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:

\(f(v_k)\)

velocity_component

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

Symbol:

v_k

Latex:

\(v_k\)

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(-1 * m * v_k^2 / (2 * k_B * T))

Latex:

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