.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/md/lj_fluid/plot_03_hard_sphere_freezing.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_md_lj_fluid_plot_03_hard_sphere_freezing.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_md_lj_fluid_plot_03_hard_sphere_freezing.py:


Alder and Wainwright's hard spheres: packing-driven order with no attraction
===============================================================================

Alder and Wainwright's 1957 computer experiments showed that particles
with *no attraction at all* -- hard spheres -- order into a crystal-like
arrangement once they are packed densely enough: freezing driven by
excluded volume (entropy), not by energy. This package integrates smooth
forces rather than hard-sphere collision events, so the hard spheres are
approximated by steeply repulsive particles: the Lennard-Jones potential
cut off at its own minimum :math:`r=2^{1/6}\sigma`
(:attr:`~chemistrykit.md.systems.lj_fluid.LennardJones.r_min`) and shifted,
which leaves only a repulsive wall. Comparing the radial distribution
function g(r) of this purely repulsive
:class:`~chemistrykit.md.systems.lj_fluid.LJFluid` at low density and at
a density near hard-sphere freezing shows sharp positional order
appearing from packing alone.

.. GENERATED FROM PYTHON SOURCE LINES 21-53

.. code-block:: Python

    import matplotlib.pyplot as plt

    from chemistrykit.md.systems.lj_fluid import LennardJones, LJFluid
    from chemistrykit.md.systems.thermostats import VelocityRescalingThermostat
    from chemistrykit.md.visualizers.md_plots import plot_radial_distribution_function

    r_wca_cutoff = LennardJones().r_min  # 2**(1/6) sigma: purely repulsive, hard-sphere-like
    T_target = 1.0
    dt, n_steps = 0.003, 4000

    fig, ax = plt.subplots(figsize=(7, 5))
    peak_heights = {}
    for density, color in [(0.30, "steelblue"), (1.05, "darkorange")]:
        fluid = LJFluid.from_lattice(
            n_per_side=6,
            density=density,
            temperature=T_target,
            cutoff=r_wca_cutoff,
            rng=0,
            skin=0.3,
            rebuild_every=10,
        )
        thermostat = VelocityRescalingThermostat(target_temperature=T_target, interval=5)
        fluid.run(dt=dt, n_steps=n_steps, thermostat=thermostat, sample_every=200)
        r, g = fluid.radial_distribution_function(n_bins=120)
        plot_radial_distribution_function(r, g, ax=ax, color=color, label=f"density = {density}")
        peak_heights[density] = float(g.max())

    ax.set_title("Hard-sphere-like particles: dilute gas vs. densely packed")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/md/lj_fluid/images/sphx_glr_plot_03_hard_sphere_freezing_001.png
   :alt: Hard-sphere-like particles: dilute gas vs. densely packed
   :srcset: /api/gallery/md/lj_fluid/images/sphx_glr_plot_03_hard_sphere_freezing_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 54-65

The dilute configuration's g(r) is essentially featureless beyond the
repulsive core -- close to the flat g(r)=1 of an ideal gas, since
particles rarely encounter one another. At more than three times the
density, with exactly the same purely repulsive potential (no
attractive term added), the first-neighbor peak becomes far taller and
sharper: particles are packed closely enough that excluded volume alone
forces pronounced local order, exactly the entropic (packing-driven)
ordering mechanism Alder and Wainwright found in hard-sphere computer
experiments -- reproducing it here only qualitatively, since a
convincing first-order fluid-solid transition needs a much larger
system and a much longer run than this short pedagogical example:

.. GENERATED FROM PYTHON SOURCE LINES 65-70

.. code-block:: Python


    for density, height in peak_heights.items():
        print(f"density = {density}: g(r) first-peak height = {height:.2f}")

    plt.show()




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

 .. code-block:: none

    density = 0.3: g(r) first-peak height = 1.62
    density = 1.05: g(r) first-peak height = 3.34





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

   **Total running time of the script:** (0 minutes 1.227 seconds)


.. _sphx_glr_download_api_gallery_md_lj_fluid_plot_03_hard_sphere_freezing.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_hard_sphere_freezing.ipynb <plot_03_hard_sphere_freezing.ipynb>`

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

      :download:`Download Python source code: plot_03_hard_sphere_freezing.py <plot_03_hard_sphere_freezing.py>`

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

      :download:`Download zipped: plot_03_hard_sphere_freezing.zip <plot_03_hard_sphere_freezing.zip>`


.. only:: html

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

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