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

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

.. _sphx_glr_api_gallery_statphys_molecular_dynamics_plot_maxwell_boltzmann_relaxation.py:


Boltzmann's H-theorem: relaxation to the Maxwell-Boltzmann distribution
============================================================================

This example simulates a 2D periodic gas of :math:`N=200` particles
interacting pairwise through the Lennard-Jones potential

.. math::

    V(r) = 4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12} -
    \left(\frac{\sigma}{r}\right)^{6}\right],

integrated forward with the symplectic Velocity Verlet scheme. Boltzmann's
H-theorem is one of the most celebrated (and originally controversial)
results in statistical mechanics: even though this underlying Newtonian
dynamics is perfectly time-reversible, the coarse-grained velocity
distribution evolves irreversibly toward the Maxwell-Boltzmann equilibrium
form, with the H-function

.. math::

    H = \int_0^\infty f(v) \ln \frac{f(v)}{2\pi v} \, dv

(Boltzmann's :math:`\int f \ln f \, d^2v` written in terms of the speed
distribution :math:`f(v)`, with :math:`2\pi v` the 2D Jacobian) decreasing monotonically on average as the gas thermalizes, since the
Maxwell-Boltzmann distribution is the one that minimizes :math:`H` at
fixed energy. This example starts the gas from a highly artificial
"delta-function" speed distribution -- every particle moving at the same
speed, in a random direction -- and watches it relax.

.. GENERATED FROM PYTHON SOURCE LINES 31-37

.. code-block:: Python


    import matplotlib.pyplot as plt

    from physicskit.statphys.chapters.molecular_dynamics import LennardJonesGas
    from physicskit.statphys.visualizers.particle_render import plot_velocity_histogram








.. GENERATED FROM PYTHON SOURCE LINES 38-40

Relax a far-from-equilibrium gas and track H(t)
---------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 40-65

.. code-block:: Python

    gas = LennardJonesGas(
        n_particles=200,
        box_size=25.0,
        temperature_init=1.2,
        initial_velocity_distribution="delta",
        dt=0.003,
        seed=0,
    )
    history = gas.run(n_steps=4000, steps_per_record=40)

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].plot(history["time"], history["H"])
    axes[0].set_xlabel("time")
    axes[0].set_ylabel("H(t)")
    axes[0].set_title("Boltzmann H-function: monotonic decrease")

    axes[1].plot(history["time"], history["temperature"], label="temperature")
    ax2 = axes[1].twinx()
    ax2.plot(history["time"], history["total_energy"], color="crimson", label="total energy")
    axes[1].set_xlabel("time")
    axes[1].set_ylabel("temperature")
    ax2.set_ylabel("total energy", color="crimson")
    axes[1].set_title("Thermalization and energy conservation")
    plt.tight_layout()




.. image-sg:: /api/gallery/statphys/molecular_dynamics/images/sphx_glr_plot_maxwell_boltzmann_relaxation_001.png
   :alt: Boltzmann H-function: monotonic decrease, Thermalization and energy conservation
   :srcset: /api/gallery/statphys/molecular_dynamics/images/sphx_glr_plot_maxwell_boltzmann_relaxation_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 66-68

Initial vs. final speed distribution against Maxwell-Boltzmann
--------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 68-73

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(6, 4))
    plot_velocity_histogram(gas, ax=ax)
    ax.set_title(f"Final speed distribution (t = {gas.time:.1f})")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/statphys/molecular_dynamics/images/sphx_glr_plot_maxwell_boltzmann_relaxation_002.png
   :alt: Final speed distribution (t = 12.0)
   :srcset: /api/gallery/statphys/molecular_dynamics/images/sphx_glr_plot_maxwell_boltzmann_relaxation_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.294 seconds)


.. _sphx_glr_download_api_gallery_statphys_molecular_dynamics_plot_maxwell_boltzmann_relaxation.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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