In 1911, Jüttner proposed the generalization, for a relativistic gas, of the Maxwell–Boltzmann distribution of velocities. Here, we want to discuss, among others, the Jüttner probability density function (PDF). Both the velocity space and, consequently, the momentum space are not flat in special relativity. The velocity space corresponds to the Lobachevsky one, which has a negative curvature. This curvature induces a specific power for the Lorentz factor in the PDF, affecting the Jüttner normalization constant in one, two, and three dimensions. Furthermore, Jüttner distribution, written in terms of a more convenient variable, the rapidity, presents a curvature change at the origin at sufficiently high energy, which does not agree with our computational dynamics simulations of a relativistic gas. However, in one dimension, the rapidity satisfies a simple additivity law. This allows us to obtain, through the central limit theorem, a new, Lorentz-invariant, PDF whose curvature at the origin does not change for any energy value and which agrees with our computational dynamics simulations data. Also, we perform extensive first-principle simulations of a one-dimensional relativistic gas constituted by light and heavy particles.
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October 2022
Research Article|
October 14 2022
Relativistic gas: Lorentz-invariant distribution for the velocities
Special Collection:
Complex Systems and Inter/Transdisciplinary Research
Evaldo M. F. Curado
;
Evaldo M. F. Curado
a)
(Conceptualization, Investigation, Methodology, Software, Writing – original draft, Writing – review & editing)
1
Centro Brasileiro de Pesquisas Fisicas
, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
2
National Institute of Science and Technology for Complex Systems
, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
a)Author to whom correspondence should be addressed: evaldo@cbpf.br
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Carlos E. Cedeño
;
Carlos E. Cedeño
b)
(Conceptualization, Investigation, Methodology, Software, Writing – original draft, Writing – review & editing)
1
Centro Brasileiro de Pesquisas Fisicas
, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
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Ivano Damião Soares
;
Ivano Damião Soares
c)
(Conceptualization, Investigation, Methodology, Software, Writing – original draft, Writing – review & editing)
1
Centro Brasileiro de Pesquisas Fisicas
, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
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Constantino Tsallis
Constantino Tsallis
d)
(Conceptualization, Investigation, Methodology, Software, Writing – original draft, Writing – review & editing)
1
Centro Brasileiro de Pesquisas Fisicas
, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
2
National Institute of Science and Technology for Complex Systems
, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
3
Santa Fe Institute
, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
4
Complexity Science Hub Vienna
, Josefstädter Strasse 39, 1080 Vienna, Austria
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a)Author to whom correspondence should be addressed: evaldo@cbpf.br
b)
Electronic mail: cecedeno@cbpf.br
c)
Electronic mail: ivano@cbpf.br
d)
Electronic mail: tsallis@cbpf.br
Note: This article is part of the Focus Issue on Complex Systems and Inter/Transdisciplinary Research.
Chaos 32, 103110 (2022)
Article history
Received:
June 04 2022
Accepted:
September 12 2022
Citation
Evaldo M. F. Curado, Carlos E. Cedeño, Ivano Damião Soares, Constantino Tsallis; Relativistic gas: Lorentz-invariant distribution for the velocities. Chaos 1 October 2022; 32 (10): 103110. https://doi.org/10.1063/5.0101935
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