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Verify thermal speeds for kappa distributions #186

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lemmatum opened this Issue Dec 13, 2017 · 7 comments

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lemmatum commented Dec 13, 2017

#171 was recently merged and it contains tentative functionality for kappa distribution functions and their thermal speeds. These should be verified, especially the mean magnitude thermal velocity for the kappa distribution function. I've made efforts to test and get references for different parts of this code, which I list below.

I found references for 1D and 3D kappa velocity distribution functions and both those references have the same definition for the "most probable" thermal speed. I've not found any references for the root mean square (RMS) and mean magnitude thermal speeds.

Numerical testing and back of the envelope math suggests that the RMS speed for the kappa distribution function is identical to the RMS speed for the Maxwellian distribution function. I also numerically tested the mean magnitude thermal speed for the kappa distribution function and it gives an answer which is somewhat close to the Maxwellian mean magnitude thermal speed but is still a bit off. I think this may have to do with numerical errors preventing integration of values at large velocities, yet those values are weighed heavily both due to the mean magnitude being integral of v * f(v) and kappa distributions having higher energy tails. Not sure why this would be less pronounced in the RMS case since it goes as v**2 * f(v).

Here is the script I used for the numerical tests. This was placed one directory above the PlasmaPy root directory and run within virtualenv. Note that as you make kappa smaller the distribution is more suprathermal and you have to integrate further out (make infApprox larger), and as kappa gets larger it converges to the Maxwellian distribution function (as expected).

Reference materials:

@lemmatum lemmatum added this to To Do in Plasma parameters via automation Dec 13, 2017

@lemmatum lemmatum added this to the v0.1 milestone Dec 13, 2017

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StanczakDominik commented Dec 13, 2017

Thanks for writing this up! @namurphy this could be helpful to share with your knowledgeable friend once they're back from the conference :)

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samurai688 commented Dec 19, 2017

Hmm, I'm not knowledgeable about Kappa distributions, but I poked around a bit and just found the following paper: http://www.tp4.ruhr-uni-bochum.de/~ioannis/publications/2009A10.pdf

It gives the same definition of most probable speed, and in equation (5) they give a "mean energy per particle" expression. I got super confused after that part, I think there was some argument about the notion of temperature in the kappa distribution, and how to compare things at a given temperature.

But if we are just concerned with the distributions, I think in theory we should be able to convert their equation (5) to a speed and see if it matches the numerical mean magnitude thermal speed.

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lemmatum commented Dec 20, 2017

@samurai688 Equations (4) and (5) would suggest that rms speed squared for a kappa distribution would then be v_kappa_rms ** 2 = 2 / m * 3/2 * E_0 * (k / (k - 3/2)). Then substitute in E_0 = m * v **2 / 2, you get v_kappa_rms ** 2 = 3 / 2 * (v ** 2) * (k / (k - 3/2)), where v ** 2 = v_Boltzmann **2 * (k - 3/2) / k. So we get v_kappa_rms ** 2 = 3 / 2 * v_Boltzmann ** 2, which is just v_kappa_rms ** 2 = v_Boltzmann_rms ** 2.
QED, the RMS speeds for kappa and Boltzmann distributions are the same. There is probably a similar relationship for mean magnitude.

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lemmatum commented Jan 13, 2018

@namurphy Do we have an expert to look at this?

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nshaffer commented Jan 26, 2018

I'm not an expert, but I've taken courses from of a Kappa evangelist. Attempting to talk about the "thermal speed" or the "temperature" of a Kappa distribution is a pedagogical quagmire. There is no unique definition for the temperature of a Kappa distribution (or non-Maxwellian distributions in general). You can define temperatures in a variety of ways, though. A common way is to assert that some moment of a Maxwellian is equal to that same moment of a Kappa and define the Kappa's "temperature" as the T(w, kappa) that satisfies that equality.
moments_kappa_mb
The arXiv article by Pierrard and Lazar uses the <v^2> moment (i.e., the "kinetic" temperature), but they also caution that you can't just use that definition for T as a drop-in replacement for the temperature dependence of, say, the Debye length.

My broader point here is that as soon as one decides that non-Maxwellian distributions are on the table, it's ambiguous to talk about their thermal speeds. It's just not a sufficiently fundamental concept unless you go to great pains in documentation to say what your really mean by "thermal speed".

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antoinelpp commented Jan 26, 2018

The temperature should always be defined via the standard deviation. As the kinetic energy is define with $k_B T = 3/2 E_k$, and here the kinetic energy $E_k$ doesn't take into account the global convection, ie $E_k = m/2 <(v-)^2>$. It is this definition which is used to define the temperature in the fluid equations.

Thermal speed seems to be ambiguous even for Maxwellian distribution. I always define the thermal speed by the mean of the magnitude of the velocity, ie. $v_{th} = <|v|>$.

@nshaffer for cold plasma applications, the thermal speed is quite usefull, and I think we need to define a function for it. We just need to chose a definition that will be consitent for every distribution functions.

@StanczakDominik StanczakDominik modified the milestones: v0.1, v0.2 Apr 22, 2018

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lemmatum commented Apr 22, 2018

I think we're getting caught in the weeds a bit @nshaffer @antoinelpp , the point here is to have thermal speeds for the kappa distribution which are defined in the same fashion as they are for the Maxwellian thermal velocity (and to have a clearly stated warning that these don't physically work the same way as Maxwellian velocities and temperatures). At present, we have 3 methods for Maxwellian thermal speed: most_probable, rms, and mean_magnitude. These are mathematically defined with respect to a distribution based on the velocity moments. I've already verified that most_probable method for kappa thermal velocity is correctly implemented. My earlier comment in this thread goes through a derivation for rms, so I believe that is implemented correctly. Now we just need to verify mean_magnitude.

If we want some other velocities outside of this scope, we should leave that for a separate issue.

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