@naturallyInconsistent so is bose-einstein condensation not well understood then?
hm well thermodynamics and statistical mechanics is still all hazy to me :P
@naturallyInconsistent the rule that it is reasonable to model a system which is really a canonical ensemble as a grand canonical ensemble, even without taking the thermodynamic limit, so long as you fix the expected number of total particles to be the total number of particles in your system.
hm actually this question has led to quite an interesting search. this letter Fourth Statistical Ensemble for the Bose-Einstein Condensate mentions that (to my understanding) the grand canonical ensemble description of BEC is inconsistent with empirical observations of BEC.
@SillyGoose But this is not correct, nor necessary. If a system is so small in number of treated particles that the thermodynamic limit is not a reasonable approximation, stat therm might well not apply. But yes, there are cases where stat therm applies but the thermodynamic limit is not a reasonable approximation, in which case, yes, it is necessary to be careful which ensemble, and in fact, which choice of entropy function, etc, to use.
There is no global energy conservation in the GR. Thus, maybe TOE could make perpetuum mobile. Maybe there would be a machine, capable to create things from nothing.
For example, it could build a copy of itself, only in the vacuum of the space.
Essentially, it would be a virus.
A virus, conquering the vacuum of the whole Universe.
@peterh It is never too late to begin to speculate new uses when the new physics theories come out and allows us to discuss such speculations meaningfully, but it is often too early to start such speculations before we get to a point whereby such speculations become meaningful.
@naturallyInconsistent I have only extrapolated things we know. :-) What I am not sure: I not sure, if there is no global energy conservation in GE, then a TOE based perpetuum mobile should be possible.
@peterh The standard cosmology that we currently have, we are heading towards the big freeze. While the total energy budget is increasing due to more vacuum to have more vacuum energy, overall the energy is getting more and more spread out, and much more useless (remember, 2nd law of thermodynamics still applies), so there is no mechanism yet for having perpetuum mobile.
@naturallyInconsistent It is not about such escatological things like FLRW and big freeze. It is just TOE
TOE means: "theory of everything". The theory revealing the hidden connection behind all the 4 interactions. Well, already only 3. Thus, we could manipulate gravitation by electromagnetic means
Now here happens the funny thing that there is no global energy conservation in GR.
@PM2Ring Looks it is a really hellish substance. Surprisingly, it is being mixed up with gallium, stabilizing it in delta. But, alloying with neptunium could stabilize it in the more dense alpha-phase
Beside that, neptunium itself is also fissile. And a nuclear bomb needs to dense fissile material as possible. Why it is not alloyed with neptunium? Good question
@peterh I am not sure what do you mean by "Good question" at the end. Have you considered how stupendously expensive neptunium and its refinement in order to alloy it with the already expensively refined plutonium would be?
@naturallyInconsistent I think it is not so expensive. It can be reprocessed from spent nuclear fuel. Yes that is expensibe, but roughly like plutinium.
@peterh Comparing using the prices of the elements on wikipedia, you are correct; it is actually a tenth the price of plutonium. But consider that gallium is in the 3 digits in price and neptunium is above 6 digits in price, one really has to wonder, is the slight increase in density worth the price?
@peterh oh, the wiki page for allotropes of plutonium gives the answer. Gallium stablises plutonium in the least dense delta phase, makes it malleable for machining and ease of making the weapons. During the detonation, the plutonium-gallium alloy quickly turns into the densest alpha phase, making the sudden transition from sub-critical to over-critical easy. Neptunium stablises the densest alpha phase, making the above good property unavailable.
@peterh As naturallyInconsistent said, for a fission core we want to stabilise the least dense phase, to maximise the change in density when the core is compressed.
@user85795 A hollow pit is even more efficient, but requires much higher precision in the conventional explosive lenses that are used to compress the pit. See en.wikipedia.org/wiki/Pit_(nuclear_weapon)
@PM2Ring I dont think either of you are correct. You arent even considering similar things. You are using geodesic deviation equation, which is correct if you want to get the tidal acceleration, but gs was trying to argue that, without the 2nd particle in a pair to measure the tidal acceleration from, the one particle ought to feel an acceleration of that form. That is the wrong form, since one ought to derive from the Lagrangian / Hamiltonian, but you aren't even talking about the samething.
In the end, though, it seems like gs used linear approximation, in which case both answers agree
@naturallyInconsistent We're talking about the tidal acceleration on a body in (radial) freefall. Of course, as well as the radial stretching, a sphere will be compressed into an ellipsoid of the same volume. But at this stage, I just want the radial stretching. I'm not interested in a "I'm right, you're wrong" contest. I just want the correct equation.
@PM2Ring You using the geodesic deviation equation is much more likely to be correct, as I said earlier. His argument starts from a wrong form of the acceleration. He should be using either the Euler-Lagrange or Hamilton's equations to get the accelerations correctly, rather than guesstimating from the metric's terms.
@naturallyInconsistent Ok. I'm fairly confident with simple Schwarzschild metric stuff for light trajectories. But I've rarely done stuff with timelike trajectories, and when I do so, I'm definitely not confident that I'm not making some obvious blunder. Everything is so much easier when $d\tau=0$. ;)
I think @PM2Ring already said it; you want a metastable phase with a relatively low density that changes into a phase with a higher density during operation of the core.
The production rate for Np in a reactor is much lower than that for Pu – where should it come from in the first place? Np in quantitative quantities would therefore always be more expensive than the Pu itself.
Plutonium has a bunch of allotropes. https://en.wikipedia.org/wiki/Allotropes_of_plutonium Also the radioactivity tends to mess up its crystal structure.
@RyderRude Using l'Hôpital may involve circular reasoning. The usual way to find the derivative of sine from 1st principles is via the limit of sin(x)/x at zero.
@PM2Ring There was someone who asserted that they found a crazy arrangement of axioms and definitions so that the limit may be obtained in a non-circular way. It was so convoluted that I cannot even judge if they had sneakily inserted a circularity. It is definitely not worth the effort.
I usually reduce $\lim_{x\to0}\dfrac{1-\cos x}x$ to the sinc limit so that much less work drawing stuff out is done.
@SillyGoose but it is not; the limitations are understood, and how to improve it is also obvious, as they found. As mentioned before, the conveniences of the equalities afforded to us in the thermodynamic limit is wonderful, so why would you want to deviate from it?
class time, human attention time, are limited, scarce resources.
it doesn't accurately describe BEC for a (what seems to be standard) system to observe BEC in
in particular it inaccurately predicts a property of BEC that is not a very exotic property (in the sense that any decent "model" should predict such a property)
well i dont get why textbooks don't just present the "obvious improvements"?
if cold atoms is the particular system that most people experimentally work with BECs, why introduce BECs using a model which is just inappropriate for this system
it could just be patched up by supposing the system actually is grand canonical, which via the stack question i linked seems to be a possibility. then BECs can be pedagogically introduced without misleading people
@naturallyInconsistent does this sentence mean that stereotypical BEC is characterized by a transition from a substance to becoming superfluid and superconductive?
@SillyGoose It is analytic of simple form, linear in this chemical potential term that has a simple Taylor's expansion form at low T (or the equivalent nonlinear series in fugacity form)
@SillyGoose from normal system, to either one case
well u are saying that BEC is characterized by a substance becomign superfluid or superconductive (or perhaps both). unless either of these characteristics are equivalent to macroscopic occupation of the groudn state, then i don't see how one can characterize BEC by both what you said nad what i said
@SillyGoose a BEC is definitely a phase in which all the bosons occupy the ground state, in practice since this happens at $T = 0$ you always have a fraction of constituents which are not in the ground state
@naturallyInconsistent It is relatively cheap because it can be easily separated chemically. (It's so simple, we do it as a student experiment with Np-239).
Np is practically a waste by-product. It would be more expensive if there was significant demand for it.
Np is soluble at a few percent in delta-Pu, but the Np alloy does not stabilize the delta-Pu phase. (Yeah, I had to look it up in my Katz/Seaborg/Morss.)
The delta structure is stabilized down to room temperature by dissolution of 2 % Ga.
It's even possible to preserve a metastable delta structure with 1 % Ga and quenching down to room temperature.
@Obliv the 2200 lbs car with the same horsepower as the lighter bike (which is at 96 horsepower) will generate higher impact force if it hits a concrete wall despite maybe slower than the bike?
@user85795 generally more horsepower means higher top speed, correct?
