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

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

.. _sphx_glr_api_gallery_statphys_renormalization_plot_block_spin_rg_flow.py:


Kadanoff block-spin renormalization group flow
====================================================

Real-space renormalization group (RG) transformations reveal *why*
critical phenomena are universal. Starting from an Ising configuration
(:math:`H = -J \sum_{\langle i,j \rangle} s_i s_j`, :math:`s_i = \pm 1`)
equilibrated at temperature :math:`T`, Kadanoff's construction repeatedly
replaces each non-overlapping :math:`2\times2` block of spins by a single
effective spin via majority rule,

.. math::

    s'_{I} = \mathrm{sign}\!\left(\sum_{i \,\in\, \text{block } I} s_i\right),

with an exact tie broken by a fair coin flip -- mapping the lattice onto a
smaller one that behaves as if sampled at some effective temperature
:math:`T'`. Iterating this map drives almost any starting configuration to
one of three fixed points: a fully ordered lattice (:math:`T \ll T_C`), a
fully disordered one (:math:`T \gg T_C`), or -- exactly at :math:`T_C` --
a self-similar configuration that looks statistically the same at every
coarse-graining step, tracked here via the order parameter :math:`|\langle
s \rangle|`, the visual and quantitative signature of scale invariance at
a critical point.

.. GENERATED FROM PYTHON SOURCE LINES 26-35

.. code-block:: Python


    import matplotlib.pyplot as plt

    from physicskit.statphys.chapters.ising_lattice import Ising2D
    from physicskit.statphys.chapters.renormalization import BlockSpinRG
    from physicskit.statphys.visualizers.rg_render import plot_rg_flow

    T_C = Ising2D().T_C








.. GENERATED FROM PYTHON SOURCE LINES 36-38

RG flow from three starting temperatures
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-48

.. code-block:: Python

    for T, label in [(0.5 * T_C, "T << T_C"), (T_C, "T = T_C"), (2.0 * T_C, "T >> T_C")]:
        rg = BlockSpinRG(L=64, T=T, J=1.0, n_equil_sweeps=400, seed=0)
        grids = rg.iterate(n_steps=4)
        orders = [rg.order_parameter(g) for g in grids]

        axes = plot_rg_flow(grids, titles=[f"L={g.shape[0]}\n|m|={m:.2f}" for g, m in zip(grids, orders)])
        plt.gcf().suptitle(f"Block-spin RG flow, {label} = {T:.2f}")
        plt.tight_layout()

    plt.show()



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


    *

      .. image-sg:: /api/gallery/statphys/renormalization/images/sphx_glr_plot_block_spin_rg_flow_001.png
         :alt: Block-spin RG flow, T << T_C = 1.13, L=64 |m|=1.00, L=32 |m|=1.00, L=16 |m|=1.00, L=8 |m|=1.00, L=4 |m|=1.00
         :srcset: /api/gallery/statphys/renormalization/images/sphx_glr_plot_block_spin_rg_flow_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /api/gallery/statphys/renormalization/images/sphx_glr_plot_block_spin_rg_flow_002.png
         :alt: Block-spin RG flow, T = T_C = 2.27, L=64 |m|=0.72, L=32 |m|=0.80, L=16 |m|=0.88, L=8 |m|=0.88, L=4 |m|=0.88
         :srcset: /api/gallery/statphys/renormalization/images/sphx_glr_plot_block_spin_rg_flow_002.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /api/gallery/statphys/renormalization/images/sphx_glr_plot_block_spin_rg_flow_003.png
         :alt: Block-spin RG flow, T >> T_C = 4.54, L=64 |m|=0.00, L=32 |m|=0.00, L=16 |m|=0.12, L=8 |m|=0.19, L=4 |m|=0.25
         :srcset: /api/gallery/statphys/renormalization/images/sphx_glr_plot_block_spin_rg_flow_003.png
         :class: sphx-glr-multi-img






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

   **Total running time of the script:** (0 minutes 0.150 seconds)


.. _sphx_glr_download_api_gallery_statphys_renormalization_plot_block_spin_rg_flow.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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