Velocity component distribution
===============================

For a system containing a large number of identical non-interacting non-relativistic
classical particles in thermodynamic equilibrium, the velocity component distribution
is a function :math:`f(v_k)` such that :math:`f(v_k) dv_k` gives the fraction of particles with
speeds in the interval :math:`dv_k` around velocity component :math:`v_k`.

**Notation:**

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

**Notes:**

#. Applicable for any velocity component in Cartesian coordinates.

**Conditions:**

#. 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_velocity_vector>`__.

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

.. py:data:: velocity_component_distribution

    Distribution function of velocity component :math:`v_k`.

Symbol:
    :code:`f(v_k)`

Latex:
    :math:`f(v_{k)}`

Dimension:
    :code:`1/velocity`


.. py:data:: velocity_component

    Velocity component in Cartesian coordinates, :math:`k = x, y, z`.

Symbol:
    :code:`v_k`

Latex:
    :math:`v_{k}`

Dimension:
    :code:`velocity`


.. py:data:: particle_mass

    :attr:`~symplyphysics.symbols.basic.mass` of a particle.

Symbol:
    :code:`m`

Latex:
    :math:`m`

Dimension:
    :code:`mass`


.. 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(v_k) = sqrt(m / (2 * pi * k_B * T)) * exp(-m * v_k^2 / (2 * k_B * T))`


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