UY1: The Maxwell-Boltzmann distribution

The probability of finding a particle in the speed interval v to v + dv is given by:

$$P (v) = 4 \pi v^{2} \left( \frac{m}{2 \pi k_{B} T} \right)^{\frac{3}{2}} e^{- \frac{m v^{2}}{2 k_{B} T}}$$

, where

  • $4 \pi v^{2}$ is the density of states
  • $- \frac{m v^{2}}{2 k_{B} T}$ is the ratio of kinetic energy to thermal energy
  • $\left( \frac{m}{2 \pi k_{B} T} \right)^{\frac{3}{2}} e^{- \frac{m v^{2}}{2 k_{B} T}}$ is the Boltzmann distribution


a) P(v) = 0 at v = 0 and as v tends to infinity.

b) P(v) reaches a maximum at some intermediate value of v.
– The position of this peak and the shape of the distribution function varies with temperature: As temperature increases, the peak occurs at higher v and the width of the distribution becomes broader: the average speed increases and a larger fraction of molecules can attain very high speeds.

Various measures of the characteristic speed

maxwell boltzmann 2– vmp is the most probable speed at the peak of the distribution function.
– vav is the average speed of the distribution.
– Note: vrms > vav > vmp


The most probable speed 

The most probable speed occurs at:

$$\frac{d P(v)}{dv} = 0$$


$$v_{mp} = \sqrt{\frac{2 k_{B} T}{m}}$$
The average speed

The average speed is given by:

$$\bar{v} = \int\limits_{0}^{\infty} P(v) \, v \, dv$$


$$\bar{v} = \sqrt{\frac{8 k_{B} T}{\pi m}}$$
The root-mean-square speed

The root-mean-square speed is given by:

$$v_{\text{rms}} = \left( \int\limits_{0}^{\infty} P(v) \, v^{2} \, dv \right)^{\frac{1}{2}}$$


$$v_{\text{rms}} = \sqrt{\frac{3 k_{B} T}{m}}$$


Next: Theorem Of Equipartition Of Energy

Previous: The Boltzmann Distribution

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