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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_msd_1.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_plot_msd_1.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_plot_msd_1.py:


===============================
Mean-square displacements (MSD)
===============================

Generate a number of random walks and compute their MSD.

Compute also the MSD for a constant velocity motion.

For a random walk, the MSD is linear: :math:`MSD(\tau) \approx 2 D \tau`

For a constant velocity motion, the MSD is quadratic:
:math:`MSD(\tau) = v \tau^2`

We show in the figures the numerical result computed by `tidynamics.msd`
('num.') and the theoretical value ('theo.').

For the constant velocity case, we also display a "pedestrian approach" where
the loop for averaging the MSD is performed explicitly.

.. GENERATED FROM PYTHON SOURCE LINES 21-83



.. image-sg:: /auto_examples/images/sphx_glr_plot_msd_1_001.png
   :alt: Examples for the mean-square displacement
   :srcset: /auto_examples/images/sphx_glr_plot_msd_1_001.png
   :class: sphx-glr-single-img





.. code-block:: default


    import numpy as np
    import tidynamics
    import matplotlib.pyplot as plt

    plt.rcParams['figure.figsize'] = 5.12, 3.84
    plt.rcParams['figure.subplot.bottom'] = 0.18
    plt.rcParams['figure.subplot.left'] = 0.16

    # Generate 32 random walks and compute their mean-square
    # displacements

    N = 1000
    mean = np.zeros(N)
    count = 0
    for i in range(32):
        # Generate steps of value +/- 1
        steps = -1 + 2*np.random.randint(0, 2, size=(N, 2))
        # Compute random walk position
        data = np.cumsum(steps, axis=0)
        mean += tidynamics.msd(data)
        count += 1

    mean /= count
    mean = mean[1:N//2]

    time = np.arange(N)[1:N//2]

    plt.plot(time, mean, label='Random walk (num.)')

    plt.plot(time, 2*time, label='Random walk (theo.)')

    time = np.arange(N//2)

    # Display the mean-square displacement for a trajectory with
    # constant velocity. Here the trajectory is taken equal to the
    # numerical value of the time.

    plt.plot(time[1:], tidynamics.msd(time)[1:],
             label='Constant velocity (num.)', ls='--')
    plt.plot(time[1:], time[1:]**2,
             label='Constant velocity (theo.)', ls='--')

    # Compute the the mean-square displacement by explicitly
    # computing the displacements along shorter samples of the
    # trajectory.

    sum_size = N//10
    pedestrian_msd = np.zeros(N//10)
    for i in range(10):
        for j in range(N//10):
            pedestrian_msd[j] += (time[10*i]-time[10*i+j])**2
    pedestrian_msd /= 10
    plt.plot(time[1:N//10], pedestrian_msd[1:], ls='--',
             label="pedestrian")

    plt.loglog()
    plt.legend()
    plt.xlabel('time')
    plt.ylabel('mean square displacement')
    plt.title('Examples for the mean-square displacement')
    plt.show()


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

   **Total running time of the script:** ( 0 minutes  0.373 seconds)


.. _sphx_glr_download_auto_examples_plot_msd_1.py:

.. only:: html

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




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

      :download:`Download Python source code: plot_msd_1.py <plot_msd_1.py>`

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

      :download:`Download Jupyter notebook: plot_msd_1.ipynb <plot_msd_1.ipynb>`


.. only:: html

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

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