A Lorentz invariant velocity distribution for a relativistic gas

Evaldo M. F. Curado; Ivano D. Soares

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arXiv · 1406.0777

A Lorentz invariant velocity distribution for a relativistic gas

Evaldo M. F. Curado·Ivano D. Soares

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Abstract

We derive a Lorentz invariant distribution of velocities for a relativistic gas. Our derivation is based on three pillars: the special theory of relativity, the central limit theorem and the Lobachevskyian structure of the velocity space of the theory. The rapidity variable plays a crucial role in our results. For $v^2/c^2 \ll 1$ and $1/\beta=kT/2 m_0 c^2 \ll 1$ the distribution tends to the Maxwell-Boltzmann distribution. The mean $\langle v^2 \rangle$ evaluated with the Lorentz invariant distribution is always smaller than the Maxwell-Boltzmann mean and is bounded by $\langle v^2 \rangle/c^2=1$. This implies that for a given $\langle v^2 \rangle$ the temperature is larger than the temperature estimated using the Maxwell-Boltzmann distribution. For temperatures of the order of $T \sim {10^{12}}~ K$ and $T \sim {10^{8}}~ K$ the difference is of the order of $10 \%$, respectively for particles with the hydrogen and the electron rest masses.

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Evaldo M. F. Curado, Ivano D. Soares. 2014-06-03. A Lorentz invariant velocity distribution for a relativistic gas. https://arxiv.org/abs/1406.0777

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