.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_acf_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_acf_1.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_plot_acf_1.py:


=====================
Correlation functions
=====================

Generate the velocity for a Ornstein-Uhlenbeck process and compute its
autocorrelation function.

We also display the force velocity correlation as an example of using the
routine `correlation`.

.. GENERATED FROM PYTHON SOURCE LINES 12-81



.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /auto_examples/images/sphx_glr_plot_acf_1_001.png
         :alt: Velocity autocorrelation
         :srcset: /auto_examples/images/sphx_glr_plot_acf_1_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_plot_acf_1_002.png
         :alt: Force-velocity correlation
         :srcset: /auto_examples/images/sphx_glr_plot_acf_1_002.png
         :class: sphx-glr-multi-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 data for a Ornstein-Uhlenbeck process

    gamma = 2.7
    T = 0.1
    dt = 0.02
    v_factor = np.sqrt(2*T*gamma*dt)

    N = 32768
    v = 0
    for i in range(100):
        noise_force = v_factor*np.random.normal()
        v = v - gamma*v*dt + noise_force

    v_data = []
    noise_data = []

    for i in range(N):
        noise_force = v_factor*np.random.normal()
        v = v - gamma*v*dt + noise_force
        v_data.append(v)
        noise_data.append(noise_force)
    v_data = np.array(v_data)
    noise_data = np.array(noise_data)/np.sqrt(dt)

    # Compute the autocorrelation function and plot the result

    acf = tidynamics.acf(v_data)[:N//64]

    time = np.arange(N//64)*dt

    plt.plot(time, acf, label='VACF (num.)')

    plt.plot(time, T*np.exp(-gamma*time), label='VACF (theo.)')

    plt.legend()
    plt.title('Velocity autocorrelation')
    plt.xlabel(r'$\tau$')
    plt.ylabel(r'$\langle v(t) v(t+\tau) \rangle$')

    # Compute the force velocity correlation and plot the result

    plt.figure()

    time = np.arange(N)*dt
    twotimes = np.concatenate((-time[1:][::-1], time))

    plt.plot(twotimes, tidynamics.correlation(noise_data, v_data),
             label='num.')
    plt.plot(time, 2*T/gamma*np.exp(-gamma*time), label='theo.')

    plt.xlim(-5/gamma, 5/gamma)

    plt.ylim(-2*T/gamma/10, 1.1*2*T/gamma)

    plt.legend()
    plt.title('Force-velocity correlation')
    plt.xlabel(r'$\tau$')
    plt.ylabel(r'$\langle F(t) v(t+\tau) \rangle$')

    plt.show()


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

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


.. _sphx_glr_download_auto_examples_plot_acf_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_acf_1.py <plot_acf_1.py>`

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

      :download:`Download Jupyter notebook: plot_acf_1.ipynb <plot_acf_1.ipynb>`


.. only:: html

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

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