Maxwell speed distribution

The Maxwell velocity distribution provides probability that the speed of a molecule of mass m lies in the range v to v+dv is given by

$P(v)dv=4\pi v^{2}dv{(\frac{m}{2\pi k_{B}T})}^{3/2}\exp (-m{v}^2/2k_{B}T)$

where T is the temperature and $k_B$ is the Boltzmann constant. The maximum of this distribution is located at

$v_{\mathrm{m}\mathrm{a}\mathrm{x}}=\sqrt{\frac{2k_{B}T}{m}}$

The mean speed is given by

$\overline{v}=\frac{2}{\sqrt{\pi }}{v}_{\mathrm{m}\mathrm{a}\mathrm{x}}$

and the root-mean-square speed by

$\sqrt{\overline{v^2}}=\sqrt{\frac{3}{2}}{v}_{\mathrm{m}\mathrm{a}\mathrm{x}}$

References

Related reading

• J. S. Rowlinson "The Maxwell-Boltzmann distribution", Molecular Physics 103 pp. 2821 - 2828 (2005)
• Elyas Shivanian and Ricardo Lopez-Ruiz "A New Model for Ideal Gases. Decay to the Maxwellian Distribution", arXiv:1105.4813v1 24 May (2011)

External resources

• Initial velocity distribution sample FORTRAN computer code from the book M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989).