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| visits | member for | 1 year, 5 months |
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28m |
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Why distinguish between row and column vectors? Why not define vectors to be things that act on covectors to product numbers instead? Because infinite-dimensional spaces need not be reflexive |
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May 13 |
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Physical interpretation of Poisson bracket properties @dmckee: notation is domain-specific, and it's quite common to use curlies for Poisson brackets, both in introductory and advanced literature |
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May 10 |
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If particles can find themselves spontaneously arranged, isn't entropy actually decreasing? @LubošMotl: care to comment on the revision to my answer? |
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May 10 |
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Are there problems solvable with Newtonian physics, GR and QM? two historically relevant examples would be the perihelion precession of mercury (GR vs Newton) and black body radiation (classical Rayleigh–Jeans law vs quantum Planck's law) |
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May 8 |
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If particles can find themselves spontaneously arranged, isn't entropy actually decreasing? I'd also like to throw in some nice quotes from this paper: As the thermodynamic entropy is not measurable except when the process is reversible, the second law remains useless as a computational tool. and It is (has?) not been possible to show that the statistical entropy is identical to the thermodynamic entropy in general. |
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May 8 |
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If particles can find themselves spontaneously arranged, isn't entropy actually decreasing? @LubošMotl: Entropy is well-defined for time-dependent processes. Indeed, it has to be well-defined because the second law of thermodynamics says how it changes during such processes - I don't think that necessarily follows: In particular, there are formulations of the 2nd law that explicitly state There exists for every system in equilibrium a property called entropy, and for irreversible processes the 2nd law only makes a statement about initial and final equilibrium states |
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May 8 |
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If particles can find themselves spontaneously arranged, isn't entropy actually decreasing? @Arnaud: it is indeed hard to define entropy if you don't at least assume local equilibrium; take the dual to your experiment: confine the gas to one side of the box, remove the wall and let it expand; because the rate of expansion is fixed, at each point in time you could re-introduce the wall (freeze the instantaneous system parameter volume) and define the entropy of the expanding gas as the entropy of that equilibrium system |
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Apr 29 |
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English translation of Helmholtz' paper: “On the Physical Significance of the Principle of Least Action” I just took a quick look at the paper, and the notation is funny - the same letters are used, but with different meanings: the coordinates are $p$ instead of $q$, which in turn is used for the velocities $\dot q$; momenta are $c$ instead of $p$, potentials are $F$ instead of $V$, the Lagrangian is called $H$ and has the opposite sign, ie corresponds to $-L$; kinetic energy is called $L$ instead of $T$; the Hamiltonian is called $H'$ instead of $H$ |
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Apr 28 |
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Rate of spontaneous tachyon emission @BenCrowell: the point is that there is no local environment of available tachyons: the interaction is non-local and the environment is basically the whole spacetime (or rather the subset at space-like distance); I should probably edit my answer to make this more explicit; I've yet to think about how general relativity (where there's not necessarily a single critical frame) changes the picture |
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Apr 28 |
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Forces as One-Forms and Magnetism velocity-dependent forces cannot be represented as 1-forms on $M$, but rather as 1-forms on $TM$ (the differential of the Lagrangian) or, factoring out a force of inertia, as sections of the pullback of $T^*M$ over $TM\to M$ |
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Apr 27 |
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Is this statement about quantum mechanics valid? see also plato.stanford.edu/entries/qt-quantlog |
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Apr 22 |
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Why do we still need to think of gravity as a force? @ejrb: it somewhat depends on your framework how special gravity is - eg in string theory gravity is less special than in loop quantum gravity, and the teleparallel reformulation of general relativity makes it into a proper force instead of an inertial one |
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Apr 12 |
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Why isn't temperature measured in Joules? @Kaz: degree of freedom is not used as a unit - we just count them and divide the energy by that number |
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Apr 12 |
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Why isn't temperature measured in Joules? I agree in principle, but keep in mind that if equipartition holds, measuring temperature in units of energy does make sense: it corresponds to average energy per degree of freedom, and as the latter is a unit-less cardinal number, the densities end up with the same unit |
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Apr 12 |
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Nonseparable Hilbert space @SudipPaul: quote 1 implies that in general, you can no longer define operators on just a countable subset; quote 2 implies that Hahn-Banach becomes non-constructive: Zorn's lemma will tell you that there is a linear extension, but you have no way to actually get it; quote 3 implies that there's no (countable!) Hilbert basis, which is often assumed in QM when you treat infinite spaces like finite ones; quote 4 implies that restriction to separable spaces doesn't have a known down-side as far as physics go |
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Apr 9 |
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How hot is the water in the pot? +∞ for actually modelling the system; small nit: naming a rate of energy $Q$ is a bit unfortunate... |
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Apr 9 |
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How hot is the water in the pot? @Taro: for $c \gg t$ the denominator will be approximately $1 + f(t/c)^b$ which explains why $f$ and $c$ can be varied side by side without changing the result and suggests that we haven't found the right physical parameters yet |
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Apr 9 |
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How hot is the water in the pot? @Taro: you can always try to hit your problem with a numeric hammer like five parameter logistic regression $$T = d + (a - d) / (1 + (t / c)^b )^f$$ and you'll get a fit; $a\approx22$ turns out to be the room temperature, $d\approx91$ the steady-state temperature, $b\approx1.5$ might be specific for water or your setup; $c$ and $f$ turn out to be fudge factors of order $10^5$ with high variability (they can be used to balance each other to some degree), which suggests that the functional dependence isn't quite right yet |
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Apr 9 |
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How hot is the water in the pot? @Taro: the basic idea behind using hyperbolic tangent is logistic growth; fitting the measured pot temperatures with $A \tanh B(t−C) + D$ looks good to me |
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Apr 6 |
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How hot is the water in the pot? @Taro: reality is messy, and we're dealing with an effective theory; if fudge factors help you get a better fit, use them even if you can't derive them from first principles; explaining where they come from comes after that |
