[knzhou]

I've seen this blow up in several places now, but there seriously is nothing to this paper.

Essentially all they're pointing out is that the set of atomic transition energies (1) is positive, (2) has a smooth distribution, (3) goes to zero at zero and infinity, (4) has a maximum somewhere in between, (5) is skewed right. All of these things are completely mundane and well-understood, and not at all unique to the "Planck distribution". Statisticians probably know of tens of other distributions with these properties, which would fit their curve about as well.

Their claim is like saying that any function that goes from -1 to 1 smoothly must be a logistic function, or any function that goes to 0 at infinity but slowly must be a power law. That's not a paper, that's a hunch. If the researchers wanted to be serious, they could have run a statistical test to quantify how well the data fit the Planck distribution (just like tests of normality are routinely done in statistics). But they didn't, and the reason probably is because the test would fail.

[MooMooMilkParty]

Posting this verbatim from my reddit comment, but I ran a few quick fits and the (arbitrarily scaled) Planck distribution looks better than standard fits of some other distributions. Of course I ran absolutely no statistics on these or tried any sort of real analysis that should have been included in the paper in the first place, so take it with multiple grains of salt:

I fit a few distributions to the data and actually the Planck distribution at 9000K seems to be noticeably better. I did no further statistics and haven't even looked at the fitted parameters for this. Note: I am an earth scientist, not a physicist. https://imgur.com/Qg4ixF2

Edit: More distributions: https://imgur.com/IrF6lsV

Edit: Filtering out sodium and potassium to try to account for some of the low-wavelength counts doesn't seem to help fitting the distributions either: https://imgur.com/dHDUS9Z

You can get the data here: https://physics.nist.gov/PhysRefData/ASD/lines_form.html

And here's the (garbage) code to reproduce my plots:

    %pylab inline
    import pandas as pd
    import scipy
    import scipy.stats
    
    mpl.style.use('seaborn-muted')
    mpl.rcParams['figure.figsize'] = (18, 12)
    mpl.rcParams['font.size'] = 16
    df = pd.read_csv('nist.csv')
    
    def filter_string(x):
        x = str(x).replace('=', '').replace('"', '').replace('*', '').replace('+', '').replace('(', '').replace(')', '')
        if len(x) == 0:
            x = 'NaN'
        return x
    
    asl_comp = df['ritz_wl_vac(nm)'].apply(filter_string).astype(float)
    asl_ver  = df['obs_wl_vac(nm)'].apply(filter_string).astype(float)
    sp_num = df['sp_num'].astype(float)
    
    df_neutral = df[sp_num == 1]
    asl_comp = df_neutral['ritz_wl_vac(nm)'].apply(filter_string).astype(float)
    asl_ver  = df_neutral['obs_wl_vac(nm)'].apply(filter_string).astype(float)
    sp_num = df_neutral['sp_num'].astype(float)
    
    asl_ver = asl_ver[~asl_ver.isna()]
    
    def planck(wav, T=9000.0):
        h = 6.626e-34
        c = 3.0e+8
        k = 1.38e-23
        a = 2.0*h*c**2
        b = h*c/(wav*k*T)
        intensity = (a / ( (wav**5)) * (1 / (np.exp(b) - 1.0) ))
        return intensity
    
    
    dist_names = ['beta', 'lognorm', 'pearson3', 'gumbel_r']
    x = np.arange(0, 1500, 5)
    intensity = planck(x / 1e9)
    size = len(asl_ver)
    
    yvals, xvals = np.histogram(asl_ver, bins=300, normed=True)
    
    for dist_name in dist_names:
        dist = getattr(scipy.stats, dist_name)
        param = dist.fit(asl_ver)
        pdf_fitted = dist.pdf(x, *param[:-2], loc=param[-2], scale=param[-1])
        plt.plot(x, pdf_fitted, label=dist_name, linewidth=4,)
    
    plt.hist(asl_ver, bins=300, density=True, color='grey');
    plt.plot(x, intensity / 1.1e17, linewidth=4, color='red', linestyle='--', label='scaled planck (9000k)')
    plt.legend()
    plt.xlabel(r'$\lambda (nm)$')
    plt.ylabel('Density')

[knzhou]

Neat! Still, I wouldn't say the Planck distribution is noticeably better. The reason the other 4 distributions appear to have the peak too low is because they're doing a tradeoff: they do that to better fit the values of the left hand side of the graph.

The Planck fit gets the peak height right, but mainly because it has no chance of fitting the left hand side at all (since it increases much more slowly than the other distributions), so it doesn't even try.

This is why I said it would be better to have actual statistical tests -- we shouldn't have to have this kind of qualitative discussion when it's already a solved problem.

[mcnamaratw]

Can you link the discussion on Reddit?

[banana_giraffe]

https://www.reddit.com/r/Physics/comments/gf1kbd/if_you_over...

[mcnamaratw]

Seemingly multiple people downvoted this. Why??

[stared]

I agree. To me, it sounds like random matrix spectra.

Just because it has the shape of the Planck distribution, it does mean it comes from it.