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

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

where T is the temperature and $k_{B}$ is the Boltzmann constant. The maximum of this distribution is located at

$v_{\rm {max}}={\sqrt {\frac {2k_{B}T}{m}}}$

The mean speed is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{v} = \frac{2}{\sqrt \pi} v_{\rm max}}

and the root-mean-square speed by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\overline{v^2}} = \sqrt \frac{3}{2} v_{\rm max}}

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).