@TobiasFünke I was thinking about it a few minutes ago: except for the obvious reasons that wouldn't make it work, I imagined that we establish small research group here to escape the ruthless world of academia and... We end up establishing a similar dynamics with tyrant advisors and stuff like that :P
as in, serious research requires expertise and extreme effort. if we just build some theory for fun, we know it is probably wrong. so we wouldn't be motivated to do it
and we can't do serious research here either. some people here do expertise in things, but they dont expertise in a common thing
I want to know why the non-interacting KE in KSDFT, given by the sum of the KEs of the KS orbitals, is the minimum non-interacting KE for any N electron wave function that produces that specific density
@Allie it depends on what exactly you mean, I would say. Suppose $n$ is the ground state density of the non-interacting system, and suppose the ground state is non-degenerate, and thus can be expressed as a single Slater determinant. Then from the definition of the functional it should be easy to show that it is indeed the sum over the corresponding single particle kinetic energies.
But is this what you mean?
You are working in Levys constrained search formalism, correct?
See e.g. Levy's paper. It can be shown that the ground state is a wave function which minimizes the functional (here $T[n]$) for the given ground state density. Let's denote the ground state by $\psi_0$ and its density $n_0$. Then by this theorem and by definition, you have $T[n_0]=\min\limits_{\psi\mapsto n} \langle \psi,T\psi\rangle = T[n_0]=\langle \psi_0,T\psi_0\rangle$. Now if the ground state $\psi$ is non-degenerate, and we are in a non-interacting system, it is a Slater determinant
and thus you can express the expectation value as a sum over single-particle kinetic energies
It was just a quick thought now. I hope it is more or less correct, but you should really double check. better safe than sorry :p
welcome. but as I said: please double check :p I don't want to spread fake news. If you have further questions we can discuss later, just ping me. Then I might have more time to think about it
@TobiasFünke I know there are many websites and papers, including wiki, asserting that KS DFT does Slater determinant, but that step is computationally intensive and way too strong when not doing molecules (they do, however, often do it for atomic and molecular codes), and so KS DFT does not need to do the Slater determinant step. It is literally not in the code, and rightly so, since in condensed matter, we know that the Slater exchange is too strong.
@Allie btw, to properly answer your question one must start with a proper definition of the functionals (including domains). as I said, we might discuss later ^^
see e.g. the discussion in Eschrig's DFT notes or Parr and Yang section 7.3.; they explain it nicely, I think. Hope this helps.
@TobiasFünke I'd fix this so: HK1 guarantees that there is a 1-1 mapping between $V_\text{ext}$ and ground state density $n$; HK2 says that $n$ is the minimiser of the universal Hamiltonian functional $\mathcal H[n]$. Now, you have many tolerable contenders for the XC functionals, i.e. not so far off that they are show-stopping in themselves, and you want to handle the biggest problem, the KE. Then consider minimising, instead, $$\mathcal H[n]-T[n]+T(\{\psi_s\})-\lambda(n-\sum_r|\psi_r|^2)$$
Where $\lambda$ is the Lagrange's undetermined multiplier enforcing the constraint that the fictitious non-interacting wavefunctions $\psi_r$ having to reproduce the ground state density $n(\vec r)$
Clearly, since $n$ minimises the universal Hamiltonian function $\mathcal H[n]$, if this thing is meant to be a sensible replacement problem at all, it should also minimise this new system. That's what the bait-and-switch is trying to assert.
(note that this minimisation is trivial: almost everything else is a function of $n$ and is thus taken to be constant, and only $T(\{\psi_s\})$ is going to change when you do the constrained variation. And yet you also know that this KE term is a quadratic with strict lower bound and no upper bound. If there is a stationary point at all, it has to be a minimum. And there are no other stuff to vary other than the $\lambda\sum_r|\psi_r|^2$ term. Obviously the whole thing is to be minimised.)
Yes, exactly! I also searched some solid state books
Just in solid state I've seen it arising differently and I'm trying to connect the things.
In this case, I'm given this plot after finding the Bogoliubov spectrum $E_k=\sqrt{(\varepsilon_k-\mu)^2+|\Delta|^2}$
Where actually $\Delta=0$ outside of a small neighbourhood of the Fermi energy, so I suppose that's similar to when in solid state you compute corrections at the edges of the Brillouin zone
I can't believe that my Prof replied the same day I sent the e-mail. Everyone told me it took weeks to even get a reply and they had to text every two days
@HerrFeinmann About your point earlier, I think a research career in mathematics probably gives people a lot more room to explore. In hep-th, it seems like you have to just churn out papers for the first decade or so of your career if you want any chance of being competitive for jobs
@naturallyInconsistent Sorry, I really don't see what you are trying to do here (it is late and I am exhausted from work, so this is due to me). But in general I don't see any problems in proving assertions using the fact that Slater determinant exist in the corresponding Hilbert space.
...and if you minimize a functional of $N$ orthonormal single-particle states which yield a density via the usual formula, I guess it is equivalent to minimize over the set of Slater determinants in some sense
@TobiasFünke thanks for suggesting "THE UNREASONABLE EFFECTIVENSS OF MATHEMATICS IN THE NATURAL SCIENCES"
i am reading it now and so far it is very nice. i like this sentiment a lot, as it's smth ive been thinking about recently:
"How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?" It has to be admitted that we have no definite evidence that there is no such theory."
i also wonder to what extent these things are useful to think about. but the quote i put above is related to some recent concerns i have because i have been reading a paper that proposes a model. to keep it vague, after reading this paper i have the impression "sure this works, but it's rather simple, so surely it's not a unique way to model this situation, so to what extent is this a good model?" and im trying to think about if this is even a good question or not.
i just feel like if a model is good, it should have some "uniqueness" in that it captures things in a nontrivial way. but i also have the mentality that a model is not "right" or "wrong" but just a way to look at things, so these two mentalities are at a tug-of-war
so what is reasonable criteria for a good model :P
@TobiasFünke debateable. have you heard the chinese room thought experiment? i think this is what it's called.
i guess this is a natural thing to wonder about tho. i mean as a budding physicist we learn all of these ways to model things, then eventually one gets thrown into the world and told "now make your own models!"
@Relativisticcucumber I heard of it but I don't see a relation
@Relativisticcucumber I think it is really important to think about such things. In general, I think that philosophy of science is underrepresented in the standard physics curriculum. I mean, you don't have to study it for years, but some basic understanding and an awareness of problems is important
well this thought experiment is like "if all i do is apply a book of rules i become indistinguishable from someone who natively speaks the language" so i think it can highlight that, actually memorization done to a very high degree can lead to producing results that are indistinguishable from other types of understandings. i mean i think we can imagine a "physics book room" situation that works in quite the same way, no?
@TobiasFünke that does seem reasonable. ill continue to ponder this until i have some thoughts of my own on the matter.
separately, i seek thoughts on another controversial topic :P i am trying to improve my geography so that i do not fit the bad stereotypes of americans who dont know geography (admittedly before this endeavor i was quite bad) for those of you who aren't american, what do you all think about what standard knowledge of geography is elsewhere, and what constitutes a good working knowledge in other regions?
right now i am endeavoring to be able to label all countries on a world map. i am at about 80% accuracy, but i still have minimal knowledge of cities, capitals, flags, etc. i also need to look at an unlabeled map, it's still kind of hard for me to say "x country has neighbors a, b, and c" reliably :P so this is also smth i aim to improve
In school we were taught geography at least 5 years or so, if I remember well (or even longer if one wanted to). Starting from basics like topography (what you mean), i.e. naming countries, cities, rivers, perhaps mountains and so on; then later to topics such as climate/whether, and also more human-centered
@Relativisticcucumber if you're interested there's a bunch of games I kinda like! maybe you already know them? worldle and there's one like geoguessr but it's a static photo you guess the year it was taken too called whentaken
@ACuriousMind shockingly I haven't read really anything else from le guin! I'm not really into a lot of sci fi in general, and I tried earthsea because my partner thought I would love it but I didn't get far. I will say that I'm not the biggest fan of her writing style from "the dispossessed" at the start - I found it a bit slow. I thought it picked up around 1/4-1/3 of the way through.
what interested me most about the book was the themes and ideas she explored about political systems (anarcho-syndicalism and capitalism and alien USSR analogue), and how that interacted with "doing physics".
if that sounds like something you're interested in, then yes; if not, probably not? may I ask why you didn't like the left hand of darkness @ACuriousMind ?
My question is (ACM, if you think this is controversial, feel free to remove): how you cope with it? The existence of pseudoscientific "knowledge" is probably the thing that I struggle the most to accept in this world, so I wonder if any of you has such strong feelings about it and most of all, any advice to cope with it. Few things can trigger my rage like hearing/seeing such things
@TobiasFünke Well, in the hope this won't become semantics, I think the definition on wiki is fairly good:
>Pseudoscience consists of statements, beliefs, or practices that claim to be both scientific and factual but are incompatible with the scientific method.
In this thread a definition of the internal symmetry is given: https://physics.stackexchange.com/questions/263489/what-does-it-mean-to-say-internal-symmetry
Considering the explanation given, does it mean that i.e. translational transformations/ rotational transformations and time transformations act on the spacetime and not on the qft?
But I'm probably talking about more extreme versions like astrology, to mention one. So, let's stay in the "bulk" and avoid "surface effects", considering the things that are unmistakably pseudoscience
"An internal symmetry is a transformation acting only on the fields, therefore not transforming spacetime points, and leaving the lagrangian or the physical results invariant. Example of internal symmetries are gauge symmetries. These are local symmetries, which means the transformations are in general spacetime dependent in the sense they are, in general, different for each spacetime point."
So internal symmetries don't act on the spacetime point, gauge transformations are internal symmetries, gauge transformations can be local, dependent on spacetime. Isn't this a contradiction ?
@TobiasFünke Ok, let me put it in this way. Since my plan is not to make a philosophical dissertation, I think there are some things that all of us here would consider as pseudoscience, at least de facto, like astrology (?)
Astrology is a good example, for the sake of discussion (but I could not resist to point out that it is really non-trivial :p), I think, because it is becoming a trend (again)
If you don't want to call them pseudoscience, let's invent another term. This is a discussion at the physics level of rigor, so no need to define the domain, just have a vague idea of what we're talking about :P
Humans are not rational creatures (?), and no-one is exempt from fallacies, stereotypes and so on. You must always question yourself and your believes, over and over again. I think too few people (try to) do so, and thus believe in the uttermost non-sense or hold very...questionable (?) beliefs. Does that make sense?
But I think there is a fundamental difference. Even if it's not the most common situation, religion and science are compatible, in the sense that they are independent. The Bible and other religious texts are not interpreted literally (I'm thinking about things like Earth being 10k years old, to mention one) and you just have no real superposition between religion and science. For example, one may argue that a superior entity is irrelevant for science, but you can't prove there is or isn't a God
On the other hand, astrology does go against science
I guess if you are enraged by pseudoscience, you will not be in a state to effectively discuss and convince people who believe in it otherwise. I know it's not a 'decision' to have strong feelings, though.
@TobiasFünke That is precisely why I ask, because I realize that it affects me deeply, but I realize that in most cases people who "fall for it" do not have a scientific education
@qwerty I guess it's more about situations in which someone close, e.g. a friend with whom you share some other aspect of your life, mentions that kind of stuff. It affects me in general, even in unrelated discussions
@TobiasFünke eh, well, they might not have strong beliefs, maybe they just got introduced to some fringe ideas and don't know better. being able to talk and say "listen, this is why it's incorrect" is helpful imo.
I'm sure @ACuriousMind has a way to cope with it :P
Well, anyways, changing topic: @TobiasFünke do you know any place featuring a similar graph about the energy gap of superconductivity? I've found it in the slides of my lectures. It reminds me of energy bands in solid state physics but the context is wildly different and I don't even know how one would get this in SC
@qwerty Well - The Left Hand of Darkness also sounded interesting to me, a cultural ambassador from Earth (or a society like Earth) sent to a society of hermaphrodites with an estrous cycle who have entirely different views on sex and gender - but ultimately I didn't enjoy it. I thought all the protagonists were stupid and/or immoral and didn't really agree with what I perceived to be the themes of the book, though I could see how in 1969 this was a remarkable book to write either way.
To go in more detail would require me to spoil the whole book :P
@Relativisticcucumber if Wigner had provided a resolution it would have been called the reasonable effectiveness... :P
@imbAF No, it is not. A Lorentz transformation acts "on spacetime" in the sense that it sends $x$ to $\Lambda x$. An internal symmetry does no such thing - the transformation $\psi(x)\mapsto \mathrm{e}^{\mathrm{i}\alpha(x)}\psi(x)$ does not transform the spacetime point $x$.
However, to say that it is a characteristic of gauge symmetries to not act on spacetime points is questionable; the "spacetime symmetries" of GR are gauge symmetries; it's however correct that gauge symmetries outside of GR are almost always internal
(and the GR case is rather subtle to treat correctly)
@HerrFeinmann you might have found a resolution -- i didn't read the discussion that follows, but i'd ask "why do you care"? not in a flippant sense, i'm sincerely posing this q -- in my opinion, i think there are tons of ideas floating around, good and bad, and it is not my business what other people choose to believe, nor to police what people say. i believe that people oftentimes believe and hear what they want to believe. [...]
[...] for someone who truly seeks knowledge and information, the world is at their fingertips. for someone who doesn't, why is it my place to judge that or correct them? ultimately other people's lives are other people's lives.
