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12:01 AM
is someone good at explaining entropy here?
specifically is there a relationship between work, entropy and position?
12:26 AM
Well entropy is a meassure of "chaos".
Consider a volume of gas in a box, if the particles go faster and have more space to move, thus the system have bigger entropy.. you can check also mathematical definitions of entropy and Boltzmann relationship
I wouldn't use chaos to describe entropy because chaos theory is its own subject. Entropy defined as a state function of a thermodynamic system in classical mechanics can depend on a number of variables
In statistical mechanics it's a measure of the amount of states a system can have.. I don't have that deep of an understanding but if wiki doesn't satisfy you I'm sure someone here can answer @geocalc33
you might find what you're looking for in the case of the carnot cycle en.wikipedia.org/wiki/Carnot_cycle
Boltzmann entropy is $S=k\log(W)$
I would like to start here?
I don't understand $W$ at all
I think $W$ is the number of possible microstates of a system.
so $W$ is a scalar?
Yes, it's a natural number.
12:40 AM
This relationship you wrote, is written on his grave :) Just an interesting fact. Another interesting fact would be both he and his student commited sucide.
Wow that's quite sad..
I'm going to brush up on thermo & stat mech tomorrow in his honor :)
@geocalc33 do you know what $W$ is
the number of micro states for a gas's macro state?
Do you vaguely know what a fermion is, vaguely know what a quantum state is
12:53 AM
What is a quantum state
Here's an example. Consider $n = \sum_i n_i$ identical fermions, meaning not more than one particle can occupy a state. Lets say they are hydrogen atom electrons with the discrete hydrogen atom energy spectrum $E_m = - 1/m^2$. Lets say $n_i$ particles are in the energy level $E_i$. Thus the total energy is $E = \sum_i n_i E_i$.
We can treat the particles with energy $E_i$ as a subsystem of the total system. The $i$'th energy level has degeneracy $g_i = 2 i^2$, meaning there are $2i^2$ wave functions with the same energy $E_i = - 1/i^2$. In other words, the $n_i$ particles with energy $E_i$ can take any of the $g_i = 2 i^2 $ states with that energy, but each of the $n_i$ particles can only occupy one state each, we can't have more than one in the same state. This means that we must have $n_i \leq g_i$.
The question is, how many way can we put $n_i$ particles into $g_i$ states (think of them as boxes) such that each box has at most one particle, and further the order in which we put them in doesn't matter. The first particle has $g_i$ possible choices, the second particle has $g_i - 1$ possible boxes, the third has $g_i - 2$ etc... so that we can have $g_i (g_i - 1) ... (g_i - n_i + 1) = \frac{g_i}{n_i!}$ ways to put $n_i$ particles into $g_i$ boxes.
But this assumes the order matters, we overcounted by $n_i!$ permutations. In other words, we get $W_i = {g_i \choose n_i} = \frac{g_i!}{n_i! (g_i - n_i)!}$ total ways of putting $n_i$ particles into $g_i$ boxes (states) where the particles are identical and the order we put them in doesn't matter. This is a measure of the 'disorder' of the system, it quantifies how many different ways can produce the same macroscopic state with energy $n_i E_i$, the total energy of the $i$'th subsystem.
Further when we include another subsystem, say the $j$'th subsystem, we get $W_i W_j = \frac{g_i!}{n_i! (g_i - n_i)!} \frac{g_j!}{n_j! (g_j - n_j)!}$ as a measure of the disorder, i.e. it's a multiplicative measure of the disorder of the system (like probability is multiplicative).
The entropy is an additive 'measure' of this, since things like energy etc... are additive over independent subsystems. To turn a multiplicative quantity into an additive quantity we can take a logarithm, hence $S = \ln(W_i W_j) = \ln(W_i) + \ln(W_j) = S_i + S_j$.
It's sometimes written as $S = k \ln W$ where $W = \Pi_i W_i$ for the subsystems with $W_i$'s and $k$ is a constant used to change degrees to kelvins and vice versa when we define temperature in terms off entropy
@RyanUnger if you want to try make sense of the above, $E_n = - 1/n^2$ is the discrete hydrogen atom spectrum, and $2n^2$ is the $2n^2$ different wave functions with that energy (i.e. diffferent wave functions due tot different values of $l,m,s$ in $(n,l,m,s)$ for $s$ spin, $l$ total angular momentum, $m$ the $z$ component etc...) so the states with energy $E_n$ are the different wave functions
can entropy be multiplicative?
1:09 AM
A logarithm acts as $\ln(ab) = \ln(a) + \ln(b)$ which is additive not multiplicative
okay got it
That's the setup for deriving the 'Fermi distribution' which reduces to the 'Boltzmann distribution' in a certain limit
that was actually really helpful even if I only understood some of it
That might be more confusing if you haven't done some qm
I have done some (very basic qm)
it was thermo-fluid-qm fit into one semester
qm part was mainly looking at schrodinger's wave function
oh yes I remember that sort of problem :)
Hello guys, i have this small issue i am facing right now.. consider a two body problem where they interact with eachother using a central potential.. the relative energy is then given as $ 0.5 \mu \dot r^2 + V(r) $ where as $ r= (r_1-r_2 )^2 $ and $\dot r = \dot r_1 - \dot r_2 $ .. where does the energy have a minmum..? only by diriving in relation to the distance r or do we need to also consider the derivation of $ \dot r $ since it is a function of two variables.
4 hours later…
4:58 AM
@MadSpaces Presumably you mean the total energy is $E = 0.5 \mu \dot r^2 + V(r)$ since that's obviously the sum of the PE + KE. But the total energy is normally constant so there is no minimum.
3 hours later…
8:01 AM
A bit of a soft question: when would you say quantum computing caught up with classical computing (in specific tasks biased in favor of QC)? I'm pretty sure it was earlier than Google's work claiming supremacy.
2 hours later…
9:59 AM
Mar 25 '16 at 19:58, by ACuriousMind
Reminds me of this quote: "Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously."
A couple of us suggested that Anna change her thousands separator, but she didn't take the hint. I'd edit it myself, but it feels a bit rude. What to do... physics.stackexchange.com/questions/667000/…
10:30 AM
@PM2Ring stylistic choice, not worth editing for imo
@ACuriousMind Ok, but as Eric points out, she's being inconsistent by using . as both the thousands separator and the decimal point in the same post.
10:49 AM
Someone should post the info to MSE for Physics.SE regarding unpinning the accepted answer. (The deadline passed a couple of days ago). The consensus looks like we're mostly in favour of unpinning. I'm happy to post it, but I think it would look more official for a diamond mod to do it. Of course, that MSE question is primarily asking us what we think the default should be across the network, but the devs will also use that post to learn what each site wants for its own setting.
11:45 AM
@PM2Ring yeah, I get that, I still don't think it's really worth an edit because while inconsistent it's not actually ambiguous - in the place where it's used as a thousand separator interpreting as a decimal point makes no sense and vice versa
@PM2Ring I made an edit.
12:03 PM
@ACuriousMind Fair enough. That's basically why I didn't edit it when I originally noticed it.
@ACuriousMind Thanks.
also thousand separators and decimal separators are culture dependant
it's fine
it's usually clear from context
If we say ohm's law is empirical
what is the actually logic behind the phenomena
what do you mean by "logic"
You can derive it from the usual conduction models in some limit, if that is what you want to know
I just wanted to derive it "logically"
do you mean from first principles?
You can look up the derivation from the Drude model, for instance
12:17 PM
Just now I was going through it
@DanielUnderwood the classic one is en.wikipedia.org/wiki/Anscombe%27s_quartet
but no t-rex there means it isn't as interesting
I only like datasets that make dinosaurs
you should have worked in paleontology
12:44 PM
that might have been a short career for me
"what do you mean I can't turn the fossils into a robot?"
if you wanted dinosaur robots you should have become an imagineer
1:00 PM
So, this is confusing
I’m reading a paper, and it cites a 1991 AmJPhys paper
But as far as I can tell, this paper does not exist
Like, you go to the listed issue/page and it’s just…not there
And multiple other papers also have that citation
“ H. Urbanke, American Journal of Physics, 59 53 (1991).”, where art thou
1:16 PM
I guess someone went back and upvoted enough old posts of mine to get me up to 50k reputation
Figured it out; the citations should have been to page 503, not 53
got it
Rather silly that multiple papers seem to have copy/pasted that :P
Haha yeah for real
1:41 PM
How is power consumed by a non ohmic material have a relation with its total circuit resistance ?
@Semiclassical it happens all the time, even with more important information than the page number (authors' names, years etc)
How do you define resistance for a non-ohmic material in the first place? @Ishwaran
@fqq yeah, I’ve probably done it myself at some point
most people copy the references from their own previous papers (if they have cited it before) or from other papers (especially if they do not read the paper they are citing, which is very often the case) so errors propagate
I guess the editing process should fix that, but it's not always efficient at it
If one defines resistance as voltage/current, then the power dissipated is still current * voltage = current^2 * resistance = voltage^2 / resistance
But now the resistance will itself be a function of current
@Semiclassical I typed the question incorrectly @Semiclassical, its ohmic.... Today my teacher solved a problem. A ohmic circuit consumes 100w when 220v is passed... He calculated the amount of power consumed when 110v is passed in terms of Voltage and Resistance (i.e V^2/R) so how is Power related to its circuit resistance
1:53 PM
Q: Bekenstein bound and "dimensionality" of information (entropy)

Robin SaundersThe Bekenstein bound says that the maximum entropy which can be contained in a (spherical) volume of space with a given amount of energy is proportional to the amount of energy multiplied by the "length scale factor" (radius) of the volume - up to some dimensionless constant, it is just the afore...

this person says entropy is dimensionless?
from what I know the units are usually J/K
@Ishwaran um. it’s related in the way you just said
@Semiclassical you mean by V^2/R
Yes. V = I R so P = I V = I^2 R = V^2/R
One expression for power can be more convenient than another depending on the information given, but they’re all saying the same thing
(And even if you don’t have an ohmic material one still has P=IV)
@Semiclassical yes, I can understand that Power is inversely proportional when v is constant , and directly when I is constant, but from what you said, it all depends on the consistency of V and I ?
2:06 PM
Remember your dollar signs :p
@Semiclassical Remember that not all resources are digitized
@Semiclassical "Power is inversely proportional when v is constant and directly when I is constant"
Ohm's law is also wrong
But it is true in many situations
@Slereah If a law is not universal.. isnt it wrong ?
Plenty of laws aren't universal
Most of them in fact
universal laws are usually pretty unwieldy
2:10 PM
Ohm's law is correct if the current is small
It’s a law for ideal resistors
But there’s strictly speaking no such thing as an ideal resistor
@Semiclassical What is a ideal resistor.
It resists ideally
anyways... its empirical ryt
2:11 PM
One which satisfies Ohm’s law for all currents
@Semiclassical bit circular
It’s like saying that a circle’s circumference is exactly 2pi*R. It’s true for a mathematical circle, but a mathematical circle doesn’t exist in nature
Ohm's law is true in the cases where it is true
@Slereah “it’s shaped like itself”
@Slereah You mean it resists "exactly" to its ratings ?
@Slereah What are the true cases of ohms law ?
2:14 PM
The point, as Slereah noted above, is that every material will stop being ohmic if you put enough current through it
Ohm's law is the limit where... drift velocity is small?
@Semiclassical so its accurate enough only to some extent ?
I forget
what happens for Ohm's law
Not even a theoretical limit, really
@Slereah depends on the material probably
I had a coworker who used to work for power plant companies
Ohm's law doesn't apply when you deal with big transformers with 100kV going through them
2:16 PM
And it doesn’t really work at all for a small lightbulb
Unless you use very small currents I guess
I think the differential resistance tends to be positive for most materials but not all
@Slereah I remember i saw some ohms law dealing with integrals, is that the exact thing ?
@Ishwaran You have to use the Maxwell equations when doing Ohm's law
It will usually involve some integrals yes
usually flux integral across a wire
Ohms law in that case is the statement that the volume current density is proportional to the electric field
@Slereah sighs
Which is perfectly fine as a “guess” for how the current flow is driven by the electric field
But it’s still not universal in the way that Maxwell’s equations are
2:21 PM
a good counter example of Ohm's law is a diode
Diodes will not react linearly to potentials
otherwise they would allow currents both ways
@Semiclassical Maxwell's equations are exact ?
They don’t even respond symmetrically to voltages , yeah
@Ishwaran more or less. It’s a bit subtle b/c Maxwell’s equations describe classical electromagnetism
I remember my seniors saying it is also not an example of exact formulas
@Semiclassical nice
2:23 PM
We probably don't know any exact formulas
So if you want to address quantum systems then Maxwell is no longer applicable
Otherwise we would know the Grand Unified Theory
Quantum electrodynamics works differently
Maxwell is perfectly exact as a description of what it means to do classical E&M
It’s just that the world isn’t actually classical :P
So you're saying the Maxwell equation is true if it was true
2:26 PM
What is i^.i^ and i^.-i^
Classical electromagnetism is a weird beast
Especially if you go to light speed
I guess there is a bit of difficulty saying in what sense Maxwell’s equations are more ‘universal’ than Ohm’s law
@Semiclassical quantum systems ? :|
I’d defend it on the grounds that you can consider classical systems which obey Maxwell but not Ohms law
@Ishwaran yep
Eg how do you handle the fact that an electron acts like a microscopic magnetic dipole
I even dont know a bit abt quantum mechanics sighs
2:30 PM
i^. -i^= 1.-1.cos180
=1 ?
@cOnnectOrTR12 expressions like i^i tend to be ambiguous. For instance, one can write i=exp(i*pi/2). In that case, seemingly i^i=exp(-pi/2).
@cOnnectOrTR12 I don't know what you think you're writing there, but that's just punctuation salad and not a formula
But also i = exp(i* 5pi/2)
So seemingly i^i also should equal exp(-5pi/2), and so forth
@Ishwaran science/physics is not about "exact formulas", it's about formulas that are useful in modelling reality
@ACuriousMind I am doing a problem on finding net flux through a cube. Field direction is i^ and area is -i^. What is the flux
2:35 PM
I have no idea what "i^" or "-i^." is supposed to mean
@ACuriousMind you mean.. accurate enough for our reality ?
"minus i to the power of dot" doesn't mean anything
@ACuriousMind i cap and minus i cap
Evidently you mean that the field is in the +x direction while the surface is oriented in the -x direction
2:36 PM
@Ishwaran I mean that almost every formula you derive from physics relies on some idealization, some assumption that will rarely be 100% true in reality
but if the error between the easy idealized formula and the impossible-to-compute "real" formula is tolerable for whatever application you have in mind, that doesn't matter
What is dot product of i cap and -i cap ?
@ACuriousMind So we cannot "actually" derive "exact" formulas in physics like we do in mathematics ?
@Ishwaran what does "exact" mean?
2:38 PM
Can you draw a mathematical circle with a pencil?
@ACuriousMind That is 100% accurate
@ACuriousMind 1 ?
@Semiclassical even a computer cant :|
Right. For one, pencils dont make infinitely thin lines
mathematics is not concerned with the real world - it starts from axioms and derives logically sound conclusions from it, but when you try to apply it to reality (as in physics) you need to start mapping the abstract mathematical numbers/objects/whatever to something "real"
2:41 PM
Nevertheless we do use formulas for a circle all the time . Just because a mathematical circle doesn’t exist in nature, doesn’t mean it’s a useless concepu
@Ishwaran all the laws are provisional.
@ACuriousMind is it 1 or -1
@cOnnectOrTR12 I see
@cOnnectOrTR12 I still have no idea what you're asking. Please stop pinging me.
@Ishwaran but "100% accurate" - how can you tell when something is that?
@ACuriousMind Is that ?
what actual difference does is make whether something is accurate within your margins of error (and you always have margins of error in the real world!) or "100% accurate"?
how do you know that a law that held true yesterday will hold true tomorrow?
all "laws" are extrapolation from finite data
2:46 PM
So the formulas evolve towards accuracy by the time passes ?
Ceci n'est pas une pipe
I would suggest before asking whether specific laws of physics are "exact" or "true" that you figure out what exactly it means to call something "true" or "exact"
Even a single observation can disprove a theory. And then we need to make amendments in it or abandon it completely
That’s a bit debatable. Solitary observations generally don’t mean much
A single observation that disproves a well-tested theory is often very good evidence...that your observation method is flawed :P
@Ishwaran but what does accuracy mean?
The idea of "accuracy" implies that there is a single absolute true value for something, and we're deviating from it
2:49 PM
@Semiclassical Can you tell what is the dot product of i cap and -i cap? I am stuck here
use a map to change coordinates $f:\Bbb R^3 \to M$ where $M$ is the strictly positive part of $\Bbb R^3$ (i.e. x,y,z greater than 0). This map is just a map from $\Bbb R^3$ to another part of $\Bbb R^3.$ Does that mean you can still equip $M$ with the metric from $\Bbb R^3$?
Look up the definition of dot products and basis vectors, and then figure it out yourself
If you understood what you were writing, you’d be able to answer it yourself. Do that first.
I don't fully understand the machinery behind this
@ACuriousMind Gotta use Bayesianism to test the hypothesis
The hard part is deciding the respective probabilities of "Your theory is wrong" versus "Your experimental setup is wrong"
@ACuriousMind truth as a “mirror of reality” is a bit BS, yes
2:53 PM
"one's mind is a reflection of one's room"
I can recollect How Newton's law of gravitation was proved wrong
In practice truth is more a matter of whether something can be relied upon, and to what extent it fails at this
@Ishwaran and still there are many situations where it is perfectly "accurate"
Is Newton’s law of gravitation reliable in all circumstances? Hardly. Is it reliable in many real world problems? oh boy yes
thank you Newton
2:55 PM
no one needs relativity or QM to compute how much fuel the need to get a car up a hill
saying Newton's laws are "wrong" is perhaps technically correct but I don't consider it a very useful statement
@ACuriousMind unless the person telling you the route is a real tricky blighter ;P
@ACuriousMind Lazy man's mantra
Can't you do the relativistic version of the equation
Relativistic gasoline fluid to move the relativistic car
Remember that relativistic aberration may distort your vision of the road
there are extremely many ridiculous and not so ridiculous statements that are blatantly, utterly wrong. The interesting thing about Newton's laws is that it is not like these things. And you might say it's more "accurate" but that's a bit strange because there are some situations where it's accurate to really tiny fractions and others where it's wildly off-base
“You are 1/alpha miles from your destination, as estimated by the most recent CODATA report.”
2:58 PM
what happens when they embed a $\Bbb R^{3,1}$ spacetime into a $\Bbb R^4$
@geocalc33 who's doing that and what does that even mean?
Those are the same manifolds
No need to embed them
Except by the identity
those are just two 4d spacetimes with different metrics (assuming by $\mathbb{R}^4$ you mean Euclidean space), you can't embed them into each other
oh okay
but you can have Riemannian submanifolds embedded in $\Bbb R^{3,1}$?
@ACuriousMind I asked a question In astronomy se, probably a month ago... i remember an answer states newton's law of gravity is accurate enough to launch satellites to mars..
3:00 PM
@geocalc33 How do you embed a Riemannian manifold into another manifold
Embedding is a strictly topological notion
sure, satellites aren't all that fast and Earth/Mars gravity isn't all that strong in the grand scheme of things
No need to define the metric
There’s ways to relate the two manifolds but just “embed one into the other” says far too little
@Slereah not true
@ACuriousMind He said embedding, not isometric embedding :p
3:01 PM
if one talks about embeddings of (pseudo)-Riemannian manifolds, it is often implied that the embedding is an isometry onto its image
That's why you should never imply
No need to be coy
also there is no isometric embedding of Minkowski spacetime into Euclidian space, if that is what you want to know
Because of Sylvester's law or something
Ah yes, the other inertia
@Slereah I wanted to re-creationally embed a foliation of some manifold diffeomorphic to Minkowski (SR) into R^2
yeah I can't do this isometrically
but you can embed the foliation back into R^2 and let it obey the usual euclidean topology/metric
4:07 PM
Q: Will reading Feynman Vol. III and Townsend's "Modern Approach" enable me to (in)validate basic ideas and communicate with physicists?

Andrew CheongSometimes while reading physics news or just taking a walk, I conceive some novel-seeming idea or "picture" about the universe. (Familiar to many here I'm sure.) Of course it's just fantasy—and not knowing enough to reject it. Still, I'm drawn to them, and almost to distraction—it's frustrating t...

1 hour later…
5:14 PM
what is the rotation of a general unit vector $ \nabla \times \vec{e} $?
I suspect it is null, but i do not know how to prove it? Since generally, for any vector $ r $ is the rotation zero
I'm not sure how you're differentiating a unit vector
you can differentiate vector fields, not vectors
i did not deffereintiate anything.
Or do you mean something else ? (deriviation? )
Oh... wait i see what you mean
But why cant we differentiate it .. some unit vectors move with the coordinate system... so they change with time .. or coordinates
ah, another victim of physicists not doing differential geometry properly :P
well, in any case, if your "unit vector" moves, then it's just another vector field $e(x,y,z)$, no?
and you can take its derivatives just like with anything else
there's nothing special about it (and in particular there's nothing guaranteed about its rot or grad)
Oh .. so it does not have to equal zero?
I wouldn't think so, no
5:30 PM
Alright. thanks!
@ACuriousMind Can we create and study gravitational waves here on earth?
Or would the effects of such little mass accelerating not produce measurable waves?
not really the GR expert around here, but the only grav. waves strong enough to detect with current technology come from stuff like black hole mergers
Hi. I have a question about twin paradox. What would happen if one twin jumped onto his spaceship and traveled at 0.9 c for 20 years according to the earth. Then returned back. How would the twins perceive each other ?
I believe you're right, but I don't know the maths to be certain.
Regardless, we have tons of data out there coming in now. Binary neutron star systems & blackholes all over
Exciting times
Ronald I read something like, if two black holes collided 3000 miles from us, the amplitude of the gravitational wave would be in terms of nanometers. The values are not exact but close I suppose. So it is not as we think about it.
5:49 PM
@Xfce4 we have literally this question in like hundred variations on the main site already
Acuriousmind yes but I will ask a chain of question. That is why
I think you mean light years, mate. 3000 miles isn't very far from us ;)
I mean 3000 miles man. It is assumed you are not destroyed by the event.
Oh you're probably right actually, the waves detected by ligo were smaller than 10^-18
and nanometers are 10^-9, but the strain magnitude is inversely proportional to the distance
so it'd probably be smaller than that
6:16 PM
@RonaldVilliers how about you read my amazing answer on the topic
The values I mentioned are not exact but the main point is the amplitude was way less than I (and probably most of us) would expect. So, it would be almost impossible to create them strong enough that we can work on them -by classical methods ofc.
A: Why can't we produce gravitational waves in the lab?

SlereahIn 1973, Grishchuk and Sazhin proposed in their paper, "Emission of gravitational waves by an electromagnetic cavity", a method by which to generate gravitational waves for experiments, using the argument that while the generation would be very weak, it would also not suffer from the decay in $r^...

6:42 PM
Oh that's a neat idea, I think it'd be a very complex setup indeed, you'd need to make up for the mass through acceleration, right?
Thru which I mean amplifying the tidal force by additive interference
Since Grav Rad. travels at c, you'd need multiple distanced cavities though,
It was a nice proposition but I don't know if it's reasonably feasable
4 hours later…
11:02 PM
@geocalc33 the answer in the post you made following our discussion is wrong, of course one can do the 'exponentiation' and although there are subtleties it's roughly a way to get the 'canonical distribution' approach to thermodynamics, see this book

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