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

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

.. _sphx_glr_api_gallery_statphys_kpz_plot_kpz_roughening.py:


Universal interface roughening: the Kardar-Parisi-Zhang universality class
=================================================================================

Kardar, Parisi, and Zhang's 1986 equation for a growing interface,

.. math::

    \frac{\partial h}{\partial t} = \nu \nabla^2 h
    + \frac{\lambda}{2}(\nabla h)^2 + \eta(x, t),

predicts universal scaling for the interface width :math:`w(t) =
\sqrt{\langle (h - \bar h)^2 \rangle}`: early-time growth :math:`w \sim
t^\beta` with :math:`\beta = 1/3`, crossing over to a size-limited
saturation :math:`w_{\text{sat}} \sim L^\alpha` with :math:`\alpha = 1/2`
once :math:`t` exceeds the correlation time :math:`\tau \sim L^z`
(:math:`z = 3/2`). Rather than integrate this delicate nonlinear SPDE
directly, this example uses the Restricted Solid-On-Solid (RSOS) growth
automaton -- a purely local, parameter-free deposition rule long known to
lie in the same universality class -- and recovers both exponents directly
from simulation.

.. GENERATED FROM PYTHON SOURCE LINES 23-31

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.statphys.chapters.kpz_growth import KPZInterface
    from physicskit.statphys.utils.finite_size_scaling import power_law_exponent
    from physicskit.statphys.visualizers.kpz_render import plot_width_growth








.. GENERATED FROM PYTHON SOURCE LINES 32-37

Early-time growth: w(t) ~ t^(1/3)
------------------------------------------------------------
A single large interface (L=400) is grown far short of its own
saturation time (~L^1.5), so the width is still in the free, unsaturated
growth regime throughout.

.. GENERATED FROM PYTHON SOURCE LINES 37-45

.. code-block:: Python

    interface = KPZInterface(L=400, seed=0)
    times, widths = interface.run_growth_curve(t_max=1500, n_points=30)
    beta_fit, _ = power_law_exponent(times, widths)

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    plot_width_growth(times, widths, ax=axes[0], beta=1.0 / 3.0)
    axes[0].set_title(f"Growth regime: fitted beta = {beta_fit:.3f} (theory 1/3)")




.. image-sg:: /api/gallery/statphys/kpz/images/sphx_glr_plot_kpz_roughening_001.png
   :alt: Growth regime: fitted beta = 0.315 (theory 1/3)
   :srcset: /api/gallery/statphys/kpz/images/sphx_glr_plot_kpz_roughening_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Growth regime: fitted beta = 0.315 (theory 1/3)')



.. GENERATED FROM PYTHON SOURCE LINES 46-52

Saturation: w_sat(L) ~ L^(1/2)
------------------------------------------------------------
Several smaller interfaces are grown well *past* their own saturation
time (5 L^1.5 sweeps), and their long-time-averaged width is recorded --
the finite-size-limited plateau the growth regime above crosses over
into.

.. GENERATED FROM PYTHON SOURCE LINES 52-78

.. code-block:: Python

    L_values = np.array([32, 64, 96, 128])
    w_sat = []
    for L in L_values:
        iface = KPZInterface(L=int(L), seed=1)
        iface.grow(n_sweeps=int(5 * L**1.5))  # run well past the L^1.5 saturation time
        samples = []
        for _ in range(20):
            iface.grow(n_sweeps=int(0.2 * L**1.5))
            samples.append(iface.width())
        w_sat.append(np.mean(samples))
    w_sat = np.array(w_sat)
    alpha_fit, _ = power_law_exponent(L_values, w_sat)

    axes[1].loglog(L_values, w_sat, marker="o", linestyle="none", label="simulation")
    amplitude = w_sat[0] / L_values[0] ** 0.5
    axes[1].loglog(L_values, amplitude * L_values**0.5, linestyle="--", label=r"$L^{1/2}$")
    axes[1].set_xlabel("L")
    axes[1].set_ylabel(r"$w_{\text{sat}}(L)$")
    axes[1].set_title(f"Saturation regime: fitted alpha = {alpha_fit:.3f} (theory 1/2)")
    axes[1].legend()

    plt.tight_layout()
    plt.show()

    print(f"Growth exponent beta = {beta_fit:.3f} (KPZ theory: 1/3)")
    print(f"Roughness exponent alpha = {alpha_fit:.3f} (KPZ theory: 1/2)")



.. image-sg:: /api/gallery/statphys/kpz/images/sphx_glr_plot_kpz_roughening_002.png
   :alt: plot kpz roughening
   :srcset: /api/gallery/statphys/kpz/images/sphx_glr_plot_kpz_roughening_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Growth exponent beta = 0.315 (KPZ theory: 1/3)
    Roughness exponent alpha = 0.481 (KPZ theory: 1/2)





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

   **Total running time of the script:** (0 minutes 0.119 seconds)


.. _sphx_glr_download_api_gallery_statphys_kpz_plot_kpz_roughening.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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