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 \(f(v_k)\) such that \(f(v_k) dv_k\) gives the fraction of particles with speeds in the interval \(dv_k\) around velocity component \(v_k\).

Notation:

1. \(k_\text{B}\) (k_B) is boltzmann_constant.

Notes:

1. Applicable for any velocity component in Cartesian coordinates.

Conditions:

1. Number of particles is big enough that the laws of thermodynamics can be applied.

2. Particles are identical, non-interacting, non-relativistic, and classical.

3. The ensemble of particles is at thermodynamic equilibrium.

Links:

1. Wikipedia.

velocity_component_distribution

Distribution function of velocity component \(v_k\).

Symbol:

f(v_k)

Latex:

\(f(v_{k)}\)

Dimension:

1/velocity

velocity_component

Velocity component in Cartesian coordinates, \(k = x, y, z\).

Symbol:

v_k

Latex:

\(v_{k}\)

Dimension:

velocity

particle_mass

mass of a particle.

Symbol:

m

Latex:

\(m\)

Dimension:

mass

equilibrium_temperature

Equilibrium temperature of the ensemble.

Symbol:

T

Latex:

\(T\)

Dimension:

temperature

law

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

Latex:

\[f(v_{k)} = \sqrt{\frac{m}{2 \pi k_\text{B} T}} \exp{\left(- \frac{m v_{k}^{2}}{2 k_\text{B} T} \right)}\]