.. 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_07_einstein_mean_squared_displacement.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_07_einstein_mean_squared_displacement.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_md_lj_fluid_plot_07_einstein_mean_squared_displacement.py:


Einstein's diffusion law: mean-squared displacement grows linearly in time
=============================================================================

Einstein's 1905 analysis of Brownian motion showed that a particle
performing a random walk wanders off, on average, not in proportion to
time but to its square root: the mean-squared displacement grows
linearly,

.. math::

   \langle |\vec r(t) - \vec r(0)|^2 \rangle = 6Dt \quad (t\to\infty,\ 3\text{D}),

with the self-diffusion coefficient :math:`D` as the slope. In a
simulated Lennard-Jones liquid every atom performs such a random walk,
kicked about by its neighbors. This example unwraps the periodic
trajectory (:func:`~chemistrykit.md.systems.transport.unwrap_trajectory`),
computes the time-origin-averaged mean-squared displacement
(:func:`~chemistrykit.md.systems.transport.mean_squared_displacement`),
and reads :math:`D` from the linear regime
(:func:`~chemistrykit.md.systems.transport.einstein_diffusion_coefficient`).

.. GENERATED FROM PYTHON SOURCE LINES 25-59

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.md.systems.lj_fluid import LJFluid
    from chemistrykit.md.systems.thermostats import VelocityRescalingThermostat
    from chemistrykit.md.systems.transport import einstein_diffusion_coefficient, mean_squared_displacement

    T, dt, sample_every = 1.0, 0.005, 4
    fluid = LJFluid.from_lattice(n_per_side=6, density=0.7, temperature=T, cutoff=2.5, rng=0)
    fluid.run(dt=dt, n_steps=2000, thermostat=VelocityRescalingThermostat(T, interval=10), sample_every=2000)
    result = fluid.run(dt=dt, n_steps=4000, sample_every=sample_every)

    max_lag = 500
    msd = mean_squared_displacement(result.positions, box_length=result.box_length, max_lag=max_lag)
    lag_t = np.arange(max_lag + 1) * dt * sample_every
    D = einstein_diffusion_coefficient(lag_t, msd, fit_from=2.0)

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    axes[0].plot(lag_t, msd, color="steelblue", label="simulated MSD")
    axes[0].plot(lag_t, 6.0 * D * lag_t, color="black", linestyle="--", label=f"6Dt, D = {D:.3f}")
    axes[0].set_xlabel("t")
    axes[0].set_ylabel(r"$\langle|\Delta \vec r|^2\rangle$")
    axes[0].set_title("Mean-squared displacement")
    axes[0].legend()

    axes[1].loglog(lag_t[1:], msd[1:], color="steelblue", label="simulated MSD")
    v2 = 3.0 * T  # <v^2> = 3 k_B T / m
    axes[1].loglog(lag_t[1:40], v2 * lag_t[1:40] ** 2, color="gray", linestyle=":", label=r"ballistic $\langle v^2\rangle t^2$")
    axes[1].loglog(lag_t[40:], 6.0 * D * lag_t[40:], color="black", linestyle="--", label="diffusive 6Dt")
    axes[1].set_xlabel("t")
    axes[1].set_title("Log-log: ballistic, then diffusive")
    axes[1].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/md/lj_fluid/images/sphx_glr_plot_07_einstein_mean_squared_displacement_001.png
   :alt: Mean-squared displacement, Log-log: ballistic, then diffusive
   :srcset: /api/gallery/md/lj_fluid/images/sphx_glr_plot_07_einstein_mean_squared_displacement_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 60-63

At very short times an atom simply flies freely (MSD grows as
:math:`t^2`); after a few collisions its motion becomes a random walk
and the MSD settles onto Einstein's straight line:

.. GENERATED FROM PYTHON SOURCE LINES 63-67

.. code-block:: Python


    print(f"self-diffusion coefficient (reduced units): D = {D:.4f}")

    plt.show()




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

 .. code-block:: none

    self-diffusion coefficient (reduced units): D = 0.0949





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

   **Total running time of the script:** (0 minutes 2.587 seconds)


.. _sphx_glr_download_api_gallery_md_lj_fluid_plot_07_einstein_mean_squared_displacement.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_07_einstein_mean_squared_displacement.ipynb <plot_07_einstein_mean_squared_displacement.ipynb>`

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

      :download:`Download Python source code: plot_07_einstein_mean_squared_displacement.py <plot_07_einstein_mean_squared_displacement.py>`

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.. only:: html

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