Single-particle state distribution
==================================

For a system of identical fermions in thermodynamic equilibrium, the average number of fermions
in a single-particle state :math:`i` is given by the Fermi—Dirac distribution.

**Notation:**

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

**Notes:**

#. If the energy states are degenerate, i.e. two or more particles are on the same energy level,
   the average number of fermions can be found by multiplying by the degeneracy :math:`g_i` of
   the energy level.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics#Fermi%E2%80%93Dirac_distribution>`__.

.. py:currentmodule:: symplyphysics.laws.thermodynamics.fermi_dirac_statistics.single_particle_state_distribution

.. py:data:: occupancy_of_state

    Occupancy of a single-particle state :math:`i`.

Symbol:
    :code:`N_i`

Latex:
    :math:`N_{i}`

Dimension:
    :code:`dimensionless`


.. py:data:: energy_of_state

    :attr:`~symplyphysics.symbols.basic.energy` of state :math:`i`.

Symbol:
    :code:`E_i`

Latex:
    :math:`E_{i}`

Dimension:
    :code:`energy`


.. py:data:: total_chemical_potential

    Total :attr:`~symplyphysics.symbols.thermodynamics.chemical_potential` of the system.

Symbol:
    :code:`mu`

Latex:
    :math:`\mu`

Dimension:
    :code:`energy`


.. py:data:: temperature

    :attr:`~symplyphysics.symbols.thermodynamics.temperature` of the system.

Symbol:
    :code:`T`

Latex:
    :math:`T`

Dimension:
    :code:`temperature`


.. py:data:: law

    :code:`N_i = 1 / (exp((E_i - mu) / (k_B * T)) + 1)`


    Latex:
        .. math::
            N_{i} = \frac{1}{\exp{\left(\frac{E_{i} - \mu}{k_\text{B} T} \right)} + 1}