Stretched exponential

[henrikjacobsenfys]

Implementing the Fourier transform of a stretched exponential. It's not trivial to make it fast and correct at the same time. Claude did that for me. I have updated the text and read through the whole thing.

[codecov]

Codecov Report

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@@            Coverage Diff             @@
##           master     #245      +/-   ##
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+ Coverage   98.47%   98.51%   +0.03%     
==========================================
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  Lines        5512     5646     +134     
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+ Hits         5428     5562     +134     
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Files with missing lines	Coverage Δ
src/easydynamics/__init__.py	100.00% <100.00%> (ø)
src/easydynamics/sample_model/__init__.py	100.00% <100.00%> (ø)
...c/easydynamics/sample_model/components/__init__.py	100.00% <100.00%> (ø)
...s/sample_model/components/stretched_exponential.py	100.00% <100.00%> (ø)

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[rozyczko]

Looks good to merge

[rozyczko]

evalaute -> evaluate

[rozyczko]

Instad -> Instead

[rozyczko]

width returns a DescriptorNumber, whereas every other component's width is a Parameter.
Anything that does comp.width.fixed, comp.width.make_dependent_on(...) or puts it in a FitBinding will fail.

[rozyczko]

I ran Fable on this PR and it found an issue I can't comment on, since I don't know science well enough.

=====================================================================

width overstates the true half width by orders of magnitude at small β, so the numerical convolution's grid-resolution warning never fires and coarse grids give wrong results silently (medium)

width is exposed as $\Gamma = \hbar/\tau$
(stretched_exponential.py:493-515) so that the shared
width-versus-grid checks in
NumericalConvolutionBase._check_width_thresholds can see how wide the
component is. But $\Gamma$ is only the HWHM at $\beta = 1$. Below that,
the peak sharpens dramatically — the height grows as
$\Gamma(1 + 1/\beta)$ while the area stays fixed — and the actual HWHM
collapses:

$\beta$	HWHM / $\Gamma$	peak height relative to a Lorentzian of the same $\Gamma$
1.0	1.00	1.0
0.7	0.56	1.3
0.5	0.22	2.0
0.3	1.5e-2	9.3
0.2	2.7e-4	120
0.1	< 1e-8	3.6e6

So a component with $\tau = 5$ ps, $\beta = 0.2$ reports
width = 0.13 meV (comfortably above a 0.1 meV grid step, no warning),
while its real HWHM is $3.5 \times 10^{-5}$ meV. The numerical
convolution then samples a sub-grid spike and returns garbage. Measured
with a 0.05 meV Gaussian resolution on $[-2, 2]$ meV, default
upsample_factor=5:

$\beta$	grid points	convolved peak	area on $[-2,2]$	warnings
0.3	81	3.71	0.818	none
0.3	401	2.71	0.693	none
0.3	40001 (converged)	2.64	0.684	none
0.2	81	28.7	3.85	none
0.2	401	7.40	1.18	none
0.2	4001	3.03	0.630	none
0.2	40001 (converged)	2.75	0.595	none

At $\beta = 0.2$ on an 81-point grid the area is wrong by a factor of
six and nothing tells the user. $\beta = 0.6$ is fine on every grid, so
this only bites in the small-$\beta$ regime — but that is exactly the
regime where people reach for a KWW instead of a Lorentzian.

Note this is not a bug in _kww_shape; the direct evaluate is exact.
It is a mismatch between what width promises to the convolution
machinery and what the profile actually looks like.

Suggested fix (pick one):

1. Make width report an actual half width. There is no closed form,
   but a bisection on _kww_shape for the $w$ where
   $G_\beta(w) = \tfrac12,\Gamma(1 + 1/\beta)$ costs ~40 single-point
   quadratures (microseconds) and width is not on the evaluation hot
   path. Then the existing small-width warning fires exactly when it
   should. Keep the docstring note that this is derived and read-only.
2. Failing that, document loudly in the class docstring that $\Gamma$ is
   not the HWHM for $\beta < 1$, that the numerical convolution needs
   upsample_factor raised for small $\beta$, and add a regression test
   that the convolution converges (compare 401 vs 4001 points at
   $\beta = 0.3$).

Either way, add a test that a narrow small-$\beta$ component on a coarse
grid produces the "small compared to the spacing" warning — today it
does not.