Sandbox::https://github.com/PlasmaPy/PlasmaPy/pull/1726

JaydenR2305 · 2022-09-03T14:26:46Z

Currently, for reasonable electron densities (~1e10 cm^-3 - 1e20 cm^-3), chemical_potential returns a number extremely close to zero (~1e-16). This behavior is reasonable and expected. However, there are some occasions where the current least squares regression method will return an abnormally large value, demonstrating numerical instability. This instability can sometimes cause the docstring test to fail

I propose we use some alternative to a least squares method. The method tested in the figures is the Broyden–Fletcher–Goldfarb–Shanno algorithm. The figures were generated at 11000 Kelvin using the two different methods to calculate the chemical potential, and the resulting residual.

For densities greater than 1e20 cm^-3, the graphs produce identical results

The first regime is where the chemical potential is ~0. The algorithms generally agree on zero but sometimes the least squares method produces unexpected results

The residuals for both functions are the same suggesting they have an approximately equal error

Both algorithms produce the same results for electron densities greater than 1e20

I have added a changelog entry for this pull request.
If adding new functionality, I have added tests and
docstrings.
I have fixed any newly failing tests.

codecov · 2022-09-03T14:56:10Z

Codecov Report

Merging #1726 (5896625) into main (fe2216b) will decrease coverage by 0.00%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##             main    #1726      +/-   ##
==========================================
- Coverage   98.09%   98.09%   -0.01%     
==========================================
  Files          87       87              
  Lines        8100     8099       -1     
==========================================
- Hits         7946     7945       -1     
  Misses        154      154

Impacted Files	Coverage Δ
plasmapy/formulary/quantum.py	`100.00% <100.00%> (ø)`

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rocco8773 · 2022-09-06T21:54:57Z

My suspicion is that you're running into the same issue discussed in astropy/astropy#9721.

I'd suggest exploring something like...

@validate_quantities(
    n_e={"can_be_negative": False, "pass_equivalent_units": True},
    T={"can_be_negative": False, "equivalencies": u.temperature_energy()},
)
def chemical_potential(n_e: u.m**-3, T: u.K) -> u.dimensionless_unscaled:
    # deBroglie wavelength
    lambdaDB = thermal_deBroglie_wavelength(T)
    # degeneracy parameter
    degen = (np.cbrt(n_e) * lambdaDB).to(u.dimensionless_unscaled)**3

    ...

Using "pass_equivalent_units" will allow n_e to NOT be converted to u.m, n_e will stay in the units it originally was passed with. This means, once you get to the degen line and multiply np.cbrt(n_e) * lambdaDB the .to() method only has to convert one ratio of u.cm/u.m instead of a power of three. This should reduce the floating point arithmetic error and make the calculation more stable.

rocco8773 · 2022-09-06T22:03:50Z

~~This needs to be merged before #1727~~

~~I need to think about the order of merger a little more...~~

@JaydenR2305 Can you add the same changes you made in #1727, sans change log entry, to this PR? If all the checks pass, then I'd be comfortable merging #1727 without all of its checks passing.

JaydenR2305 · 2022-09-06T23:17:39Z

Looks like 5.0 still uses asscalar somewhere..

JaydenR2305 · 2022-09-06T23:59:02Z

My suspicion is that you're running into the same issue discussed in astropy/astropy#9721.

I'd suggest exploring something like...
@validate_quantities(
    n_e={"can_be_negative": False, "pass_equivalent_units": True},
    T={"can_be_negative": False, "equivalencies": u.temperature_energy()},
)
def chemical_potential(n_e: u.m**-3, T: u.K) -> u.dimensionless_unscaled:
    # deBroglie wavelength
    lambdaDB = thermal_deBroglie_wavelength(T)
    # degeneracy parameter
    degen = (np.cbrt(n_e) * lambdaDB).to(u.dimensionless_unscaled)**3

    ...
Using "pass_equivalent_units" will allow n_e to NOT be converted to u.m, n_e will stay in the units it originally was passed with. This means, once you get to the degen line and multiply np.cbrt(n_e) * lambdaDB the .to() method only has to convert one ratio of u.cm/u.m instead of a power of three. This should reduce the floating point arithmetic error and make the calculation more stable.

This seems to be producing identical results

rocco8773 · 2022-09-07T00:10:05Z

Looks like 5.0 still uses asscalar somewhere..

You are correct, see my comment #1727 (comment).

rocco8773

In the docstring, can you make the executive decision to either consistently use the subscript $_a$ or drop it all together? I think either approach is valid. I just want it to be consistent across the docstring, since switching back-and-forth and lead to confusion for users. If you choose to keep the subscript, then can you update the sentence explaining the equation to read something like:

"... where $\chi_a$ is the degeneracy parameter of particle 'a'..."

If we keep the subscript, then we at least need to explain what it means.

rocco8773

Do you know which section in Bontiz discusses the chemical potential? If so, can you included in the first line of the docstring Notes section?

JaydenR2305 · 2022-09-08T00:57:12Z

Do you know which section in Bontiz discusses the chemical potential? If so, can you included in the first line of the docstring Notes section?

Good call! I added the specific equation number from Bonitz and improved the overall explanation. The documentation was somewhat vague on how the calculation is performed so I tried to make things more explicit

namurphy

LGTM! It looks like you addressed all of @rocco8773's comments, and I only have a very minor suggestion for the changelog entry. I'm happy to get this merged soon. Thank you for doing this!

lemmatum · 2022-09-08T05:24:36Z

Cool, I've never heard of the BFGS algorithm. I have some reading up to do!
My only comment is it'd be good to add a test where the chemical potential is non-zero. Otherwise I think the changes look good.

Thanks for digging into this!

lemmatum · 2022-09-08T05:26:21Z

        self.n_e_fail = 1e23 * u.cm**-3
        self.T = 11604 * u.K
-        self.True1 = 6.133671348607095e-11
+        self.True1 = 0


Add another test where the result is non-zero.

Co-authored-by: Nick Murphy <namurphy@cfa.harvard.edu>

…enR2305/PlasmaPy into chemical-potential-stability

rocco8773 · 2022-09-08T20:25:09Z

+Changed `~plasmapy.formulary.quantum.chemical_potential` to use the
+Broyden-Fletcher-Goldfarb-Shanno algorithm to implicitly solve for beta.


Suggested change

Changed `~plasmapy.formulary.quantum.chemical_potential` to use the

Broyden-Fletcher-Goldfarb-Shanno algorithm to implicitly solve for beta.

Changed `~plasmapy.formulary.quantum.chemical_potential` to use the

Broyden-Fletcher-Goldfarb-Shanno algorithm to implicitly solve for

the β(\ :math:`= 1/(k_B T)`\ ) parameter.

Only so beta is not mistaken for the plasma beta, which would likely be common in this community.

Whoops, that shouldn't say beta anyways. It should be the dimensionless chemical potential

rocco8773 · 2022-09-08T20:29:39Z

    The definition for the ideal chemical potential is implicit, so it must
    be obtained numerically by solving for the Fermi integral for values
    of chemical potential approaching the degeneracy parameter. Since values
    returned from the `~plasmapy.formulary.mathematics.Fermi_integral`
-    are complex, a nonlinear Levenberg-Marquardt least squares method is
-    used to iteratively approach a value of :math:`μ` which minimizes
-    :math:`I_{1/2}(β μ_a^{ideal}) - χ_a`
+    are complex, the Broyden–Fletcher–Goldfarb–Shanno algorithm is
+    used to iteratively approach a value of :math:`μ^{ideal}` which minimizes
+    :math:`\lvert I_{1/2}(β μ^{ideal}) - χ \rvert`.


I'm thinking as a follow up PR, it would be good to break out the code for the ideal chemical potential into its own function. @JaydenR2305 What do you think about this and would you be willing to do it? Again, this would be a followup PR and I'm not suggesting to do it in this PR.

Sure. That means this function would just be a wrapper for that function multiplied by beta?

Yeah, in the end that would be the case.

rocco8773 · 2022-09-08T20:43:22Z

+    def test_knowns(n_e, T, expected_beta):
        """
-        Tests Fermi_integral for expected value.
+        Tests chemical_potential for expected value.
        """
-        methodVal = chemical_potential(self.n_e, self.T)
-        testTrue = u.isclose(methodVal, self.True1, rtol=1e-16, atol=0.0)
-        errStr = f"Chemical potential value should be {self.True1} and not {methodVal}."
+        methodVal = chemical_potential(n_e, T).value
+        testTrue = np.isclose(methodVal, expected_beta, rtol=1e-16, atol=0.0)
+        errStr = (
+            f"Chemical potential value should be {expected_beta} and not {methodVal}."
+        )
        assert testTrue, errStr


Can you add another assert to ensure the returned unit is dimensionless_unscaled? It'll likely look something like...

assert testTrue, errStr assert chemical_potential.unit == u.dimensionless_unscaled

Also, could you please refactor this to meet PEP8 naming standards? e.g. methodVal -> method_val

rocco8773

Looks good to me! Thank you for working on this.

This one attempts to use the :wikipedia: role, though I'll need to check that it works.

Change method to bfgsb

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github-actions Bot added plasmapy.formulary Related to the plasmapy.formulary subpackage plasmapy.formulary.quantum Related to the plasmapy.formulary.quantum module labels Sep 3, 2022

JaydenR2305 added 3 commits September 3, 2022 10:46

Fix tests

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Add changelog

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JaydenR2305 added 3 commits September 3, 2022 16:19

Revert to numba 0.56.0

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Add changelog

79f48a4

Update docstring

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JaydenR2305 marked this pull request as ready for review September 4, 2022 13:30

pheuer requested review from StanczakDominik, lemmatum and namurphy September 6, 2022 13:25

JaydenR2305 added 3 commits September 6, 2022 14:54

Revert numba strict version

40ddc2e

Bump minimum version of astropy to 5.0

7a6eaca

Update changelog

8a45488

Merge pull request #1 from JaydenR2305/numpy-asscalar

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JaydenR2305 and others added 2 commits September 6, 2022 19:29

Try 5.0.1

df66881

Merge pull request #2 from JaydenR2305/numpy-asscalar

d34f516

Merge branch 'main' into chemical-potential-stability

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rocco8773 requested changes Sep 7, 2022

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Improve docstring

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Fix typo

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namurphy approved these changes Sep 8, 2022

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Comment thread changelog/1726.trivial.rst Resolved

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JaydenR2305 and others added 4 commits September 8, 2022 05:52

Update changelog/1726.trivial.rst

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Co-authored-by: Nick Murphy <namurphy@cfa.harvard.edu>

Add test for non-zero potential

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Merge branch 'chemical-potential-stability' of http://github.com/Jayd…

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…enR2305/PlasmaPy into chemical-potential-stability

Appease pre-commit

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namurphy requested a review from rocco8773 September 8, 2022 17:08

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JaydenR2305 added 2 commits September 8, 2022 20:31

Improve testing

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rocco8773 approved these changes Sep 9, 2022

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rocco8773 merged commit 62dd840 into PlasmaPy:main Sep 9, 2022

JaydenR2305 deleted the chemical-potential-stability branch September 9, 2022 23:57

namurphy added a commit to namurphy/PlasmaPy that referenced this pull request Oct 25, 2022

Revise changelog for PlasmaPy#1726

b210fa2

This one attempts to use the :wikipedia: role, though I'll need to check that it works.

		Changed `~plasmapy.formulary.quantum.chemical_potential` to use the
		Broyden-Fletcher-Goldfarb-Shanno algorithm to implicitly solve for beta.

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