.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/polymer/chain_statistics/plot_03_kuhn_freely_jointed_chain.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_polymer_chain_statistics_plot_03_kuhn_freely_jointed_chain.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_polymer_chain_statistics_plot_03_kuhn_freely_jointed_chain.py:


Kuhn's random-walk chain: sampled freely jointed chains
==========================================================

Kuhn (1934) modeled a flexible polymer as a random walk of :math:`n`
freely jointed segments of length :math:`b`.
:func:`~chemistrykit.polymer.systems.chain_statistics.freely_jointed_chain`
samples such conformations directly; averaging their squared end-to-end
distance reproduces the exact result :math:`\langle R^2\rangle=nb^2`
(:class:`~chemistrykit.polymer.systems.chain_statistics.IdealChain`), and
the end-to-end vector's components become Gaussian with variance
:math:`nb^2/3`, as the central limit theorem predicts.

.. GENERATED FROM PYTHON SOURCE LINES 16-33

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.polymer.systems.chain_statistics import IdealChain, freely_jointed_chain

    b = 1.0
    ideal = IdealChain()
    n_values = np.array([10, 30, 100, 300, 1000])

    print("  n    <R^2> sampled   n b^2 (Kuhn)")
    R2_sampled = []
    for n in n_values:
        X = freely_jointed_chain(int(n), b, n_chains=4000, rng=int(n))
        R2 = np.sum(X[:, -1] ** 2, axis=-1).mean()
        R2_sampled.append(R2)
        print(f"{n:5d}   {R2:12.1f}   {ideal.mean_square_end_to_end(n, b):10.1f}")





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

 .. code-block:: none

      n    <R^2> sampled   n b^2 (Kuhn)
       10            9.9         10.0
       30           30.1         30.0
      100          100.3        100.0
      300          296.7        300.0
     1000         1005.3       1000.0




.. GENERATED FROM PYTHON SOURCE LINES 34-36

The end-to-end x-component over many 100-segment chains is Gaussian with
variance n b^2 / 3.

.. GENERATED FROM PYTHON SOURCE LINES 36-41

.. code-block:: Python

    n = 100
    X = freely_jointed_chain(n, b, n_chains=20000, rng=0)
    Rx = X[:, -1, 0]
    print(f"\nvar(R_x) = {Rx.var():.2f}  (Kuhn: n b^2 / 3 = {n * b**2 / 3:.2f})")





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

 .. code-block:: none


    var(R_x) = 33.94  (Kuhn: n b^2 / 3 = 33.33)




.. GENERATED FROM PYTHON SOURCE LINES 42-67

.. code-block:: Python

    fig, axes = plt.subplots(1, 3, figsize=(14, 4.2))
    walk = freely_jointed_chain(300, b, rng=7)[0]
    axes[0].plot(walk[:, 0], walk[:, 1], lw=0.8)
    axes[0].plot(*walk[0, :2], "go", label="start")
    axes[0].plot(*walk[-1, :2], "rs", label="end")
    axes[0].set_aspect("equal")
    axes[0].set_title("One freely jointed chain (xy projection)")
    axes[0].legend()

    axes[1].loglog(n_values, R2_sampled, "o", label="sampled")
    axes[1].loglog(n_values, ideal.mean_square_end_to_end(n_values, b), "k--", label=r"$nb^2$")
    axes[1].set_xlabel("n (Kuhn segments)")
    axes[1].set_ylabel(r"$\langle R^2\rangle$")
    axes[1].set_title("Random-walk scaling")
    axes[1].legend()

    grid = np.linspace(-40, 40, 300)
    s2 = n * b**2 / 3
    axes[2].hist(Rx, bins=60, density=True, alpha=0.6, label="sampled")
    axes[2].plot(grid, np.exp(-(grid**2) / (2 * s2)) / np.sqrt(2 * np.pi * s2), "k", label="Gaussian")
    axes[2].set_xlabel(r"$R_x$")
    axes[2].set_title("End-to-end component, n=100")
    axes[2].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/polymer/chain_statistics/images/sphx_glr_plot_03_kuhn_freely_jointed_chain_001.png
   :alt: One freely jointed chain (xy projection), Random-walk scaling, End-to-end component, n=100
   :srcset: /api/gallery/polymer/chain_statistics/images/sphx_glr_plot_03_kuhn_freely_jointed_chain_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.380 seconds)


.. _sphx_glr_download_api_gallery_polymer_chain_statistics_plot_03_kuhn_freely_jointed_chain.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_kuhn_freely_jointed_chain.ipynb <plot_03_kuhn_freely_jointed_chain.ipynb>`

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

      :download:`Download Python source code: plot_03_kuhn_freely_jointed_chain.py <plot_03_kuhn_freely_jointed_chain.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_03_kuhn_freely_jointed_chain.zip <plot_03_kuhn_freely_jointed_chain.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

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