.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statmech/maxwell_boltzmann/plot_02_stern_experiment.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_statmech_maxwell_boltzmann_plot_02_stern_experiment.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_statmech_maxwell_boltzmann_plot_02_stern_experiment.py:


Verifying the speed distribution: Stern's molecular-beam experiment
=======================================================================

Otto Stern's 1920 molecular-beam apparatus gave the first direct
experimental measurement of a gas's actual molecular speeds, rather than
only their statistical consequences (pressure, diffusion, viscosity) -- a
rotating-drum time-of-flight technique later refined by Zartman and Ko
(1930-1934) into a quantitative confirmation of the full Maxwell-Boltzmann
functional form, not just its mean.
:meth:`~chemistrykit.statmech.MaxwellBoltzmannSpeedDistribution.sample`
draws exactly such a synthetic "beam" of molecular speeds; binning it into
a histogram and comparing it to the exact analytic pdf reproduces the kind
of check Stern, and later Zartman and Ko, made directly against a real gas.

.. GENERATED FROM PYTHON SOURCE LINES 18-39

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.statmech import MaxwellBoltzmannSpeedDistribution

    mass = 2.18e-25  # potassium-like, kg -- close to the alkali-metal beams Stern's and Zartman/Ko's apparatus actually used
    temperature = 470.0  # K, a plausible oven temperature for an alkali-metal beam source

    dist = MaxwellBoltzmannSpeedDistribution(mass=mass, temperature=temperature)
    sampled_speeds = dist.sample(200_000, rng=0)

    fig, ax = plt.subplots(figsize=(7, 5))
    ax.hist(sampled_speeds, bins=80, density=True, alpha=0.5, color="steelblue", label="synthetic 'beam' (200,000 draws)")
    v = np.linspace(0.0, float(sampled_speeds.max()), 400)
    ax.plot(v, dist.pdf(v), color="crimson", linewidth=2, label="exact Maxwell-Boltzmann pdf")
    ax.set_xlabel("speed (m/s)")
    ax.set_ylabel("probability density")
    ax.set_title("Sampling the Maxwell-Boltzmann distribution (a Stern-type beam measurement)")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/statmech/maxwell_boltzmann/images/sphx_glr_plot_02_stern_experiment_001.png
   :alt: Sampling the Maxwell-Boltzmann distribution (a Stern-type beam measurement)
   :srcset: /api/gallery/statmech/maxwell_boltzmann/images/sphx_glr_plot_02_stern_experiment_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 40-44

The sampled mean speed converges to the exact analytic mean speed as
the number of "molecules" in the synthetic beam grows -- exactly the
statistical convergence a real, finite molecular-beam measurement is
limited by:

.. GENERATED FROM PYTHON SOURCE LINES 44-50

.. code-block:: Python


    for n in (100, 10_000, 1_000_000):
        speeds = dist.sample(n, rng=0)
        print(f"n = {n:>9,}: sampled mean speed = {speeds.mean():.2f} m/s  (exact: {dist.mean_speed():.2f} m/s)")

    plt.show()




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    n =       100: sampled mean speed = 284.19 m/s  (exact: 275.32 m/s)
    n =    10,000: sampled mean speed = 274.88 m/s  (exact: 275.32 m/s)
    n = 1,000,000: sampled mean speed = 275.24 m/s  (exact: 275.32 m/s)





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.087 seconds)


.. _sphx_glr_download_api_gallery_statmech_maxwell_boltzmann_plot_02_stern_experiment.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_stern_experiment.ipynb <plot_02_stern_experiment.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_stern_experiment.py <plot_02_stern_experiment.py>`

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    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_