Speed distribution
==================

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

**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_speed>`__.

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

.. py:data:: speed_distribution_function

    :attr:`~symplyphysics.symbols.classical_mechanics.speed` distribution function.

Symbol:
    :code:`f(v)`

Latex:
    :math:`f(v)`

Dimension:
    :code:`1/velocity`


.. py:data:: particle_speed

    Particle :attr:`~symplyphysics.symbols.classical_mechanics.speed`.

Symbol:
    :code:`v`

Latex:
    :math:`v`

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) = sqrt(2 / pi) * (m / (k_B * T))^(3/2) * v^2 * exp(-m * v^2 / (2 * k_B * T))`


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