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May
28
awarded  Popular Question
May
2
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May
2
comment Spectrum of CMB vs. duration of last scattering
Maybe you could consider that, radiation emitted at higher temperature also suffer from more redshift, though I haven't done the calculation myself.
Feb
26
answered Ideal gas in a vessel: kinetic energy of particles hitting the vessel's wall
May
12
awarded  Teacher
Feb
25
comment Similarity between the Coulomb force and Newton's gravitational force
That comes from approximations to general relativity, as mentioned in other answers. I have no opinion on it. It is kind of a "top-to-bottom" approach: start from a complete (and correct as we now know) theory and try to find what it would look like in some special situations, and then find some similarity with the Maxwell theory. What I was talking about in my answer is that the simplistic "bottom-to-top" approach does not work: one cannot start from analogy between Newton and Coulomb's law to get a fully relativistic theory for gravity. That's why Einstein introduced his "fancy" ideas.
Feb
24
revised Similarity between the Coulomb force and Newton's gravitational force
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Feb
24
answered Similarity between the Coulomb force and Newton's gravitational force
Feb
24
revised Possible ambiguity in using the Dirac Delta function
edited the link to the book "Physical Kinetics"
Feb
24
comment Possible ambiguity in using the Dirac Delta function
@Qmechanic Thanks for such a very detailed answer and I really appreciate your exposition on the transformation of the delta function! But, just at your equation (6) you said "it is natural to choose ... $\delta(\mathrm{M}\cdot \mathrm{n})$", and my immediate question is, is it also natural to choose $\delta(\mathrm{M}\cdot \mathrm{n}/|\mathrm{M}|)=\delta(\cos\theta)$"? Note that the two choices give different power index for $M$ (in equation (7)), which is essential (since it is not a constant and may later be integrated), while the factor $m^{-3}$ for the monatomic case is not.
Feb
23
revised Possible ambiguity in using the Dirac Delta function
added 274 characters in body
Feb
22
revised Possible ambiguity in using the Dirac Delta function
The final sentence emphasizing that I am asking about the "translation" from physics to mathematics.
Feb
22
comment Possible ambiguity in using the Dirac Delta function
@Qmechanic Thanks for answering. I have no problem understanding the mathematical manipulations here. What I am asking is not the mathematics, but the translation from "physics" to mathematics. Specifically for the footnote to equation (1.1) in "Physical Kinetics", when translating the sentence "$\mathbf{M}$ should be perpendicular to the molecular axis" into delta function, one could use $\delta(\mathbf{M}\cdot\mathbf{n})$, or $\delta(\mathbf{M}\cdot\mathbf{n}/|\mathbf{M}|)=\delta(\cos\theta)$. Which one is correct (or both?) and why?
Feb
22
comment Possible ambiguity in using the Dirac Delta function
Yes, the result of the integral containing a delta function should equal the one with a constraint (though I have never seen anyone ever checked for this). But in some physics books the presence of delta function also amounts to change the dimension of the integral into the "correct" one (since the delta function is NOT dimensionless), for example, in the book page I mentioned above.
Feb
22
comment Possible ambiguity in using the Dirac Delta function
As far as I can see, they simply took $\delta(M\cos\theta)=\delta(\cos\theta)/M$. If $M=0$ is important for the integral, then this should be clearly stated from the beginning. I understand what you mean by "not getting additional factors", but isn't this a "tautology"? You see, I was asking "what should be put inside $\delta()$", and the answer is "not getting additional factors"? How do we know which are intrinsic and which are additional?
Feb
22
comment Possible ambiguity in using the Dirac Delta function
Furthermore, with regard to the footnote in the book I mentioned in the main post, then it seems to me that $\delta(\cos\theta)$ is "less unnecessarily complicated" than $\delta(M\cos\theta)$, while the authors of that book apparently preferred the more "complicated" one. The dimension of the two results (corresponding to two different choices) are of course different, but here it is not clear to me what should be the correct dimension.
Feb
22
comment Possible ambiguity in using the Dirac Delta function
Well, then the question becomes, what do you exactly mean by "unnecessary complications" (this is actually what I meant in my original question)? What are the guidelines for avoiding "unnecessary complications"? Is $\delta(x)$ "just right" for the constraint $x=0$, while $\delta(2x)$ overly complicated? Sorry this might sound picky, but I am serious and this is really something I am curious about and uncomfortable with.
Feb
22
awarded  Editor
Feb
22
revised Possible ambiguity in using the Dirac Delta function
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Feb
22
comment Possible ambiguity in using the Dirac Delta function
I can calculate it by letting $t=x^2$, but maybe here the problem is that $\delta(x)$ (and its integration) is defined for $(-\infty,\infty)$, while $x^2$ is always $\ge0$. The point of my question is, what kind of guideline one should follow when translating a physical constraint into a $\delta\text{-function}$?