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6

This phenomenon has been studied by Vella and Mahadevan and written up in the American Journal of Physics (http://scitation.aip.org/content/aapt/journal/ajp/73/9/10.1119/1.1898523). It's called the Cheerios effect. If the cereal pieces clump together away from the edges of the bowl, they gravitate toward a slight concavity in the surface caused by water ...


4

The equipartition theorem is a mathematical consequence of very specific kind of Hamiltonians. It states that any 'squared' term of deegree of freedom in the Hamiltonian gets $\frac{1}{2}k_bT$ of energy (it is a statement about the energy distribution for this kind of Hamiltonians). For example - classical ideal gas Hamiltonian - ...


4

You're missing a minus in the entropy definition - $S=-Tr(\rho\ln\rho)$ Entropy of a unitarily evolving system (doesn't matter in which picture) is conserved (The entropy is a trace of a function of the density matrix "operator" thus it depend solely on the eigenvalues of it's input operator, but the eigenvalues of the density matrix don't change under ...


4

Yes, neutrinos should obey Fermi-Dirac statistics and yes, the Pauli Exclusion Principle should operate for neutrinos. But let's examine how dense the neutrino population has to be for this to be important. The Fermi momentum is given by $$ p_F = \left( \frac{3}{8\pi}\right) h n_{\nu}^{1/3} $$ where $n_{\nu}$ is the neutrino number density. In order to be ...


4

In macroscopic units it should be $$S=-R\alpha \log(\alpha e^{-S_1/R})-R(1-\alpha)\log\Big(1-\alpha)e^{-S_2/R}\Big) \\=\alpha \Big(S_1-R\log\alpha\Big)+(1-\alpha)\Big(S_2-R\log(1-\alpha)\Big),$$ where $R$ is the universal gas constant. In the pure case, this reduces to the textbook formula. But such a formula cannot be true in general. The general formula ...


2

The variables involved here are classical and you resolve them classically. They enter into the operator because they are parameters of the wavefunction. So let's do this a little more broadly. For continuous systems, we want a family of solutions based on some parameters which I'll collectively identify as $\alpha \in A$; the solutions are then labeled ...


2

Maybe the important step is to realise that the allowed range of momenta is $$ R=[-p_\text{max}:-p_\text{min}]\cup[p_\text{min}:p_\text{max}]. $$ Then the first two $\Theta$'s give one if $p$ is positive and in the allowed range, whereas the last two $\Theta$'s give a contribution that's only 1 if the $p$ is in the negative allowed range. The term out front ...


2

Noether's theorem states that if a system has a continuous symmetry, there is a quantity related to this symmetry, called the Noether charge, which is conserved. It does not state anything on the fact that adding a constant term to a measurable quantity may or may not change the physical description of the system. Only some physical quantities in fact are ...


1

Something analogous to the Fermi-Dirac distribution function will probably work pretty well for the electrons in the Sun; see for example this PDF. You may need to put in some "fudge factors" for their interaction energy with the rest of the proton soup, but you're in some luck, because the interaction energy in the Sun is actually mostly lower than the ...


1

Ideal string cannot be described in terms of canonical ensemble (with Boltzmann probability distribution). The fact it is ideal means it has infinity of degrees of freedom and since each would have, according to the Boltzmann probability distribution, average energy $$\frac{1}{2}k_B T,$$ total energy of the string would be, on average, infinite. This is ...


1

I don't know if it will help you, if it doesn't, sorry for wasting your time. But this is the best explanation I have at this moment. The gas must be flowing faster because of the lower cross sectional area. Assuming no density change, the only way to maintain the same volumetric flow rate is to increase the flow speed. Now consider the perspective of a gas ...


1

I am not sure if this answer will give you an intuitive understanding of the result, but I think it may be useful as it shows the assumptions behind it. What your result means is that in an idealized situation when the volume gas or solute occupies is shrunk slightly by $\delta << V$ while its energy remains the same (let's say, isothermal compression ...


1

It can be rather involved. A lot of technical progress as been on this subject leading up to the modern conformal bootstrap work. Something you can exploit is that these functions should behave like correlation functions and thus are eigenfunctions of the conformal Casimir. That gives you differential equations which in some cases, especially in $D=2$ and ...


1

Electron and holes are Fermions (particles with spin 1/2). This means that no two particles can share the same microstate. The Fermi-Dirac distribution describes how Fermions fill the available states consistent with this property. Bosons on the other hand (particles with integer spin) can occupy the same state. The Bose-Einstein distribution describes how ...


1

equally one could use the solution of the Langevin equation: $$m\dot{v} = -\zeta v +\delta F(t)$$ witch is: $$v(t)=v(0)e^{-\zeta t \over m }+ \int \limits_0^t dt'e^{-\zeta (t-t') \over m } \delta F(t) $$ doing the ensamble average gives: $$v(t)=\overbrace{\langle 1 \rangle}^{=1} v(0)e^{-\zeta t \over m }+ \int \limits_0^t dt'e^{-\zeta (t-t') \over m } ...


1

Not sure the calculation was done correctly (on first glance). However, that is not important here, just think about what you are doing: You have N particles all of which are mutually interacting via a super-long-ranged potential (Coulomb-interaction $\sim r^{-1}$ would be considered long-ranged, you are using a parabolic $\sim r^2$ potential). So, what you ...



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