:orphan:

Examples
========

This gallery walks through every public feature of
``chemistrykit.statmech``: translational/rotational/vibrational partition
functions and the thermodynamic functions derived from them; the
Maxwell-Boltzmann speed distribution; the lattice-gas adsorption model;
quantum statistics and its classical limit; the Debye solid; exact
Ising-model results; and the second virial coefficient.

Each script in this gallery is self-contained and can be run directly with
``python examples/statmech/<section>/<script>.py``. Every script also
carries an RST module docstring as its title/description and uses ``# %%``
markers to split narrative text from code, which is exactly what
Sphinx-Gallery renders into the pages below -- the script *is* the source
of truth for what you see, not a copy of it.

Sections
--------

- **partition_functions** -- equipartition, the Einstein vibrational
  heat capacity, Gibbs's canonical partition function, and the
  Sackur-Tetrode translational entropy.
- **maxwell_boltzmann** -- the Maxwell-Boltzmann speed distribution, its
  characteristic speeds, and a Stern-type molecular-beam check.
- **lattice_gas** -- the lattice-gas derivation of the Langmuir adsorption
  isotherm, and :math:`S=k_B\ln W` by counting arrangements.
- **quantum_statistics** -- the thermal de Broglie wavelength, and the
  Maxwell-Boltzmann limit of Bose-Einstein and Fermi-Dirac statistics.
- **solids** -- Debye's :math:`T^3` law for the heat capacity of a solid.
- **ising** -- Ising's 1D chain, Kramers-Wannier duality, and Onsager's
  exact 2D solution.
- **virial** -- Mayer's second virial coefficient from a pair potential.


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Ising model
------------

Exact results for the nearest-neighbor Ising (lattice-gas) model: Ising's
one-dimensional chain, Kramers-Wannier duality, and Onsager's
two-dimensional solution.


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    <div class="sphx-glr-thumbcontainer" tooltip="Ernst Ising (1925) solved the chain of spins s_i=\pm1 with nearest-neighbor coupling J and found no spontaneous magnetization at any temperature above zero. Ising1D solves it with the 2x2 transfer matrix, whose eigenvalues \lambda_\pm give Z_N=\lambda_+^N+\lambda_-^N exactly. The plots show a checked partition function, a magnetization that is a smooth function of field at every temperature (it only becomes a step at T=0), a correlation length that grows as e^{2J/k_BT} but stays finite, and a smooth heat-capacity maximum instead of a singularity.">

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  .. image:: /api/gallery/statmech/ising/images/thumb/sphx_glr_plot_01_ising_chain_transfer_matrix_thumb.png
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  :doc:`/api/gallery/statmech/ising/plot_01_ising_chain_transfer_matrix`

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      <div class="sphx-glr-thumbnail-title">Ising's one-dimensional chain: no phase transition</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Kramers and Wannier (1941) showed that the square-lattice Ising model&#x27;s high-temperature series at coupling K=J/k_BT equals its low-temperature series at the dual coupling K^*, with">

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  .. image:: /api/gallery/statmech/ising/images/thumb/sphx_glr_plot_02_kramers_wannier_duality_thumb.png
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  :doc:`/api/gallery/statmech/ising/plot_02_kramers_wannier_duality`

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      <div class="sphx-glr-thumbnail-title">Kramers-Wannier duality and the exact Ising critical temperature</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="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. Ising2DOnsager evaluates his free energy, internal energy, and heat capacity, together with Yang&#x27;s (1952) spontaneous magnetization m=[1-\sinh^{-4}(2J/k_BT)]^{1/8}. The heat capacity diverges logarithmically at T_c, and the magnetization vanishes with exponent \beta=1/8. Neither the mean-field (Bragg-Williams) theory, shown for comparison, nor Ising&#x27;s own one-dimensional chain reproduces these features.">

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  .. image:: /api/gallery/statmech/ising/images/thumb/sphx_glr_plot_03_onsager_exact_solution_thumb.png
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  :doc:`/api/gallery/statmech/ising/plot_03_onsager_exact_solution`

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      <div class="sphx-glr-thumbnail-title">Onsager's exact solution of the two-dimensional Ising model</div>
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Lattice-gas adsorption
-----------------------

The lattice-gas derivation of the Langmuir adsorption isotherm, and
Boltzmann's :math:`S=k_B\ln W` counted directly on a lattice.


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    <div class="sphx-glr-thumbcontainer" tooltip="Irving Langmuir&#x27;s 1918 picture of adsorption: a fixed number of independent, non-interacting surface sites, each either empty or holding exactly one molecule with binding energy \epsilon. LatticeGasAdsorption treats those sites in equilibrium with an ideal-gas reservoir and recovers exactly the Langmuir isotherm \theta=P/(P+P_0) -- linear in P at low pressure, saturating at a full monolayer at high pressure -- shown here for two binding strengths.">

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  .. image:: /api/gallery/statmech/lattice_gas/images/thumb/sphx_glr_plot_01_langmuir_adsorption_thumb.png
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  :doc:`/api/gallery/statmech/lattice_gas/plot_01_langmuir_adsorption`

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      <div class="sphx-glr-thumbnail-title">Langmuir's adsorption isotherm from a lattice-gas model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Boltzmann&#x27;s 1877 insight, written by Planck as S=k_B\ln W: entropy counts the microscopic arrangements W compatible with a macroscopic state. For N indistinguishable molecules on M lattice sites, W=\binom{M}{N}, and canonical_entropy evaluates k_B\ln W exactly via 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 -k_B[\theta\ln\theta+(1-\theta)\ln(1-\theta)] as M grows.">

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  .. image:: /api/gallery/statmech/lattice_gas/images/thumb/sphx_glr_plot_02_boltzmann_entropy_counting_thumb.png
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  :doc:`/api/gallery/statmech/lattice_gas/plot_02_boltzmann_entropy_counting`

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      <div class="sphx-glr-thumbnail-title">Boltzmann's S = k ln W by counting lattice arrangements</div>
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Maxwell-Boltzmann speed distribution
--------------------------------------

The Maxwell-Boltzmann speed distribution and its characteristic speeds,
and a Stern-type molecular-beam sampling check of its shape.


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    <div class="sphx-glr-thumbcontainer" tooltip="MaxwellBoltzmannSpeedDistribution at two temperatures, with the most-probable, mean, and rms speeds marked -- their ratios are a fixed, temperature-independent property of the distribution&#x27;s shape.">

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  .. image:: /api/gallery/statmech/maxwell_boltzmann/images/thumb/sphx_glr_plot_01_maxwell_boltzmann_thumb.png
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  :doc:`/api/gallery/statmech/maxwell_boltzmann/plot_01_maxwell_boltzmann`

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      <div class="sphx-glr-thumbnail-title">The Maxwell-Boltzmann speed distribution</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Otto Stern&#x27;s 1920 molecular-beam apparatus gave the first direct experimental measurement of a gas&#x27;s actual molecular speeds, rather than only their statistical consequences (pressure, diffusion, viscosity) -- a rotating-drum time-of-flight technique later refined by Zartman and Ko (1930-1934) into a quantitative confirmation of the full Maxwell-Boltzmann functional form, not just its mean. sample draws exactly such a synthetic &quot;beam&quot; of molecular speeds; binning it into a histogram and comparing it to the exact analytic pdf reproduces the kind of check Stern, and later Zartman and Ko, made directly against a real gas.">

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  .. image:: /api/gallery/statmech/maxwell_boltzmann/images/thumb/sphx_glr_plot_02_stern_experiment_thumb.png
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  :doc:`/api/gallery/statmech/maxwell_boltzmann/plot_02_stern_experiment`

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      <div class="sphx-glr-thumbnail-title">Verifying the speed distribution: Stern's molecular-beam experiment</div>
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Partition functions
---------------------

The equipartition theorem, Einstein's vibrational heat capacity, Gibbs's
canonical-ensemble route from a partition function to every
thermodynamic function, and the Sackur-Tetrode translational entropy.


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    <div class="sphx-glr-thumbcontainer" tooltip="Einstein (1907) treated each vibration as a quantum oscillator of frequency \nu, so its heat capacity depends on \Theta_{\text{vib}}/T with \Theta_{\text{vib}}=h\nu/k_B. VibrationalPartitionFunctionHarmonic&#x27;s heat capacity is this formula: it vanishes as T\to0 (the vibration is &quot;frozen out&quot;) and approaches the classical equipartition value k_B per mode only once T is comparable to \Theta_{\text{vib}}. The second plot shows why a stiff mode (like Einstein&#x27;s diamond) stays frozen out at temperatures where a soft mode already behaves classically.">

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  .. image:: /api/gallery/statmech/partition_functions/images/thumb/sphx_glr_plot_01_vibrational_heat_capacity_thumb.png
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  :doc:`/api/gallery/statmech/partition_functions/plot_01_vibrational_heat_capacity`

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      <div class="sphx-glr-thumbnail-title">Einstein's quantum theory of the vibrational heat capacity</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="James Clerk Maxwell (1860) and, in fully general form, Ludwig Boltzmann (1871) showed that every classical quadratic degree of freedom of a system in thermal equilibrium contributes exactly \frac12k_B to its heat capacity -- the equipartition theorem. A rigid diatomic molecule therefore &quot;should&quot; carry \frac32k_B (translation) + k_B (2 rotational degrees of freedom) + k_B (1 vibrational mode, kinetic + potential) = \frac72k_B at every temperature -- a prediction 19th-century equipartition theory could not explain deviations from (Maxwell himself flagged the puzzle sharply in 1879). Only VibrationalPartitionFunctionHarmonic is an exact quantum treatment in this package, so it alone shows the real resolution Einstein gave in 1907: a mode&#x27;s contribution only &quot;switches on&quot; once k_BT exceeds its own quantum spacing. RotationalPartitionFunctionLinear, by contrast, implements only the classical high-temperature limit (as its own docstring flags), so it contributes its full k_B at every temperature here -- a deliberate approximation, harmless in practice because real rotational spacings are tiny compared to vibrational ones (as the numbers below confirm), but worth seeing explicitly rather than assuming.">

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  :doc:`/api/gallery/statmech/partition_functions/plot_02_equipartition_ladder`

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      <div class="sphx-glr-thumbnail-title">The equipartition theorem, and why only one mode actually "freezes out"</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Gibbs (1902) showed that for a system at fixed temperature every thermodynamic function follows from the canonical partition function Q=\sum_i e^{-E_i/k_BT}:">

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  :doc:`/api/gallery/statmech/partition_functions/plot_03_gibbs_canonical_partition_function`

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      <div class="sphx-glr-thumbnail-title">Gibbs's canonical ensemble: all thermodynamics from one partition function</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Sackur (1911) and Tetrode (1912) obtained the absolute entropy of an ideal monatomic gas by dividing classical phase space into cells of size h^3:">

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  .. image:: /api/gallery/statmech/partition_functions/images/thumb/sphx_glr_plot_04_sackur_tetrode_entropy_thumb.png
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  :doc:`/api/gallery/statmech/partition_functions/plot_04_sackur_tetrode_entropy`

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      <div class="sphx-glr-thumbnail-title">The Sackur-Tetrode entropy of a monatomic gas</div>
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Quantum statistics and the classical limit
--------------------------------------------

The thermal de Broglie wavelength, and how Bose-Einstein and Fermi-Dirac
statistics reduce to classical Maxwell-Boltzmann counting for ordinary
molecular gases.


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    <div class="sphx-glr-thumbcontainer" tooltip="De Broglie&#x27;s relation \lambda=h/p, averaged over a gas&#x27;s thermal spread of momenta, gives the thermal de Broglie wavelength">

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  :doc:`/api/gallery/statmech/quantum_statistics/plot_01_thermal_de_broglie_wavelength`

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      <div class="sphx-glr-thumbnail-title">De Broglie's matter waves: the thermal wavelength of a gas</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Bose and Einstein (1924-25) and Fermi and Dirac (1926) found the mean occupancy of a single-particle state of energy \varepsilon for indistinguishable bosons and fermions:">

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  :doc:`/api/gallery/statmech/quantum_statistics/plot_02_bose_einstein_fermi_dirac_limit`

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      <div class="sphx-glr-thumbnail-title">Bose-Einstein and Fermi-Dirac statistics reduce to Maxwell-Boltzmann</div>
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Heat capacity of solids
------------------------

Debye's continuum model of lattice vibrations and its low-temperature
:math:`T^3` law.


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    <div class="sphx-glr-thumbcontainer" tooltip="Debye (1912) replaced Einstein&#x27;s single vibrational frequency with the full spectrum of lattice sound waves, g(\nu)\propto\nu^2 up to a cutoff \nu_D. DebyeSolid gives">

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  .. image:: /api/gallery/statmech/solids/images/thumb/sphx_glr_plot_01_debye_t_cubed_law_thumb.png
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  :doc:`/api/gallery/statmech/solids/plot_01_debye_t_cubed_law`

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      <div class="sphx-glr-thumbnail-title">Debye's T-cubed law for the heat capacity of a solid</div>
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Virial coefficients
--------------------

The second virial coefficient of an imperfect gas computed from its
intermolecular pair potential.


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    <div class="sphx-glr-thumbcontainer" tooltip="Mayer (1937) expanded a real gas&#x27;s partition function in clusters of the Mayer function f(r)=e^{-u(r)/k_BT}-1. The first correction to the ideal-gas law, PV_m/RT=1+B_2(T)/V_m+\cdots, is">

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  .. image:: /api/gallery/statmech/virial/images/thumb/sphx_glr_plot_01_mayer_second_virial_coefficient_thumb.png
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  :doc:`/api/gallery/statmech/virial/plot_01_mayer_second_virial_coefficient`

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      <div class="sphx-glr-thumbnail-title">Mayer's cluster expansion: the second virial coefficient</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/statmech/ising/index.rst
   /api/gallery/statmech/lattice_gas/index.rst
   /api/gallery/statmech/maxwell_boltzmann/index.rst
   /api/gallery/statmech/partition_functions/index.rst
   /api/gallery/statmech/quantum_statistics/index.rst
   /api/gallery/statmech/solids/index.rst
   /api/gallery/statmech/virial/index.rst



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