Discrete distribution
=====================

Maxwell—Boltzmann distribution can be written as a discrete distribution of a single particle's
discrete energy spectrum. Maxwell-Boltzmann statistics gives the average number of particles
found in a given single-particle microstate.

**Notation:**

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

**Conditions:**

#. Particles do not interact and are classical.
#. The system is in thermal equilibrium.

**Links:**

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

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

.. py:data:: occupancy_of_state

    Occupancy of, or expected number of particles in, the single-particle microstate :math:`i`.

Symbol:
    :code:`N_i`

Latex:
    :math:`N_{i}`

Dimension:
    :code:`dimensionless`


.. py:data:: particle_count

    Total :attr:`~symplyphysics.symbols.basic.particle_count` of the system.

Symbol:
    :code:`N`

Latex:
    :math:`N`

Dimension:
    :code:`dimensionless`


.. py:data:: energy_of_state

    :attr:`~symplyphysics.symbols.basic.energy` of single-particle microstate :math:`i`.

Symbol:
    :code:`E_i`

Latex:
    :math:`E_{i}`

Dimension:
    :code:`energy`


.. py:data:: equilibrium_temperature

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

Symbol:
    :code:`T`

Latex:
    :math:`T`

Dimension:
    :code:`temperature`


.. py:data:: single_particle_partition_function

    Single-particle :attr:`~symplyphysics.symbols.thermodynamics.partition_function`, which acts as a normalizing factor of the distribution.

Symbol:
    :code:`Z`

Latex:
    :math:`Z`

Dimension:
    :code:`dimensionless`


.. py:data:: law

    :code:`N_i = N / Z * exp(-E_i / (k_B * T))`


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