.. _sphx_glr_api_gallery_md_lj_fluid:

Lennard-Jones fluid
--------------------

Simulations of the Lennard-Jones fluid: Rahman's liquid argon, the
Maxwell-Boltzmann speed distribution, hard-sphere-like packing order,
the Weeks-Chandler-Andersen repulsive reference fluid, Verlet's
integrator and neighbor list, and self-diffusion from the Einstein
mean-squared displacement and the Green-Kubo velocity autocorrelation.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Rahman&#x27;s 1964 simulation followed 864 argon atoms interacting through a Lennard-Jones potential with \epsilon/k_B = 120\ \text{K} and \sigma = 3.4\ \text{\AA}, at T = 94.4\ \text{K} and a mass density of 1.374\ \text{g/cm}^3 -- liquid argon just above its triple point. Here the same state point is simulated with LJFluid (512 atoms, reduced units), and the radial distribution function (~chemistrykit.md.systems.lj_fluid.LJFluid.radial_distribution_function) is converted back to ångströms, as Rahman compared it with neutron- and X-ray-scattering data on real liquid argon.">

.. only:: html

  .. image:: /api/gallery/md/lj_fluid/images/thumb/sphx_glr_plot_01_rahman_liquid_argon_thumb.png
    :alt:

  :doc:`/api/gallery/md/lj_fluid/plot_01_rahman_liquid_argon`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Rahman's liquid argon: g(r) of a Lennard-Jones liquid in real units</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A direct link between chemistrykit.md and chemistrykit.statmech: molecular dynamics produces an actual ensemble of particle velocities, and statistical mechanics predicts what their distribution should look like at equilibrium. After a short equilibration run, an LJFluid&#x27;s particle speeds are histogrammed and compared against chemistrykit.statmech.MaxwellBoltzmannSpeedDistribution evaluated at the simulation&#x27;s own (instantaneous, measured) temperature -- no fitting involved.">

.. only:: html

  .. image:: /api/gallery/md/lj_fluid/images/thumb/sphx_glr_plot_02_maxwell_boltzmann_check_thumb.png
    :alt:

  :doc:`/api/gallery/md/lj_fluid/plot_02_maxwell_boltzmann_check`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Cross-checking an MD trajectory against the Maxwell-Boltzmann distribution</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Alder and Wainwright&#x27;s 1957 computer experiments showed that particles with no attraction at all -- hard spheres -- order into a crystal-like arrangement once they are packed densely enough: freezing driven by excluded volume (entropy), not by energy. This package integrates smooth forces rather than hard-sphere collision events, so the hard spheres are approximated by steeply repulsive particles: the Lennard-Jones potential cut off at its own minimum r=2^{1/6}\sigma (~chemistrykit.md.systems.lj_fluid.LennardJones.r_min) and shifted, which leaves only a repulsive wall. Comparing the radial distribution function g(r) of this purely repulsive LJFluid at low density and at a density near hard-sphere freezing shows sharp positional order appearing from packing alone.">

.. only:: html

  .. image:: /api/gallery/md/lj_fluid/images/thumb/sphx_glr_plot_03_hard_sphere_freezing_thumb.png
    :alt:

  :doc:`/api/gallery/md/lj_fluid/plot_03_hard_sphere_freezing`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Alder and Wainwright's hard spheres: packing-driven order with no attraction</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Weeks, Chandler, and Andersen split the Lennard-Jones potential at its minimum r_m=2^{1/6}\sigma into a purely repulsive reference part u_0 and a smooth attractive perturbation w:">

.. only:: html

  .. image:: /api/gallery/md/lj_fluid/images/thumb/sphx_glr_plot_04_wca_repulsive_reference_thumb.png
    :alt:

  :doc:`/api/gallery/md/lj_fluid/plot_04_wca_repulsive_reference`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Weeks-Chandler-Andersen: repulsive forces set the structure of a dense liquid</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Verlet&#x27;s 1967 algorithm advances positions from the two previous positions and the current force,">

.. only:: html

  .. image:: /api/gallery/md/lj_fluid/images/thumb/sphx_glr_plot_05_verlet_integrator_thumb.png
    :alt:

  :doc:`/api/gallery/md/lj_fluid/plot_05_verlet_integrator`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Verlet's integrator: bounded energy error instead of drift</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Checking every pair of N particles costs N(N-1)/2 distance evaluations per step. Verlet&#x27;s 1967 trick: list the pairs within r_c + r_\text{skin} and reuse that list for several steps; a pair outside the list can only come inside the true cutoff r_c after the two particles have closed the skin distance between them. VerletNeighborList implements this. Here a Lennard-Jones fluid is run with a list that is never rebuilt, and at every step the listed pairs are checked against the exact set of pairs inside the cutoff (~chemistrykit.md.utils.neighbor_list.build_neighbor_list), for lists built with no skin and with a skin of 0.3\sigma.">

.. only:: html

  .. image:: /api/gallery/md/lj_fluid/images/thumb/sphx_glr_plot_06_verlet_neighbor_list_thumb.png
    :alt:

  :doc:`/api/gallery/md/lj_fluid/plot_06_verlet_neighbor_list`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Verlet's neighbor list: a skin that lets one pair list serve many steps</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Einstein&#x27;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,">

.. only:: html

  .. image:: /api/gallery/md/lj_fluid/images/thumb/sphx_glr_plot_07_einstein_mean_squared_displacement_thumb.png
    :alt:

  :doc:`/api/gallery/md/lj_fluid/plot_07_einstein_mean_squared_displacement`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Einstein's diffusion law: mean-squared displacement grows linearly in time</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Green (1954) and Kubo (1957) showed that transport coefficients are time integrals of equilibrium correlation functions. For self-diffusion the relevant function is the velocity autocorrelation C(t)=\langle\vec v(0)\cdot\vec v(t)\rangle, and">

.. only:: html

  .. image:: /api/gallery/md/lj_fluid/images/thumb/sphx_glr_plot_08_green_kubo_velocity_autocorrelation_thumb.png
    :alt:

  :doc:`/api/gallery/md/lj_fluid/plot_08_green_kubo_velocity_autocorrelation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Green-Kubo: the diffusion coefficient as the integral of the velocity autocorrelation</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/md/lj_fluid/plot_01_rahman_liquid_argon
   /api/gallery/md/lj_fluid/plot_02_maxwell_boltzmann_check
   /api/gallery/md/lj_fluid/plot_03_hard_sphere_freezing
   /api/gallery/md/lj_fluid/plot_04_wca_repulsive_reference
   /api/gallery/md/lj_fluid/plot_05_verlet_integrator
   /api/gallery/md/lj_fluid/plot_06_verlet_neighbor_list
   /api/gallery/md/lj_fluid/plot_07_einstein_mean_squared_displacement
   /api/gallery/md/lj_fluid/plot_08_green_kubo_velocity_autocorrelation