Maxwell–Jüttner distribution

Maxwell–Jüttner distribution over Lorentz factor (relativistic Maxwellian), for a gas at different temperatures. Speed is represented in terms of the Lorentz factor.

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:[2]

${\displaystyle f(\gamma )={\frac {\gamma ^{2}\beta }{\theta K_{2}(1/\theta )}}\exp \left(-{\frac {\gamma }{\theta }}\right)}$

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.

Alternatively, this can be written in terms of the momentum as

${\displaystyle f(\mathbf {p} )={\frac {1}{4\pi m^{3}c^{3}\theta K_{2}(1/\theta )}}\exp \left(-{\frac {\gamma (p)}{\theta }}\right)}$

where ${\displaystyle \gamma (p)={\sqrt {1+\left({\frac {p}{mc}}\right)^{2}}}}$. The Maxwell–Jüttner equation is covariant, but not manifestly so, and the temperature of the gas does not vary with the gross speed of the gas.[3]

Some limitations of the Maxwell–Jüttner distributions are shared with the classical ideal gas: neglect of interactions, and neglect of quantum effects. An additional limitation (not important in the classical ideal gas) is that the Maxwell–Jüttner distribution neglects antiparticles.

If particle-antiparticle creation is allowed, then once the thermal energy kT is a significant fraction of mc², particle-antiparticle creation will occur and begin to increase the number of particles while generating antiparticles (the number of particles is not conserved, but instead the conserved quantity is the difference between particle number and antiparticle number). The resulting thermal distribution will depend on the chemical potential relating to the conserved particle-antiparticle number difference. A further consequence of this is that it becomes necessary to incorporate statistical mechanics for indistinguishable particles, because the occupation probabilities for low kinetic energy states becomes of order unity. For fermions it is necessary to use Fermi–Dirac statistics and the result is analogous to the thermal generation of electron-hole pairs in semiconductors. For bosonic particles, it is necessary to use the Bose–Einstein statistics.[4]