.. _sphx_glr_api_gallery_statmech_partition_functions:

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
    :alt:

  :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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  .. image:: /api/gallery/statmech/partition_functions/images/thumb/sphx_glr_plot_02_equipartition_ladder_thumb.png
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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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  .. image:: /api/gallery/statmech/partition_functions/images/thumb/sphx_glr_plot_03_gibbs_canonical_partition_function_thumb.png
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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
    :alt:

  :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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.. toctree::
   :hidden:

   /api/gallery/statmech/partition_functions/plot_01_vibrational_heat_capacity
   /api/gallery/statmech/partition_functions/plot_02_equipartition_ladder
   /api/gallery/statmech/partition_functions/plot_03_gibbs_canonical_partition_function
   /api/gallery/statmech/partition_functions/plot_04_sackur_tetrode_entropy