.. DO NOT EDIT.
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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/md/thermostats/plot_01_nose_hoover_thermostat.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_thermostats_plot_01_nose_hoover_thermostat.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_md_thermostats_plot_01_nose_hoover_thermostat.py:


The Nose-Hoover thermostat: a friction variable that steers the temperature
==============================================================================

Nose's extended system, in Hoover's friction-coefficient form, adds a
single dynamical variable :math:`\xi` to Newton's equations:

.. math::

   \dot{\vec v}_i = \frac{\vec F_i}{m_i} - \xi \vec v_i, \qquad
   Q\dot\xi = \sum_i m_i v_i^2 - N_f k_B T_0 .

When the fluid is too hot, :math:`\xi` grows positive and drains kinetic
energy; when it is too cold, :math:`\xi` turns negative and feeds energy
back in. A Lennard-Jones fluid started well above the target temperature
is run without a thermostat (NVE) and with
:class:`~chemistrykit.md.systems.thermostats.NoseHooverThermostat` at
two thermostat masses :math:`Q`, recording the temperature and the
friction coefficient :math:`\xi`.

.. GENERATED FROM PYTHON SOURCE LINES 23-67

.. code-block:: Python

    import matplotlib.pyplot as plt

    from chemistrykit.md.systems.lj_fluid import LJFluid
    from chemistrykit.md.systems.thermostats import NoseHooverThermostat

    T_target = 1.0
    dt, n_steps = 0.004, 5000


    def make_hot_fluid():
        fluid = LJFluid.from_lattice(n_per_side=6, density=0.6, temperature=T_target, cutoff=2.5, rng=0)
        fluid.velocities *= 1.8  # start well above the target temperature
        return fluid


    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    nve = make_hot_fluid()
    result = nve.run(dt=dt, n_steps=n_steps, sample_every=10)
    axes[0].plot(result.t, result.temperature, color="gray", label="NVE (no thermostat)")

    for Q, color in [(5.0, "steelblue"), (50.0, "darkorange")]:
        fluid = make_hot_fluid()
        thermostat = NoseHooverThermostat(target_temperature=T_target, Q=Q)
        times, temps, xis = [], [], []
        for _ in range(n_steps // 10):
            fluid.run(dt=dt, n_steps=10, thermostat=thermostat, sample_every=10)
            times.append(fluid.t)
            temps.append(fluid.temperature())
            xis.append(thermostat.xi)
        axes[0].plot(times, temps, color=color, label=f"Nose-Hoover, Q = {Q:g}")
        axes[1].plot(times, xis, color=color, label=f"Q = {Q:g}")

    axes[0].axhline(T_target, color="black", linestyle=":", linewidth=0.8, label="target T")
    axes[0].set_xlabel("t")
    axes[0].set_ylabel("instantaneous temperature")
    axes[0].set_title("Temperature")
    axes[0].legend(loc="upper right")
    axes[1].axhline(0.0, color="black", linestyle=":", linewidth=0.8)
    axes[1].set_xlabel("t")
    axes[1].set_ylabel(r"friction coefficient $\xi$")
    axes[1].set_title("The extended-system variable")
    axes[1].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/md/thermostats/images/sphx_glr_plot_01_nose_hoover_thermostat_001.png
   :alt: Temperature, The extended-system variable
   :srcset: /api/gallery/md/thermostats/images/sphx_glr_plot_01_nose_hoover_thermostat_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 68-75

Without a thermostat the fluid stays hot. With Nose-Hoover the
friction coefficient first rises to cool the fluid, overshoots, and
then oscillates about zero, so the temperature swings back and forth
around the target rather than being clamped to it; a heavier
thermostat mass Q responds more slowly, with longer, gentler swings.
Averaged over the second half of each run, the temperature sits close
to the target:

.. GENERATED FROM PYTHON SOURCE LINES 75-81

.. code-block:: Python


    for line in axes[0].get_lines()[1:3]:
        temps = line.get_ydata()
        print(f"{line.get_label()}: mean T over second half = {sum(temps[len(temps) // 2 :]) / (len(temps) - len(temps) // 2):.3f}")

    plt.show()




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

 .. code-block:: none

    Nose-Hoover, Q = 5: mean T over second half = 1.000
    Nose-Hoover, Q = 50: mean T over second half = 0.999





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

   **Total running time of the script:** (0 minutes 4.702 seconds)


.. _sphx_glr_download_api_gallery_md_thermostats_plot_01_nose_hoover_thermostat.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_nose_hoover_thermostat.ipynb <plot_01_nose_hoover_thermostat.ipynb>`

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

      :download:`Download Python source code: plot_01_nose_hoover_thermostat.py <plot_01_nose_hoover_thermostat.py>`

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