.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statmech/lattice_gas/plot_02_boltzmann_entropy_counting.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_lattice_gas_plot_02_boltzmann_entropy_counting.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_statmech_lattice_gas_plot_02_boltzmann_entropy_counting.py:


Boltzmann's S = k ln W by counting lattice arrangements
==========================================================

Boltzmann's 1877 insight, written by Planck as :math:`S=k_B\ln W`: entropy
counts the microscopic arrangements `W` compatible with a macroscopic
state. For `N` indistinguishable molecules on `M` lattice sites,
:math:`W=\binom{M}{N}`, and
:meth:`~chemistrykit.statmech.LatticeGasAdsorption.canonical_entropy`
evaluates :math:`k_B\ln W` exactly via
:func:`~chemistrykit.statmech.ln_binomial`. The entropy is largest at
half filling, where the number of arrangements peaks, and per site it
converges to the Stirling-approximation mixing entropy
:math:`-k_B[\theta\ln\theta+(1-\theta)\ln(1-\theta)]` as `M` grows.

.. GENERATED FROM PYTHON SOURCE LINES 18-26

.. code-block:: Python

    import math

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import K_B
    from chemistrykit.statmech import LatticeGasAdsorption








.. GENERATED FROM PYTHON SOURCE LINES 27-28

For a tiny lattice, W can be counted by hand and matches exp(S/k_B):

.. GENERATED FROM PYTHON SOURCE LINES 28-35

.. code-block:: Python


    M_small = 6
    for N in range(M_small + 1):
        W = math.comb(M_small, N)
        S = LatticeGasAdsorption.canonical_entropy(N, M_small)
        print(f"N = {N}: W = {W:2d}, exp(S/k_B) = {np.exp(S / K_B):.4f}")





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

 .. code-block:: none

    N = 0: W =  1, exp(S/k_B) = 1.0000
    N = 1: W =  6, exp(S/k_B) = 6.0000
    N = 2: W = 15, exp(S/k_B) = 15.0000
    N = 3: W = 20, exp(S/k_B) = 20.0000
    N = 4: W = 15, exp(S/k_B) = 15.0000
    N = 5: W =  6, exp(S/k_B) = 6.0000
    N = 6: W =  1, exp(S/k_B) = 1.0000




.. GENERATED FROM PYTHON SOURCE LINES 36-37

For larger lattices, S/(M k_B) approaches the Stirling limit:

.. GENERATED FROM PYTHON SOURCE LINES 37-64

.. code-block:: Python


    theta = np.linspace(0.001, 0.999, 400)
    stirling = -(theta * np.log(theta) + (1.0 - theta) * np.log(1.0 - theta))

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    for M, color in [(10, "crimson"), (50, "darkorange"), (1000, "steelblue")]:
        N_values = np.arange(0, M + 1)
        S = np.array([LatticeGasAdsorption.canonical_entropy(N, M) for N in N_values])
        axes[0].plot(N_values / M, S / (M * K_B), marker="o" if M <= 10 else None, color=color, label=f"exact, M = {M}")
    axes[0].plot(theta, stirling, color="black", linestyle="--", label="Stirling limit")
    axes[0].axvline(0.5, color="gray", linestyle=":", linewidth=0.8)
    axes[0].set_xlabel(r"fractional filling $\theta = N/M$")
    axes[0].set_ylabel(r"$S / (M k_B) = \ln W / M$")
    axes[0].set_title(r"$S = k_B \ln W$ is largest at half filling")
    axes[0].legend()

    M = 100
    N_values = np.arange(0, M + 1)
    lnW = np.array([LatticeGasAdsorption.canonical_entropy(N, M) for N in N_values]) / K_B
    axes[1].semilogy(N_values, np.exp(lnW), color="steelblue")
    axes[1].set_xlabel("number of molecules N")
    axes[1].set_ylabel("number of arrangements W")
    axes[1].set_title(f"Microstate count on M = {M} sites")
    fig.tight_layout()

    plt.show()




.. image-sg:: /api/gallery/statmech/lattice_gas/images/sphx_glr_plot_02_boltzmann_entropy_counting_001.png
   :alt: $S = k_B \ln W$ is largest at half filling, Microstate count on M = 100 sites
   :srcset: /api/gallery/statmech/lattice_gas/images/sphx_glr_plot_02_boltzmann_entropy_counting_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 65-69

The half-filled lattice of 100 sites has about 1e29 arrangements, while
a completely full or empty lattice has exactly one (S = 0): the
overwhelming statistical weight of "disordered" states is what the second
law expresses.


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

   **Total running time of the script:** (0 minutes 0.090 seconds)


.. _sphx_glr_download_api_gallery_statmech_lattice_gas_plot_02_boltzmann_entropy_counting.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_boltzmann_entropy_counting.ipynb <plot_02_boltzmann_entropy_counting.ipynb>`

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

      :download:`Download Python source code: plot_02_boltzmann_entropy_counting.py <plot_02_boltzmann_entropy_counting.py>`

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      :download:`Download zipped: plot_02_boltzmann_entropy_counting.zip <plot_02_boltzmann_entropy_counting.zip>`


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

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