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

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

.. _sphx_glr_api_gallery_statphys_ising_plot_ising_phase_transition.py:


Onsager's exact solution: the 2D Ising phase transition
=========================================================

The 2D Ising model places a spin :math:`s_i = \pm 1` on every site of a
periodic :math:`L \times L` lattice, with Hamiltonian

.. math::

    H = -J \sum_{\langle i,j \rangle} s_i s_j,

summed over nearest-neighbor pairs. It is the simplest system with a
genuine continuous (second-order) phase transition, and the only
nontrivial one solved exactly -- by Lars Onsager in 1944. Below the
critical temperature :math:`T_C = 2J / (k_B \ln(1+\sqrt{2})) \approx
2.269\, J/k_B`, the system spontaneously magnetizes; above it, thermal
fluctuations destroy the order. This example sweeps temperature across
:math:`T_C` on an :math:`L=32` lattice with the cluster-flipping Wolff
algorithm (to avoid critical slowing down) and reproduces the textbook
signatures of the transition: the magnetization order parameter dropping
to zero, and the specific heat and susceptibility both peaking at
:math:`T_C`.

.. GENERATED FROM PYTHON SOURCE LINES 24-36

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np
    from matplotlib.colors import ListedColormap

    from physicskit.statphys.chapters.ising_lattice import Ising2D
    from physicskit.statphys.visualizers.lattice_render import plot_spin_grid, plot_thermodynamics

    # Two-valued spin configurations (s = +-1) render more clearly as light grey
    # vs. black than the default blue/red "coolwarm" colormap.
    SPIN_CMAP = ListedColormap(["black", "lightgrey"])








.. GENERATED FROM PYTHON SOURCE LINES 37-41

Sweep temperature across the transition
----------------------------------------
Sampling from high to low temperature lets each equilibration step start
from the (already close to equilibrium) previous configuration.

.. GENERATED FROM PYTHON SOURCE LINES 41-49

.. code-block:: Python


    model = Ising2D(L=32, J=1.0, kB=1.0, seed=0)
    temperatures = np.linspace(model.T_C + 1.2, max(model.T_C - 1.2, 0.5), 20)
    result = model.run_temperature_sweep(temperatures, n_equil=200, n_measure=300, algorithm="wolff")

    plot_thermodynamics(result, T_c=model.T_C)
    plt.suptitle("2D Ising model: thermodynamics across $T_C$")




.. image-sg:: /api/gallery/statphys/ising/images/sphx_glr_plot_ising_phase_transition_001.png
   :alt: 2D Ising model: thermodynamics across $T_C$
   :srcset: /api/gallery/statphys/ising/images/sphx_glr_plot_ising_phase_transition_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 0.98, '2D Ising model: thermodynamics across $T_C$')



.. GENERATED FROM PYTHON SOURCE LINES 50-52

Snapshots below, at, and above :math:`T_C`
--------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 52-60

.. code-block:: Python

    fig, axes = plt.subplots(1, 3, figsize=(10, 3.5))
    for ax, T in zip(axes, [model.T_C - 1.0, model.T_C, model.T_C + 1.0]):
        snapshot_model = Ising2D(L=64, J=1.0, kB=1.0, seed=1)
        snapshot_model.sweep(beta=1.0 / T, algorithm="wolff", n_sweeps=200)
        plot_spin_grid(snapshot_model.spins, ax=ax, title=f"T = {T:.2f}", cmap=SPIN_CMAP)
    plt.tight_layout()
    plt.subplots_adjust(top=0.88)  # tight_layout alone leaves the titles clipped by the figure edge
    plt.show()



.. image-sg:: /api/gallery/statphys/ising/images/sphx_glr_plot_ising_phase_transition_002.png
   :alt: T = 1.27, T = 2.27, T = 3.27
   :srcset: /api/gallery/statphys/ising/images/sphx_glr_plot_ising_phase_transition_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.232 seconds)


.. _sphx_glr_download_api_gallery_statphys_ising_plot_ising_phase_transition.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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