Energy distribution
===================

For a system containing a large number of identical non-interacting non-relativistic classical
particles in thermodynamic equilibrium, the energy distribution function is a function such that
:math:`f(E) dE` gives the fraction of particles with energies in the interval :math:`dE`
around energy value :math:`E`.

**Notation:**

#. :math:`k_\text{B}` (:code:`k_B`) is :attr:`~symplyphysics.quantities.boltzmann_constant`.

**Notes:**

#. Number of particles is big enough that the laws of thermodynamics can be applied.
#. Particles are identical, non-interacting, non-relativistic, and classical.
#. The ensemble of particles is at thermodynamic equilibrium.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution#Distribution_for_the_energy>`__.

.. py:currentmodule:: symplyphysics.laws.thermodynamics.maxwell_boltzmann_statistics.energy_distribution

.. py:data:: energy

    :attr:`~symplyphysics.symbols.basic.energy` of the ensemble.

Symbol:
    :code:`E`

Latex:
    :math:`E`

Dimension:
    :code:`energy`


.. py:data:: energy_distribution_function

    :attr:`~energy` distribution function.

Symbol:
    :code:`f(E)`

Latex:
    :math:`f(E)`

Dimension:
    :code:`1/energy`


.. py:data:: equilibrium_temperature

    Equilibrium :attr:`~symplyphysics.symbols.thermodynamics.temperature` of the ensemble.

Symbol:
    :code:`T`

Latex:
    :math:`T`

Dimension:
    :code:`temperature`


.. py:data:: law

    :code:`f(E) = 2 * sqrt(E / pi) / (T^(3/2) * k_B^(3/2)) * exp(-E / (k_B * T))`


    Latex:
        .. math::
            f(E) = \frac{2 \sqrt{\frac{E}{\pi}}}{T^{\frac{3}{2}} k_\text{B}^{\frac{3}{2}}} \exp{\left(- \frac{E}{k_\text{B} T} \right)}