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1:03 AM
0
Q: Question concerning the relationship between Hilbert Space, Banach bases, and Abelian Groups

IanMy Assumption Given the properties of standing waves as satisfactory to Bohr’s quantum postulate it is necessary to incorporate linear continuity. To derive such continuity from an integral of an independent variable that is divergent requires significant methodology. To do so would therefore p...

^wat
 
1:48 AM
If physics doesn't make sense, it's just advanced.
 
@DavidZ I didn't chip in on that one, because I felt I'd said my piece on the subject in
12
Q: Should Stack Overflow be a "last resort" resource?

Jonathon ReinhartI recently left this comment for a poster who was insulted by someone asking him if he could use Google: He does make a valid point, though. You'll find on StackOverflow that we, as a community, place a heavy importance on doing your own research, and basically seek to use SO as "last chance,...

 
user54412
@tpg2114 I see where you're coming from, but I feel using just translational degrees of freedom is too restricting
 
user54412
for instance there are rotational degrees of freedom for various molecules, as well as vibrational
 
@dmckee So having read that -- how do you view questions where I or you might find the answer in 10 seconds on Google, but it's because we know the words to search for while OP may not? Obviously if you just take the adjectives and nouns in the question and Google it and find the answer, they didn't do enough work
 
user54412
and even classic heat capacity changes depending on the number of such degrees of freedom (monatomic vs. diatomic ideal gasses for instance)
 
1:55 AM
@ChrisWhite I know, but when we (I say "we" from a gas dynamics standpoint) mean the other forms, we explicitly say rotational temperature, vibrational, electronic, etc.. If we say "temperature" it is either assumed all the modes are in equilibrium (and so translational temperature is fine) or we are only talking about translational energies
And yes, heat capacity can change (unless you are assuming gases are frozen). But I have only studied all of this from the perspective of gas dynamics so we always have something with mass. I've never looked at what happens with photons
I really don't know how photons are considered w.r.t. temperature/energy. I would ask a question about it but I'm worried it is clearly defined in some elementary text somewhere that is just outside of my area and it would be a bad question with an obvious answer
 
user54412
I too only work with translational temperature (in perfectly conducting, adiabatic, perfectly collisional plasmas!)
 
In the vicinity of mass-less spherical cows too
 
user54412
but for instance interstellar dust temperature is understood to basically be the temperature of the phonons excited by the last photon to strike the dust
 
We do actually have a multi-temperature capability for non-equilibrium flows and plasmas, but we treat each temperature as it's own equation
@ChrisWhite Which is what I was trying to argue in my point 3 and the comments under it -- the "temperature" of a photon is only measured when it interacts with something. So if we have a volume with photons passing through it and no matter with which to interact, the "temperature" is undefined
And so if we measure the temperature of the CMB, the "volume" that has that "temperature" is really just confined to the sensor that detected it. Not the finitely-infinite volume the photon passed through to get to the sensor.
Or it's the temperature of the volume of mass that released the photon to begin with
 
user54412
@tpg2114 It should fall out from $1/T = \partial S/\partial U$ in the statistics of an ideal gas of massless bosons. How much that's just pushing the question back is another issue. Also, that wouldn't be a bad question at all.
 
2:05 AM
Maybe I will go ahead and post the question then. It seems like something that might actually spark some different opinions
 
user54412
I guess there's the stat mech definition: write down S and U (somehow -- that was always the least clear thing in stat mech) and differentiate. Then there's the old-school thermodynamics definition: place a thermometer in equilibrium with the thing you want a temperature for. Asking how these two are equivalent (both mathematically and intuitively) seems pretty constructive -- I know it was glossed over in my courses.
 
Is there such a thing as "equilibrium" between photons?
Like, if I put a bunch of particle A in a box with a bunch of particle B, eventually the two would be in equilibrium with one another
But what if A was photons instead. The total energy would be the same, but the photons would become "absorbed" into B
Can there even be photon-photon interactions? If both A and B were photons at different "temperatures" (or energies), no equilibrium would be reached right?
I'm pretty... confused
 
2:22 AM
@tpg2114 (re: last sentence) you'd think, but remember this recent case in which that was far from obvious
 
And photons have 2 degrees of freedom, so if we go with the "temperatures based on D.O.F" naming, it wouldn't be translational temperature but something else. I'd call it "spin temperature" but I don't know if the D.O.F's are actually called spin
@ChrisWhite Question is posted (had to have fun with the title) -- feel free to edit if something isn't clear (or make suggestions)
0
Q: How hot is your photon?

tpg2114This question comes from my answer to the question Can a cubic meter of space at absolute zero have any object with mass inside? and the related discussion under it. To summarize, I stated that the temperature within a volume is related the ensemble average of the kinetic energy of the particles ...

 
user54412
 
user54412
[of](http://arxiv.org/abs/1502.01588) [Plank](http://arxiv.org/abs/1502.01589) [papers](http://arxiv.org/abs/1502.01590) [came](http://arxiv.org/abs/1502.01591) [out](http://arxiv.org/abs/1502.01592) [today](http://arxiv.org/abs/1502.01593),
 
user54412
[including](http://arxiv.org/abs/1502.01594) [this](http://arxiv.org/abs/1502.01595) [one](http://arxiv.org/abs/1502.01596) [and](http://arxiv.org/abs/1502.01597) [this](http://arxiv.org/abs/1502.01598) [one](http://arxiv.org/abs/1502.01599).
 
user54412
Great, I broke chat
2
 
2:26 AM
I haven't asked many questions on this site, but it's a pretty odd collection
 
user54412
@tpg2114 polarization is probably a better word. But it's not quite the same as a spin lattice I suppose
 
Possibly. I don't know what the +1/-1 modes are called. I assume they have a name
 
user54412
it's not like it takes energy to rotate a light beam's polarization
 
That would imply there is no energy-storing D.O.F then
And we're right back at "what the heck kind of energy is it really"
I kind of want to post a question asking what the heat capacity of a photon is now :)
@DavidZ I would agree with your opinion on the matter -- the question itself is answered obviously by searching. I think what Anna was arguing is more about the OP not understanding the result, but OP didn't say "Hey, I found this and I'm not sure how to interpret it". That, to me, would be sufficient
 
user54412
--> More seriously on Plank, this paper is the summary. Table 9 has the cosmological parameters, and there are some very pretty sky maps too.
 
2:32 AM
I'm in for the sky maps. I won't understand anything else
 
user54412
The big CMB cosmologist in my department informally declared that we've reached the end of the era of measuring the CMB temperature
 
user54412
there's stuff worth doing in polarization, but we've pretty much exhausted the useful information from just self-correlating temperature
 
user54412
Figure 20 from that paper struck me as something that could come from an MHD simulation. Then I read the caption and realized that it's the galactic magnetic field, so that makes sense.
 
Fig 20 and 22 are pretty awesome looking
 
user54412
and people think space is "empty"
 
user54412
2:36 AM
actually... I wonder if one could sensibly define a magnetic field temperature...
 
So I just had a thought... what if the energy of a photon (or a magnetic field) is really just a potential energy?
Since photon number is something not conserved, the energy contained in the field can be absorbed by things (PE->KE) or emitted (KE->PE)
And the same could possibly be said about magnetic fields
Searching for magnetic field temperature is too polluted with how magnets work at different temperatures...
@ChrisWhite Changing topics, how's the thesis coming along?
 
user54412
Not too bad. I'd say there's a >50% chance I graduate.
 
You should have reported it as a p-value against the null hypothesis that you won't graduate
 
user54412
(We have a strict 5 year limit, and I have 18 months left. It's not a question of how long the degree takes, but whether or not it happens at all.)
 
Our department is starting to implement strict time limits
But grandfathering us older folks in gently
 
user54412
2:45 AM
it definitely cuts both ways
 
The motivation in our department is that professors keep us around far too long working on industry projects since we are ridiculously cheap labor
 
user54412
it's nice to have everyone trying to get me to finish for my own good, but I'm reasonably sure I'd have a stronger career start (in academia) with an extra year
 
Our funding agencies have actually started telling our boss to graduate us or they'll stop funding him
Is it possible to stay on after you finish for ~1 year as a research engineer/post doc?
 
user54412
wow -- when funding agencies notice that, that's when it gets serious
 
user54412
@tpg2114 nope -- we're unceremoniously kicked out the door, not welcome to return except in a second post doc (maybe) or faculty position
 
2:48 AM
Yeah... it's not great. We've had several people over 10 years where they couldn't work on their thesis because of industry projects
That's not good. But I guess it is good at the same time. I mean, it's better than our... incestuous? practices. We have people who graduate, work for another 6 months, move to another lab in the department for 6 months, and maybe leave, maybe just stick around as a research engineer
There's surprisingly little turnover
 
user54412
So every week there's a lunch discussion at the Institute for Advanced Study, attended by all the local big shots in astro -- the sorts who can (or think they can) determine all sorts of policies and such. This past week was particularly political, with a former congressman (and physics PhD) attending.
 
user54412
A good deal of discussion revolved around restructuring of academia in the not-too-distant future
 
user54412
certain fields are really starting to be hit hard by the leveling off of the exponential growth they've experienced for several decades
 
user54412
and the faculty/postdoc/phd ratios are really unsustainable when taken together with the "get a phd and one day you'll become a professor" attitude
 
The perk of being funded by primarily defense-sector spending is our money is pretty secure and plentiful.
Although most of us don't even go the academia route. I think we (aerospace) tend to break the PhD to be a professor mold
I know in computational work, if you have a BS you really can't do it. An MS will have you making the grids for the guy with the PhD to actually use and analyze.
 
user54412
2:59 AM
I think there's a consensus that more PhDs outside academia is probably a good thing, but it would help if PhD programs accept this role and don't deceive students about job opportunities.
 
user54412
(That's not necessarily a complaint about my department in particular, just a general observation)
 
I actually don't have any idea what that is like. Our department doesn't let us teach, doesn't let us TA, and our ties with military research labs and industry all but hands us industry jobs. And makes it very, very hard to get academic jobs
It's always interesting how perspectives differ
My brother (at Pitt in the math dept) said he went to a seminar by a professor in the department who claims to have proven the Boussinesq hypothesis.
I want to see that paper
Not published yet though. So I'm skeptical.
 
user54412
interesting -- ping me if you hear more about that
 
I will. That's a pretty big deal if true. I'm sure he proved something, but I'm worried it's under some extremely restrictive conditions that give it very narrow scope
The turbulence modeling field is really divided on the validity of the hypothesis
 
user54412
@tpg2114 A continuum of incompressible, spherical cows?
 
3:08 AM
Heh, exactly
Spherical cows in an axisymmetric, periodic domain
 
3:30 AM
@tpg2114 I tend to be pretty forgiving of those kinds of things the first time I see. Even the second. But after a while ...
I'm aware that this isn't always a fair reaction because if the previous ones aren't coached in the same language the OP is using they still might have looked and not found it.
But I'm human, too.
 
On the spherical cow front, I'm collecting spherical cow models that get things more right than they have any right to.
I have a uniform density model of a star I use with my Virial Theorem lessons that can get the temperature of the sun more or less right. At least it hit about the mean of the core temperature and the surface temperature.
 
user54412
Impressive. Looking up the real numbers it seems the central-to-average density ratio of the Sun is actually over 100.
 
I think the most mindblowing statistic about the Sun is how long it takes for a photon from the core to actually reach space
 
user54412
@dmckee I'm trying to recall some good ones from my order-of-magnitude undergrad physics course (one of the best courses I've ever taken).
 
user54412
We estimated things from pretty much every branch of physics, some of them probably unreasonably well.
 
3:42 AM
@tpg2114 are you attending GT?
 
@AlfredCentauri Yes, for entirely too long
I did my undergrad here and just kept right on going
 
@tpg2114 I was there, on and off, from '91 through '10 over in the EE building on the corner.
 
I don't get over there very much. The only EE class I ever had was the intro to circuits for non-majors
I'm in the AE buildings, across from the hotel on North ave
Were you teaching?
 
@tpg2114 I taught that class for several years.
 
Hrm... let me hop in OSCAR and see when I took that
Summer 2006
That was the class (ECE 3710) and the lab I took in Fall 2007 (ECE 3741)
I dated a girl who was an ECE major. She got her MS there also, was working on some DSP. Voice recognition I believe?
Back in 2004-2005
 
3:53 AM
@tpg2114 I'm almost certain I taught a section of that class SU06. There were always several sections so it's possible you had my section. Actually, in almost every semester, I found that some of my 3710 students were quite a bit more capable than many of the EE majors I taught. Of course, I tried to recruit them...
 
@AlfredCentauri My instructor was... Erik I Verriest
(which if that is you and you don't want your real name known, I can delete that)
 
@tpg2114 No worries, you had a different section. Erik's name is familiar to me but I can't say that I remember meeting him.
 
So have you moved on to another institute?
 
@tpg2114 No wonder his name is familiar - he's a professor there. That's actually unusual for a prof to teach 3170 - it's normally 'non-traditional' grad students.
 
I can't say that he was the one who actually taught us. I want to say he came in day 1 and then handed us off to a grad student
But I wasn't really all that into the class so I have sort of repressed it
 
4:03 AM
@tpg2114 I'm no longer residing in GA and I'm now pursuing admission into a physics graduate program in another state. I find physics more interesting now.
 
What did you work on at Tech? And what aspect of physics are you interested in now?
Side note -- surely we have a question that explains refraction in water somewhere, right? This one can't be the first...
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Q: Chopsticks in Water

Bob the zealotWhen you put chopsticks (or any other object) in a glass cup of water, why do the chopsticks seem curved or bent? Is it because of the shape of the cup, the water, or the chopsticks itself? Note: I'm sorry I currently don't have an image of it. You can try this at home. Just put a pair of chopst...

 
4:27 AM
General question: Does anyone other than me find it ridiculously hard to log into chat? Logging into any of the SE networks is easy (my computer remembers). Once I've logged into any one, I'm generally automagically logged into every SE site in which I participate (SE remembers). Logging into chat? That is not easy. Sometimes the magic works, sometimes it doesn't.
 
user54412
@dmckee I'm apparently too tired to do it now, but I was trying to reproduce an argument that derives the radius of Jupiter from hydrostatic equilibrium (equivalently the virial theorem), the ideal gas law, and the Schroedinger equation (without any actual quantum mechanics). All you need is G, hbar, m_p, and k_B, no Jupiter mass.
 
@DavidHammen I usually don't have problems. I seem to have the opposite issue -- I can never seem to leave any chatrooms I join. I see my name in rooms for far longer than I have the tab open
@ChrisWhite $R_j \propto f(G,\bar{h},m_p, k_B)$ You can thank me for doing your math work later ;)
I am completely side tracked by looking up awesome videos and images from the GOES satellites now
 
 
4 hours later…
8:41 AM
I made it through the exam period! :D
@ChrisWhite There are courses on that in the US? Great
 
9:06 AM
@Danu: How were they? Did you make up for the bad oral?
 
9:56 AM
Yeah, pretty much
I barely passed diffgeo but that's okay since I was convinced I was going to fail. Pretty sure I'm near the top of the class on my QED/GR exams.
 
How come you didn't do well in diffgeo?
 
very hard course
plus I came in with 0 math background (topology & strong linear algebra are.... recommended to say the least)
 
user54412
@Danu Good. You're still a real physicist, not a mathematician ;)
 
Confirmed
Too bad at math to be anything else :D
I do plan on doing more mathematics though
I wanna get into some algebraic stuff
eventually get to category theory just because it sounds so exciting
anyways, I gotta go since I'm going to the Grenke chess classic today!!!
First time I'm going to a chess tournament. I hope it's fun to see the best players in the world :)
 
10:23 AM
Have fun pal @Danu :-)
 
10:59 AM
@Danu I sometimes wonder if category theory would flow more easily for me, if I had started to learn it earlier in my life. Still trying to finally internalize adjuctions, this time with Chris Henderson's "Generalized abstract nonsense: category theory and adjunctions"
In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a particular kind of category with just one object). It allows the embedding of any category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool...
I struggled many weeks with the Yoneda lemma, but now it seems so trivial to me that I can not understand anymore why I ever struggled with it.
The funny thing about adjunctions is that I have encountered so many of them already in real applications, have understood some of them quite well, even managed to read and understand the abstract stuff multiple times, but then I do something else and the next time I encounter an adjunction I have to do all the detailed work again before I understand anything.
 
 
6 hours later…
4:41 PM
@ThomasKlimpel have you seen these awesome explanations of Category theory
http://math.stackexchange.com/a/438697/82615
http://math.stackexchange.com/a/399960/82615
http://www.lepp.cornell.edu/spr/1999-09/msg0017972.html
Thinking of category theory as formalizing the basic idea of paths in the plane between points, functions between them, homotopies etc... is so fruitful that you can even think of the action/lagrangian and path integral as functors so naturally too!!!
http://math.ucr.edu/home/baez/qg-fall2004/action.pdf
 
5:01 PM
Anybody whose studied elasticity ala Landau books.google.ie/… and interested in helping me think about it please pm me, so hard to make sense of in the context of theoretical physics ! :D
 
5:55 PM
@bolbteppa If you have a question, you could just ask (in chat, or of course on SE proper).
 
I agree but I have a few questions that are things I could answer myself if I really sat down and stressed over but it's just an aside so I don't want to stress anyone unless they just like talking elasticity haha :D
For example, I wonder why one wants to express the Free energy in terms of the strain tensor as they do in Landau's elasticity to derive Hooke's law, it leads me to believe that elasticity is just a classical macroscopic equilibrium statistical mechanical theory of solids
But then I really don't have a nice way to think about the stress and strain tensor even, I've seen two ways to derive them and I just don't know how to think about them so I'd like to talk about their fundamental properties and why you derive them a certain way :)
Also I was hoping those silly tensors actually were derivable from a Lagrangian, because apparently Elasticity can be phrased in terms of Lagrangians, but the standard references treat this at the end of the books as mere asides :( My guess is they pop out of Noether's theorem the way the stress-energy tensor pops out but I can't find a book that does it this way so unless I invent it myself idk :( haha
 
Your last question sounds like it's related to my question on SE.
I've bountied it a couple of times, but there are no takers.
On my bookshelf, pretty much next to the Landau&Lifshitz book on elasticity I Weiner's Statistical Mechanics of Elasticity. I'm not sure if that might be of interest to you.
 
6:11 PM
How are electric charges dealt with in an FDTD simulation?
I know how to calculate the curl of E and B fields to model propagating waves
But what about the effect of charges at certain points in the grid on the E field
 
@user1667423 I'm sure it's not right, but a long time I programmed something similar to FDTD and in the beginning of the simulation I'd "pump" the charges in, so there'd be currents.
 
vzn
6:40 PM
user image
2
 
7:11 PM
@alarge Awesome, let me check the links out man, thanks :)
 
7:38 PM
I've always thought of elasticity as a classical mechanics concept and went looking for Lagrangians, but thinking of it as a classical thermodynamics concept completely explains why Lagrangian's aren't a nice way to go about thinking about it
I mean, Lagrangians are nice for classical mechanics (non-rel, rel, EM & GR), but once you get to QM you throw them away and use Schrodinger
 
8:38 PM
@bolbteppa ...and going from QM to QFT you start to use them again
 
Well partially
Unless you use path integrals in QFT they theoretically only arise in fourier expanding the Hamiltonian operator H in Schrodinger's equation, but that's like a math trick right? I'm guessing the Cluster-decomposition principle says any operator can be expanded in creation annihilation operators, and the whole H to Lagrangian to Lagrangian density etc... is just a math tool to do this procedure?
But path integrals are just Green functions in the Schrodinger equation which is all derived from Heisenberg uncertainty + classical mechanics + measurement, so unless you take Feynman's amplitude stuff as the foundation of QM I think you can throw them away right? I mean the claim is there is no notion of a path (hence S(path) is completely undefined)
 
@bolbteppa Well, how would you get manifest Lorentz invariance of the theory without switching to the Lagrangian formalism? How would you use Noether's theorem?
The Feynman rules of the perturbative diagrams are also for terms in the Lagrangian, no matter which approach you used to derive the rules.
 
How does Noether's theorem apply to standard quantum mechanics?
 
@bolbteppa You need it as soon as you want to talk about gauge theories, anomalous symmetries and the like
I know no derivation of a quantum anomaly that is done without, in some way, examining the transformation of the path integral measure.
The whole renormalisation group theory is also in terms of Lagrangians
I'm not saying you absolutely need to have a Lagrangian in all these cases, but it is definitely not true that Lagrangians are more important classically than quantumly.
 
8:55 PM
Apropos Yoneda's Lemma and unease with susy names, I'm reminded of this MO joke answer and comments below it: cocoa=a and coconut=nut :)
 
@Qmechanic Heh. :D
 
How is it possible that in things like the Caldeira-Leggett model you can get response functions which are imaginary for any finite finite number of modes, but then it becomes real when you have infinity modes?
 
@DanielSank Why should it not be possible?
 
Without writing a big essay haha CM has Lagrangian's that ultimately depend on one path, QM has a function of a lagrangian that ultimately depend on every path, but this function was derived under the assumption that no path exists. Of course renormalization can be done with lagrangian's and path integrals, but it's not exclusive to them, e.g. the renormalization group in the 1-D Ising model partition function :)
 
@ACuriousMind It's odd to think about a sequence of complex numbers which are purely imaginary but have a real limit!
 
9:11 PM
I'm not sure but I think Weinberg's whole point is that 2nd quantization with Lagrangians is misleading because any operator can be expressed as creation-annihilation operators because of this cluster-decomposition principle, so that whole process is just one method of explicitly doing the procedure but nothing fundamental? I'm still learning, may be wrong :) But the path integral point is pretty important, how it all can be derived assuming no path (Heisenberg + CM).
 
@DanielSank The complex plane contains many such sequences (might be a null set among all sequences, though)
 
@ACuriousMind Telling me that "is it so" doesn't really shed any light on the issue :)
 
Relating this back to elasticity, I'm not sure it's a good idea to use Hilbert stress energy tensors in elasticity, especially deriving w.r.t. a metric, seems like a conflation of classical and quantum mechanics, analogous to those papers claiming to derive quantum mechanics from entropy in a vague sense
 
@DanielSank Thought so, I don't really know what would help you
 
@ACuriousMind Perhaps some intuitive understanding of how a sequence of complex numbers can converge to a real!
 
9:17 PM
@DanielSank Well, for example, for any real sequence $a_n \to a$, the complex sequence $a_n + \frac{i}{n}$ also converges to $a$
 
Think about a sequence of complex numbers in the plane starting at z = 0 + 1*i and ending at z = 0 + 0*i, i.e. descending from the point (0,1) in the x-y plane and ending at (0,0) :)
z = (1/n)i
 
@bolbteppa That is an obviously trivial case.
@ACuriousMind Do not understand. How is $a_n + i/n$ a sequence?
What is $a_n$ in that expression?
A sequence usually depends on a discrete index, but I have no idea what $a_n$ is.
 
But what about 3 + (1/n)i?
Or something
 
That's a sequence of complex numbers. I'm asking how a sequence of purely imaginary numbers can converge to a real.
 
a_n = 2^n or whatever, or a_n = 3
What do you mean purely imaginary? z = 0 + a_n i?
 
9:21 PM
@DanielSank I am very sure that a sequence of purely imaginary numbers cannot converge to a real number other than 0
 
Uh, purely imaginary means the real part is zero.
@ACuriousMind I can disprove that statement.
 
Sad... I've been rep-capped today :(
 
(not really, but I can find a physics problem where that appears to be happening)
 
Yeah, and I gave you one, z = 0 + (1/n)i but this is too trivial for some reason, throw any sequence whose limit is zero into that and it's the exact same, z = 0 + a_n i
Same geometric picture too
 
@bolbteppa It's too trivial because it goes to zero. Come on, man.
Take a look at this
The impedance of a finite LC ladder is purely imaginary.
However, what happens if I make it infinitely long?
Then I case use induction to show that the impedance is purely real!
 
9:24 PM
Ah, well that's an artifact of the mathematics right? The use of complex numbers here is just a convenience is it not?
 
This requires assuming that the inductance and capacitance of each section goes to zero while the number of elements in the ladder goes to infinite in a particular way, but it does work out to be real.
@bolbteppa "Ah, well that's an artifact of the mathematics right?" Uh, yeah, but that's exactly what I'm asking about.
 
Hehe already sending this to infinity throws away all intuition :p
 
I don't understand why people talk about "artifacts of mathematics" as if they're to be ignored.
@bolbteppa Indeed. But that is very unsatisfactory, don't you think? I'm not going to just throw my hands up and say I can't possibly understand something because it's not obviously intuitive!
 
@DanielSank I'm not very good with circuits. Are you sure you are taking a proper mathematical limit over purely imaginary numbers that turns out to be non-zero real?
 
@ACuriousMind Of course not :)
That's what I'm trying to understand.
 
9:26 PM
So, how would the impedance of the finite ladder with $N$ steps look like?
 
It's kind of a mess.
But that's an interesting question.
I might be able to figure that out via induction.
 
Well you are talking about a classical theory, there are explicitly no complex numbers in that theory, it's just a mathematical tool so if you are getting a non-intuitive answer that may be the explanation why, I'm not saying it is but that might be why if it does occur :) I'm no good with circuits either :(
 
@DanielSank I feared that :D But how do you know it is purely imaginary, then?
 
@ACuriousMind That much you can tell easily. The impedance of an inductor and capacitor are both purely imaginary. When you combine them in series or parallel you're just adding and dividing, so no real part can emerge.
 
You are right about the intuition point :)
 
9:29 PM
@bolbteppa "Well you are talking about a classical theory, there are explicitly no complex numbers in that theory" Huh? Complex numbers, hell any numbers are just things we made up. How are they any more or less "in" the classical theory than in e.g. quantum theory? I can do quantum mechanics with imaginary numbers too you know. They're just convenient.
 
@DanielSank Why would dividing exclude a real part? 4i divided by i is 4!
 
Ok let me be more precise.
When you combine two circuit elements you're doing one of two things:
1. You're adding them: Z1 + Z2
2. You're combining them like this: Z1 Z2 / (Z1 + Z2)
If Z1 and Z2 are imaginary (which they are for inductors and capacitors) these expressions are both purely imaginary.
 
I don't agree with any of that but I wont fight hehe
 
@bolbteppa When I said " I can do quantum mechanics with imaginary numbers too you know" I meant "without"
 
@DanielSank Okay, I see. What's the argument for the infinite ladder to have real non-zero impedance?
 
9:32 PM
Ok, let's do this:
The impedance of the ladder is the impedance of the first section plus the impedance of the rest.
Since the impedance of the rest is still an infinite ladder you get:
$Z = Z_{\text{first section}} + Z$
This is obviously a very weird equation :)
 
Yes, yes it is.
Are we sure this thing has finite impedance, by the way?
 
@Qmechanic Thank you for that link, I read all the jokes :)
 
@ACuriousMind Yes.
 
@DanielSank But...the infinite ladder is just the infinite sum of the sections with one LC, yes?
 
What's going on here is that the recurrence time is going to infinity as you add more sections to the ladder. In the infinite limit the recurrence time is infinity, so any energy you shoot in never comes back. That's why it appears to have a purely real impedance (like a resistor, which dissipates energy and never gives it back).
@ACuriousMind Yes. What you do is you take the limit of infinite sections but where L and C go to zero at the same time. I realize this is a bunch of random weird hand-waving but you can make it precise. similar arguments are used in QFT where you say the vacuum is approximated by an infinitely large box.
 
9:37 PM
 
In the QFT case you have a 1/volume normalization factor out front. That makes the density of states remain finite even as the size of the box goes to infinite.
The are all related concepts.
 
@DanielSank Yeah, I know the kind of argument. So, you are not actually able to write this limit down, but based on the above, we would conclude it is a limit of imaginary numbers with a real limit, yes?
 
@ACuriousMind I think I can. Why do you say not?
"we would conclude it is a limit of imaginary numbers with a real limit, yes?" Sure seems that way.
 
@DanielSank Because you told me that you don't know the impedance of a ladder with N steps yet
 
Ah, yes indeed. I should do what you said and write down the result for N sections.
That could be very enlightening.
Thank you. This is a good direction. It's raining outside so this is a good way to relax :D
 
9:43 PM
Won't you end up with pretty horrible nested fractions (or whatever you call them), though?
For the impedance that is.
 
@alarge: I think it should be possible to use induction. I have no idea if that will give a nice expression though :)
 
So I have a feeling this might go the 1+2+3+4+... = -1/12 route
 
@alarge I always loved that...
 
Woah that's really interesting
 
Everything I've ever read about Ramanujan makes my head hurt.
 
9:45 PM
@alarge Actually, you're more right than you may realize.
 
@alarge I've got that, too. This limit has to be something not straightforward for the heuristic physical argument to describe an impossible limit.
 
I posted some questions on divergent series and got awesome answers, Kline's book on the history of math has a chapter on divergent series that makes complete sense of them too
 
On Physice.SE?
 
@bolbteppa There's nothing quantum mechanical about a metric, so I don't quite see what you're trying to say.
 
9:50 PM
Free energy is a quantum mechanical concept, equilibrium stat mech derived from a multi-body Schrodinger equation in energy representation, you're saying a quantum mechanical function is a function of a geometry and differentiating w.r.t. it (and it's not a path integral you're differentiating), maybe it's my naivety but I wouldn't trust it :)
 
@bolbteppa Gibbs free energy as well as Helmholtz free energy is a purely classical statistical concept
 
Well, you're going to have to dfferentiate wrt strain. And looking at these as coordinate transfomations, you might as well differentiate wrt metric.
 
Yeah I get you, I'd like to do what you're trying to do haha, but if you want to do that you pretty much want to set up a Lagrangian based statistical mechanics first and I don't think that's gotten anywhere :(
 
If you plug in for the free energy any of the classic free energy functionals, like the one by van der Waals, and then differentiate wrt metric, you'll get the classic result for stress. You can also show that if you restrict the variations to be Cartesian, you end up with a free divergence term, like if you were to just define force as the divergence of stress.
 
@ACuriousMind they are quantum mechanical concepts, classical statistical mechanics does not make sense if you're pedantic researchgate.net/post/…
 
10:01 PM
@bolbteppa I have no idea how what you linked would say that classical StatMech doesn't make sense
 
"There is no way around that a state in phase space needs to occupy a finite volume. There is no meaningful limit to let this volume go to zero. Therefore, strictly speaking, there is no way to do truly classical statistical mechanics."
I never said it didn't make sense, I just said they were quantum concepts
 
@bolbteppa "classical statistical mechanics does not make sense if you're pedantic"
@bolbteppa That $\hbar \to 0$ is for a constant not physically possible is not a deep insight. The limit is meant to represent that, in classical physics, every action/phase space volume that occurs is so big that it is easier to treat it as a continuum.
Also, the classical microcanonical ensemble indeed has a "minimal phase space volume" that isn't set to zero, and yet it is fully classical - it knows nothing of quantum physics.
 
"How are electric charges dealt with in an FDTD simulation?
I know how to calculate the curl of E and B fields to model propagating waves
But what about the effect of charges at certain points in the grid on the E field"

I think I have an idea
So basically the way to create charges is by pumping current from one place to another, as @alarge
So that will create one region of negative charge and one region of positive charge
Charges can essentially only be created in pairs and from the vacuum
Unless one specifies the initial conditions to mimic the 1/r^2 field produced by a net charge
and then take it from there
 
10:18 PM
@user1667423 Umm... I think you should be able to just pump a current, like if you were simulating an antenna.
 
Yeah, but I meant if I'm trying to simulate the presence of a single charge
What you said makes sense
A somewhat strange way of thinking about it
 
I know. But you should be able to just pump a current without anything canceling it out.
 
That's not what's happening in my simulation though :/
 
You mentioned something to the effect that you have to have a charge to cancel things out.
 
It'll always create a region of negative charge as well
By the very nature of Maxwell's equations I suppose
"You mentioned something to the effect that you have to have a charge to cancel things out."

Yeah it seems like it
Unless you set the initial E field condition to that produced by an electrostatic charge
It's a weird way of thinking about it
 
user54412
10:21 PM
@DanielSank Taking everything you've said so far about the LC ladder (imaginary finite-segment impedances, real infinite impedance, the idea that energy never returns in the infinite case), would it be fair to say the problem comes from switching two limits?
 
basically it seems we can only deal with charge through the electric field itself
I guess one could pump in current from "infinity"
 
I don't have any of my codes left. But I'm quite sure that by pumping stuff you should be able to reproduce the electrostatic solutions.
 
Basically make a very long current path and the charges that form will appear to be isolated
^ I think only if you start with a field that already has divergence != 0
I think the curl equations preserve divergence
Hence one must start with some divergence (positive or negative) from the very beginning
 
Yeah, you'll have to have divergence to generate charge, yes.
 
So the initial conditions of the E field specify the total charge contained by all of space
 
10:23 PM
Is that a problem in the simulation? Again, I used something not quite FDTD, but similar, and even those details I've forgotten.
 
I don't think it's a problem now that I understand it
 
user54412
@DanielSank On the one hand you found the steady state for finite ladders, then took the ladder to infinity: lim_{n->\infty} \lim_{t->\infty} <something> = lim_{n->\infty} <imaginary>. On the other hand you constructed an infinite ladder and found that at any finite time there was no energy returned: lim_{t->\infty} \lim_{n->\infty} <something> = lim_{t->\infty} <real>.
 
It's somewhat unintuitive
Initially I thought one could simply set the E field around charges in the FDTD grid
And somehow that would propagate to form the 1/r^2 field
 
@user1667423 Those sound exactly like my thoughts back when.
I still think that pumping stuff is probably the wrong way to go about it, but I never went into the literature to see what you should actually do.
 
I think it's inevitable, given that the curl equations preserve divergence
There's no way to simply have net charge pop into the simulation
Unless you pretend this happens by restricting yourself to a region and pumping in current from another region far away
Also
I just realized that pumping in current through a line through the FDTD grid from the boundary seems to mimic the effect of current coming from nowhere
but then again I haven't implemented proper boundary conditions (only Dirichlet, not PML or anything)
 
10:30 PM
What happens to the equations/simulation if you do have positive divergence? I don't think I had a problem and think that I could just push charge into the simulation. It's just if I set it to have some in the beginning, the E field would be as if it had none.
 
I'll try to set it to the 1/r^2 field as an initial condition
See what happens
 
@ACuriousMind classical statistical mechanics only works because of that finite volume assumption, it's constructed assuming classical ideas & just luckily works, for those functions to be actually classical you literally have to allow the volume go to zero (h ---> 0) conceptually the issue of class Vs. quant arises in justifying that the log of the density matrix is additive, once you get that they are the same for indistinguishable p's but it's a cheat to say classical stat mech makes sense
 
@ChrisWhite Probably, yes. But I only say that because whenever things like this come up it's always from switching the order of two limits :)
 
@ACuriousMind Good to hear
 
@alarge I really like that result but how do you make sense of it in a general physics context? I've looked so hard to find lagrangians for statistical mechanics and for elasticity and fluid dynamics, indeed I think I have a question up about this very issue, but I am more and more convinced the question doesn't make sense :(
 
10:35 PM
@ChrisWhite I don't quite see why this is the case but I think you're onto something.
 
Nice!
The initial 1/r^2 field is stable
which is obvious of course but it's nice to see it works out in the simulation
Now I could "transport" the charge by changing the current field appropriately
 
user54412
@DanielSank Which claim doesn't make sense, the n,t limit for the finite case or the t,n limit for the infinite case? Or both?
 
user54412
I'm really approaching this whole problem with vague intuition rather than rigor.
 
@ChrisWhite: I think what I did in my analysis was first take the limit $n \rightarrow \infty$ and then the limit $L,C \rightarrow 0$.
Since $L,C \rightarrow 0$ raises the resonance frequency of the modes in the finite ladder, I think that's the same thing as taking $t \rightarrow \infty$, right?
So what I did was $\lim_{t \rightarrow \infty} \lim_{n \rightarrow \infty}$.
I don't see in what way my analysis considers the reverse order though.
 
user54412
I'm saying (guessing, hoping) that secretly in the finite case there is a $t \to \infty$ limit being evaluated that turns the time-dependent response into a steady-state response, the latter of which happens to be purely imaginary.
 
10:45 PM
@bolbteppa: I must say that what you say does not make full sense to me. Where did you get these ideas from?
 
@ACuriousMind Landau & Lifshitz Statistical Physics :)
 
L&L say that classical StatMech is ill-founded?!
Are you sure that this is not just their reasoning why classical StatMech (just like all other classical physics) does not work anymore as soon as we know the world is quantum?
 
@alarge I find it interesting you differentiated the Helmholtz free energy, expressed as an integral, with respect to an arbitrary metric, and got the right result, Landau is actually doing something very similar
 
@ChrisWhite Oh snap! You're right! When you use impedances you're working with the Fourier transform of the time response function.
 
user54412
I would certainly think a world of Newtonian billiard balls has well-defined statistical mechanics
 
10:49 PM
Ho Lee Schit @ChrisWhite you da man!
 
user54412
@DanielSank Welcome praise after a week of "you broke the code" :)
 
@bolbteppa From what I remember, in Elasticity it's wrt strain (but as I mentioned earlier, you can just switch this to a metric and some extra terms)
 
@ChrisWhite No really, that was a fantastic insightful "guess" you just made there.
Why do these magic mathematically pathological but physically correct results always come from switching the order of two limits?
 
user54412
@DanielSank And to think we're often told in physics "don't worry about swapping the sum and the integral; it always works"
 
@ChrisWhite: This is really fascinating. It's like when you try to understand Fourier transforms in a mathematically rigorous setting. You have to regularize the integrals in order to get some limits to swap. This must all be related.
 
10:55 PM
0
Q: How can I improve a specific question which has been closed for being "homework-like, and not showing enough effort"?

user89The question in question: What is the minimum amount of heat energy an must an object be for it to be detectable from a light year away?' It was not a homework question, and I do not know enough physics to be able to make it more descriptive/specific. Can you help?

 
Similarly, if you put a little bit of dissipation in the LC ladder, then I bet you anything you can do the limit swap without having to wave your hands. And in that case the fact that the result is real in the end isn't surprising.
 
@alarge yeah, so you are focusing on functions of x_i terms, they are focusing on functions of dx_i = x_i' - x_i terms. They justify that the free energy depends on the strain tensor, and as a side consequence justify that dF/du is the strain, you are assuming free energy depends on the strain and the geometry of an object, I'll say it looks cool because they derive free energy with a constant F_0 term that is free energy of an undisturbed object, you derive the value of F_0 and all changes
 
...because you get some dissipation by putting in some resistance, which has real impedance.
Note that dissipation also means that the time response functions decay exponentially rather than living forever as $t \rightarrow \infty$. This is all related, I'm sure of it.
 
user54412
@DanielSank This whole exercise sounds like it would make for a good problem set in some sort of "mathematical methods of physics" course.
 
@ACuriousMind no they don't say that, but it's obvious that any classical theory of volume elements in phase space must hold when that volume shrinks to zero, the whole idea of stat mech in classical phase space is that enlarging that volume away from the ---> 0 limit is why we get statistical mechanics, and that it theoretically should give exact results in that limit, it is built on the presupposition of the classical equations of motion in the limit, so conceptually that volume is zero
For all intents and purposes that volume element is zero theoretically, it's just a mathematical trick when we make it non-zero
 
11:01 PM
@ChrisWhite Yes. Indeed.
 
It's just a game of maths, it's not actual physics, the reason it's called physics is because it gives quantum mechanical results with an entirely different conceptual justification, obviously both are linked because of Hamiltonian mechanics, so yes it really is a cheat to call free energy a classical concept
You are gonna end up saying Free energy is a classical mechanics concept if you apply it to elasticity and think of elasticity as a theory of classical mechanics
 
@bolbteppa But this does only mean that classical StatMech has a problem if the limit $\hbar\to 0$ has a problem, which it only has if there is a natural scale for $\hbar$, which again, is only there if the world is quantum. As @ChrisWhite said, classical StatMech of classical objects should be perfectly fine.
 
But that limit does have a problem, you theoretically have to send that limit to zero at any point, if you define free energy and it depends on $\hbar$ then theoretically sending it to zero gives an exact free energy function that conceptually exists in classical mechanics, you're telling me it exists but will go to zero, i.e. that magical energy functions are real physical quantities that spring up when particles enter complex patterns, but the principle of least action knows nothing about this
@alarge I really like the idea that you are defining elasticity in terms of x_i coordinates and landau is doing it with dx_i coordinates, but you have to tell me a way to get from normal statistical mechanical free energy to that crazy integral over surface area :) It's probably possible, but you might derive the result you want to get on the way already if you get me :)
 
@bolbteppa But the beauty is that the (Gibbs) free energy does not depend on the phase space volume, because it depends only on entropy, and the entropy doees not depend on the $\hbar$!
 
@ACuriousMind : venerable gentleman! In the explanation that you have me you placed a footnote, but it is with so small letters that it's impossible to see what's written there. I like your explanation and I try to read it attentively. So, won't you care to move it in the text, where the letters have normal size? (All my thanks and warm regards!)
 
11:12 PM
If the choice of the $\hbar$ in classical StatMech had an impact on any physical parameter, people would have found that problem long before quantum mechanics arrived
@Sofia Oh, sorry, I've read your comment but was then distracted. I'll do it immediately
@Sofia Done
 
@bolbteppa I'm not exactly sure what you mean.
 
@ACuriousMind entropy does depend on $\hbar$, $S = \ln(\frac{dpdq}{(2\pi \hbar)^s})$, landau stat phys vol. 5 equation 7.8, you can make it look like it doesn't, but it's equivalent to this dependence, as Landau discusses, all this sync's with the general point that these statistical mechanical functions should go to zero when the volume element shrinks and we should get back exact equations of motion, Gibbs/Helmholtz etc... are not fundamental PoLA functions, they can't be, it's all just math
 
@bolbteppa $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$, so the phase space volume shifts entropy only by a constant factor, it's unphysical.
 
@ACuriousMind yeah Landau discusses that point
 
It seems I should go and read Landau, but I am quite sure that your statements here are far stronger than what the actual reasoning warrants.
 
user54412
11:26 PM
Random thing I found while searching for images to include in an answer:
 
user54412
NASA has a bunch of time lapse maps of the Earth, including one for fires on the ground
3
 
user54412
Turns out large swaths of the planet are covered in fires at certain times of the year for harvesting crops
 
@ACuriousMind no Landau discusses how the very notion of entropy is ambiguous unless we interpret S quantum mechanically, only then is S dimensionless, otherwise S has dimensions of logarithm of action and is an explicitly ambiguous quantity (scaling an action is equivalent to changing units so the form of entropy is unit dependent), the ambiguity off the entropy explicitly illustrates the frivolous nature of these magical functions, they don't make sense classically
@alarge I'm saying there is a function called Helmholtz free energy derived from quantum statistical mechanics, it looks a certain way, you've got a different definition of Helmholtz free energy as an integral over the strain and it involves a metric etc... I just wonder how you derived this expression, Landau gives a derivation using dx_i = x_i' - x_i not a metric, i.e. changes in classical geometry
 
I had to go to my bookshelf to pick up Landau. Eq. (3.1) is what I wrote in my question as well, is it not?
 
The de_mn = dg_mn the e_mn is defined with respect to u_i = x_i' - x_i terms, g_mn is defined in terms of x_i 's right?
 
11:41 PM
It's not, there's an extra term (thus the 2)
 
Are you basically doing what Landau is doing :)
 

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