to be more clear, the mass of an electron is 0.5MeV, but 1Kelvin corresponds to 8.6*10⁻⁵eV. When we talk about a theory where m<<T, how big must be the temperature? Am I confusing something?

meow

hilbert spaces are really cool

I know it is too basic for you, people. But one of the most useful results that we use is Hilbert's closest point approximation theorem.

Nah, they have too much structure :P

I prefer sets

Lol

would like to smoke a blunt with hilbert

He had that beautiful hat.

@LittleBlue Ah, you're just measuring both in units of energy. When you say "the mass of an electron is $E$" where $E$ has units of energy, what you mean is that it has mass $E/c^2$ (since $E=mc^2$ for rest mass). When you say "the temperature is $E$", you mean the temperature is $E/k_B$ (for $k_B$ the Boltzmann constant)

too bad hes gone :(

@ACuriousMind I wonder, did even that exam take two days?!

@Allie hmm.

I didn't write the exam because I didn't need the points :P

After two years I haven't figured out how it works there. If I don't do an exam I chose two years ago, no graduation :P

@ACuriousMind Ohhhh okok, soo to actually compare them i need to multiply by k_B and, and since we are considering the natural units divide by 1?

@LittleBlue yes, if you want to translate your mass into a temperature that's what you have to do

@HerrFeinmann you need to have 180 "credit points" for a Bachelor

I think that was graduate ACM

@HerrFeinmann You do not have to commit to taking the exam just because you sit in the course

you can usually take, however, as many courses as you want (perhaps not doing the exam if not needed)

does not matter

same situation, with 120 credits instead

@TobiasFünke here too, anyways. 174 credits of exams and 6 for the graduation

one time i was told to sign up for a research credit so i did and then it turned out i was going to be charged $500 because it went over the credits guaranteed by my scholarship, and when i found out it was a day after the last day to withdraw

@ACuriousMind I see, that's actually the same spirit as here, but at this point, the only possible explanation is that you skipped other lectures :P

so i was stuck with a $500 charge

I never had time for extra courses due to my own lectures

@Allie my god.

@Allie While they are named after Hilbert because he started looking at the prototypical Hilbert space of square-integrable functions, most of the modern abstract theory of Hilbert space is due to others and the theory of operators and operator algebras in particular to people like von Neumann

But the scam here is that even though you can follow what you want without doing the exam, you have to choose beforehand the exams you want to do (at best you change every year, but you will have to follow the class that year to do the exams)

Hilbert didn't know what a Hilbert space was. He said it during a class

he asked his students what it was

lolol

So you can't follow 20 course and just do 10 :P

@ACuriousMind ok, soo 1K should be 1eV? I know its a dumb question ahahha

@HerrFeinmann I didn't have a lot of extra points. I think the only courses I took but skipped the exam were functional analysis (because I hated Sobolev spaces so much) and sheaf cohomology (because I knew I didn't need its points at all)

> There are many anecdotes showing that Hilbert was surprised by applications to quantum mechanics and by the very term "Hilbert space". Once in a conference, he asked the speaker: "What is a Hilbert space?"

@LittleBlue $k_BT\simeq8.6\cdot10^{-5}\mathrm{eV}/\mathrm{K}\cdot1\mathrm{K}=8.6\cdot 10^{-5}\mathrm{eV}$

@Allie education has become a scam

As you said at the beginning

@RyderRude capitalism

yes

@ACuriousMind it's ironic that you skipped both the thing you hated the most and the thing you probably loved the most

Well, you love De Rham cohomology, I think it's related (?)

@Allie laissez-faire.

@HerrFeinmann I like this category theory/cohomology stuff but I wouldn't say I "love(d) it the most"

it unfortunately seems that conditions have to get worse and worse for "average" people for class consciousness to become a thing

ok back to reading about thomas-fermi

Well, at the very least I remember you are very eager to talk about it! :P

@Allie aka cheap RPA

Sobolev spaces have a weird norm according to wiki

@HerrFeinmann ...if you judge by the contents of this chat, I'm also very eager to talk about GR :P

... fair point

@ACuriousMind youre not???

i think one would want to study the topology and analysis induced by this norm . maybe this is why they come under functional analysis

@Allie yep. People seem to love things other than those that directly alleviate their standard of livings. Four long years. Anyway, I digress.

in physics, one only cares about the L^2 norm, not even the L_p norm, let alone the Sobolev norm

@naturallyInconsistent I mean, I ostensibly am, but I've also made it known that I don't really like GR :P

what is the distinction between $\Delta$ and $\delta$ here?

@ACuriousMind myow life iz a lie

@RyderRude that's not true?!

@TobiasFünke lol

@TobiasFünke which branches of physics use these other norms?

what do you mean?

@RyderRude If you want to ask "Why are these spaces relevant?", why not just ask that instead of idle speculation?

@ACuriousMind this

sorry. why are these spaces relevant in physics?

Tbh I don't have an actual example at hand, but I used several times inequalities, norms of $L^p$ and so on from different spaces, to prove things. Also, one uses other norms than $L^2$ norms for sure; operator norms, trace norms etc...

L^p spaces are pretty common in probability and statistics too @RyderRude.

Also, i have a relatively stupid question.....

oh

yes

the distinctive feature of $L^2$ is that it is a Hilbert space

Yes

and the norm is induced by an inner product

Same reason as they are in mathematics: Because they are the appropriate spaces to formulate existence and uniqueness questions for PDEs in (see also this on math.SE). You pay for the relative ease of solving the PDE problems with the horror that are the embedding theorems :P

@Allie ask, ask. No question is stupid.

Regarding the screenshot above, when you see $O((\delta \varepsilon)^2)$ I assume that's referring to like the second order terms

@TobiasFünke yes

like the terms that only include $(\delta\varepsilon)^2$ and not $\delta\varepsilon$ or $\varepsilon$

@Allie yes, it means terms second-order and higher in $\delta \epsilon$. It's standard big O notation