Abstract
Two relativistic distributions which generalizes the Maxwell Boltzman (MB) distribution are analyzed: the relativistic MB and the Maxwell-J{\"u}ttner (MJ) distribution. For the two distributions we derived in terms of special functions the constant of normalization, the average value, the second moment about the origin, the variance, the mode, the asymptotic behavior, approximate expressions for the average value as function of the temperature and the connected inverted expressions for the temperature as function of the average value. Two astrophysical applications to the synchrotron emission in presence of the magnetic field and the relativistic electrons are presented.
Highlights
The equivalent in special relativity (SR) of the Maxwell-Boltzmann (MB) distribution, see [1] [2], is the so called Maxwell-Jüttner distribution (MJ), see [3] [4]
We derived in terms of special functions the constant of normalization, the average value, the second moment about the origin, the variance, the mode, the asymptotic behavior, approximate expressions for the average value as function of the temperature and the connected inverted expressions for the temperature as function of the average value
We select some approaches among others: a model for the anisotropic MJ distribution [5], an astrophysical application of the MJ distribution to the energy distribution in radio jets [6], a new family of MJ distributions characterized by the parameter η [7] and an application to counter-streaming beams of charged particles [8]
Summary
The equivalent in special relativity (SR) of the Maxwell-Boltzmann (MB) distribution, see [1] [2], is the so called Maxwell-Jüttner distribution (MJ), see [3] [4]. We select some approaches among others: a model for the anisotropic MJ distribution [5], an astrophysical application of the MJ distribution to the energy distribution in radio jets [6], a new family of MJ distributions characterized by the parameter η [7] and an application to counter-streaming beams of charged particles [8]. The above approaches do not cover the determination of the statistical quantities of the MJ distribution. L. Zaninetti of the two relativistic distributions here analyzed
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