In maxwell velocity distribution, maxwell assumed that the velocities of molecules are smaller than light, if the velocities are greater than light, then all the three components will no longer independent. Can anyone elaborate more specifically?

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    $\begingroup$ Velocity of massive particles can't be greater than light. $\endgroup$ – Kyle Kanos Aug 23 '18 at 11:03
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    $\begingroup$ Where in his derivation did he make this assumption? $\endgroup$ – Anders Sandberg Aug 23 '18 at 11:16
  • $\begingroup$ the derivations were made around the 1860's and 1870's before special relativity $\endgroup$ – jim Aug 23 '18 at 12:08
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    $\begingroup$ Relativistic effects are completely negligible at normal temperatures ("normal" meaning temperatures less than hundreds of millions of kelvins). That the Maxwell-Boltzmann distribution ignores relativistic effects means that this distribution becomes invalid when temperatures become extremely high. $\endgroup$ – David Hammen Aug 23 '18 at 12:37

The Maxwell velocity distribution is a Boltzmann distribution in which the classical, non-relativistic expression for the molecular kinetic energy is used. Hence, the speed of light does not enter at all. In a relativistic setting, it would be impossible to accelerate the molecules beyond the speed of light; any assumption that their velocity is smaller than the speed of light would automatically be satisfied.

Basically, only these assumptions enter the derivation of the Maxwell-Boltzmann distribution:

  • The kinetic energy distribution of the molecules is a thermal (Boltzmann) distribution, using the classical expression $E= \frac{1}{2} m v ^2$ for the kinetic energy.
  • When going over from the one-dimensional distribution to the three-dimensional one, one assumes that the three spatial directions are equivalent and independent.

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