.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statmech/ising/plot_03_onsager_exact_solution.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_statmech_ising_plot_03_onsager_exact_solution.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_statmech_ising_plot_03_onsager_exact_solution.py:


Onsager's exact solution of the two-dimensional Ising model
==============================================================

Onsager (1944) computed the free energy of the square-lattice Ising model
exactly: the first exact solution of a model with a genuine phase
transition. :class:`~chemistrykit.statmech.Ising2DOnsager` evaluates his
free energy, internal energy, and heat capacity, together with Yang's
(1952) spontaneous magnetization :math:`m=[1-\sinh^{-4}(2J/k_BT)]^{1/8}`.
The heat capacity diverges logarithmically at :math:`T_c`, and the
magnetization vanishes with exponent :math:`\beta=1/8`. Neither the
mean-field (Bragg-Williams) theory, shown for comparison, nor Ising's own
one-dimensional chain reproduces these features.

.. GENERATED FROM PYTHON SOURCE LINES 17-65

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.optimize import brentq

    from chemistrykit.constants import K_B
    from chemistrykit.statmech import Ising2DOnsager

    J = 100.0 * K_B
    model = Ising2DOnsager(coupling=J)
    Tc = model.critical_temperature
    print(f"T_c = {Tc:.2f} K")

    T = np.linspace(20.0, 2.0 * Tc, 500)
    T = T[np.abs(T - Tc) > 1e-6 * Tc]


    # Mean-field comparison: m = tanh(4 J m / k_B T), T_c(MF) = 4 J / k_B
    def mean_field_m(t):
        if t >= 4 * J / K_B:
            return 0.0
        return brentq(lambda m: m - np.tanh(4 * J * m / (K_B * t)), 1e-9, 1.0)


    fig, axes = plt.subplots(1, 3, figsize=(14, 4.2))
    axes[0].plot(T / Tc, model.spontaneous_magnetization(T), color="steelblue", label="Onsager-Yang (exact)")
    axes[0].plot(T / Tc, [mean_field_m(t) for t in T], color="gray", linestyle="--", label="mean field")
    axes[0].set_xlabel(r"$T / T_c$")
    axes[0].set_ylabel("spontaneous magnetization")
    axes[0].set_title(r"Order parameter, $\beta = 1/8$")
    axes[0].legend()

    axes[1].plot(T / Tc, model.internal_energy_per_spin(T) / J, color="steelblue")
    axes[1].plot(T / Tc, model.free_energy_per_spin(T) / J, color="crimson")
    axes[1].annotate("u / J", (1.5, -0.8), color="steelblue")
    axes[1].annotate("f / J", (1.5, -2.9), color="crimson")
    axes[1].axvline(1.0, color="gray", linestyle=":", linewidth=0.8)
    axes[1].set_xlabel(r"$T / T_c$")
    axes[1].set_ylabel("energy per spin / J")
    axes[1].set_title("Energy and free energy")

    axes[2].plot(T / Tc, model.heat_capacity_per_spin(T) / K_B, color="steelblue")
    axes[2].axvline(1.0, color="gray", linestyle=":", linewidth=0.8)
    axes[2].set_ylim(0.0, 4.0)
    axes[2].set_xlabel(r"$T / T_c$")
    axes[2].set_ylabel(r"$c / k_B$ per spin")
    axes[2].set_title("Logarithmically divergent heat capacity")
    fig.tight_layout()




.. image-sg:: /api/gallery/statmech/ising/images/sphx_glr_plot_03_onsager_exact_solution_001.png
   :alt: Order parameter, $\beta = 1/8$, Energy and free energy, Logarithmically divergent heat capacity
   :srcset: /api/gallery/statmech/ising/images/sphx_glr_plot_03_onsager_exact_solution_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    T_c = 226.92 K




.. GENERATED FROM PYTHON SOURCE LINES 66-68

Close to T_c the heat capacity grows like -(8/pi) K_c^2 ln|1 - T/T_c|:
each factor-of-ten step closer to T_c adds a constant amount.

.. GENERATED FROM PYTHON SOURCE LINES 68-73

.. code-block:: Python


    for eps in (1e-2, 1e-3, 1e-4, 1e-5):
        print(f"1 - T/T_c = {eps:.0e}: c/k_B = {model.heat_capacity_per_spin(Tc * (1 - eps)) / K_B:.4f}")

    plt.show()




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

 .. code-block:: none

    1 - T/T_c = 1e-02: c/k_B = 1.9698
    1 - T/T_c = 1e-03: c/k_B = 3.1098
    1 - T/T_c = 1e-04: c/k_B = 4.2486
    1 - T/T_c = 1e-05: c/k_B = 5.3873





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

   **Total running time of the script:** (0 minutes 0.113 seconds)


.. _sphx_glr_download_api_gallery_statmech_ising_plot_03_onsager_exact_solution.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_onsager_exact_solution.ipynb <plot_03_onsager_exact_solution.ipynb>`

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

      :download:`Download Python source code: plot_03_onsager_exact_solution.py <plot_03_onsager_exact_solution.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_03_onsager_exact_solution.zip <plot_03_onsager_exact_solution.zip>`


.. only:: html

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

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