Maxwell-Boltzmann Distribution

Describes the distribution of speeds of particles in an ideal gas at temperature $T$ .

The energy of a single particle moving in 3D:

E = \frac{1}{2} m (v_{x}^{2} + v_{y}^{2} + v_{z}^{2})

From the canonical ensemble, the probability of a microstate is $p \propto e^{- E / k_{B} T}$ . Since the three velocity components are independent, the joint distribution factorizes:

p (v_{x}, v_{y}, v_{z}) \propto e^{- m (v_{x}^{2} + v_{y}^{2} + v_{z}^{2}) / 2 k_{B} T}

Normalizing each Gaussian component:

p (v_{x}) = \sqrt{\frac{m}{2 π k_{B} T}} \exp (- \frac{m v_{x}^{2}}{2 k_{B} T})

Speed distribution

Converting to spherical coordinates and integrating over all directions (the $4 π v^{2}$ factor comes from the surface area of a sphere of radius $v$ ):

f (v) = 4 π v^{2} {(\frac{m}{2 π k_{B} T})}^{3 / 2} \exp (- \frac{m v^{2}}{2 k_{B} T})

This is the Maxwell-Boltzmann speed distribution.

Key speeds

Most probable speed (peak of $f (v)$ ): $v_{p} = \sqrt{\frac{2 k_{B} T}{m}}$
Mean speed: $⟨ v ⟩ = \sqrt{\frac{8 k_{B} T}{π m}}$
RMS speed: $v_{rms} = \sqrt{\frac{3 k_{B} T}{m}}$

Mean kinetic energy

⟨ E ⟩ = \frac{1}{2} m ⟨ v^{2} ⟩ = \frac{3}{2} k_{B} T

Each velocity component contributes $\frac{1}{2} k_{B} T$ — this is equipartition.