The relativistic Maxwell-Jüttner velocity distribution function

Damião, Gilberto J.; Rodrigues, Clóves G.

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Revista Brasileira de Ensino de Física

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Sumário

Artigos Gerais • Rev. Bras. Ensino Fís. 44 • 2022 • https://doi.org/10.1590/1806-9126-RBEF-2021-0271 copy

The relativistic Maxwell-Jüttner velocity distribution function

Authorship SCIMAGO INSTITUTIONS RANKINGS

In this paper, the expression of the relativistic Maxwell-Jüttner velocity distribution was deduced in a detailed and didactic way. This distribution was compared with the well-known Maxwell-Boltzmann distribution, which does not take into account relativistic effects. Applications and comparisons between the two distributions were carried out in three situations: i ) protons at solar surface temperature, ii ) electrons at Lawson’s criterion temperature, and iii ) protons at quasar temperature. It was found that the value of the ratio between the rest mass of the particle considered and the temperature of the system determines the need to consider or not a relativistic treatment for the velocity distribution function of the system.

Keywords
Maxwell-Boltzmann; Maxwell-Jüttner; relativistic distribution; velocity distribution function

	Maxwell-Boltzmann	Maxwell-Jüttner
Distribuição	$\sqrt{2 / π} ζ^{3 / 2} β^{2} e^{- ζ β^{2} / 2}$	$\frac{ζ β^{2} γ^{5} e^{- ζ γ}}{K_{2} (ζ)}$
$β_{p}$	$\sqrt{\frac{2}{ζ}}$	$\sqrt{σ_{1} + σ_{2} + σ_{3} + σ_{4}}$
$⟨ β ⟩$	$\sqrt{\frac{8}{π ζ}}$	$\frac{ζ}{K_{2} (ζ)} (\frac{Γ (3, ζ)}{ζ^{3}} - \frac{e^{- ζ}}{ζ})$
$β_{r m s}$	$\sqrt{\frac{3}{ζ}}$	$\sqrt{1 - \frac{ζ K_{1} (ζ)}{K_{2} (ζ)} + \frac{ζ π}{2 K_{2} (ζ)} [1 - ζ K_{0} (ζ) L_{- 1} (ζ) - ζ K_{1} (ζ) L_{0} (ζ)]}$

Valores esperados	Maxwell-Boltzmann	Maxwell-Jüttner	Diferença (%)
$v_{p}$	9.786,554	9.785,136	0,01
$⟨ v ⟩$	11.042,943	11.042,943	0
$v_{r m s}$	11.986,032	11.986,032	0

	Massa (kg)	Temperatura (K)	$ζ$
Dêuteron	$3, 3436 \times 10^{- 27}$	$1, 5 \times 10^{8}$	$1, 45 \times 10^{5}$
Elétron	$9, 1094 \times 10^{- 31}$	$1, 5 \times 10^{8}$	$39, 533$

Valores esperados	Maxwell-Boltzmann	Maxwell-Jüttner	Diferença (%)
$β_{p}$	0,2249	0,2305	2,4
$⟨ β ⟩$	0,2538	0,2483	2,2
$β_{r m s}$	0,2755	0,2672	3,0

Valores esperados	Maxwell-esperados	Maxwell-Jüttner	Diferença (%)
$β_{p}$	0,7113	0,7786	8,6
$⟨ β ⟩$	0,8027	0,6604	17,7
$β_{r m s}$	0,8712	0,6828	21,6

Valores esperados	Maxwell-Boltzmann	Maxwell-Jüttner	Diferença (%)
$β_{p}$	1,3553	0,9769	27,9
$⟨ β ⟩$	1,5293	0,8953	41,5
$β_{r m s}$	1,6599	0,9024	45,6

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