Maxwell–Jüttner distribution

In physics, the Maxwell–Jüttner distribution is the distribution of speeds of particles in a hypothetical gas of relativistic particles. Similar to Maxwell's distribution, the Maxwell–Jüttner distribution considers a classical ideal gas where the particles are dilute and do not significantly interact with each other. The distinction from Maxwell's case is that effects of special relativity are taken into account. In the limit of low temperatures T much less than mc²/k (where m is the mass of the kind of particle making up the gas, c is the speed of light and k is Boltzmann's constant), this distribution becomes identical to the Maxwell–Boltzmann distribution.

The distribution can be attributed to Ferencz Jüttner, who derived it in 1911. It has become known as the Maxwell–Jüttner distribution by analogy to the name Maxwell-Boltzmann distribution that is commonly used to refer to Maxwell's distribution.

As the gas becomes hotter and kT approaches or exceeds mc², the probability distribution for ${\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}$ in this relativistic Maxwellian gas is given by the Maxwell–Jüttner distribution:

where ${\displaystyle \beta ={\frac {v}{c}}={\sqrt {1-1/\gamma ^{2}}},}$ ${\displaystyle \theta ={\frac {kT}{mc^{2}}},}$ and ${\displaystyle K_{2}}$ is the modified Bessel function of the second kind.

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