Maxwell—Jüttner distribution¶
The Maxwell—Jüttner distribution is the distribution of speeds of particles in a hypothetical gas of relativistic particles. It is similar to the Maxwell—Boltzmann distribution in that it considers an ideal gas where particles are dilute and do not interact with each other, but the effects of special relativity are taken into account. In the limit of low temperatures, this distribution becomes identical to the Maxwell—Boltzmann distribution.
Notation:
\(K_2\) is the modified Bessel function of the second kind.
Conditions:
The system is in thermal equilibrium with the environment.
Particle interactions are not taken into account.
No quantum effects occur in the system.
Antiparticles cannot occur in the system.
Temperature must be isotropic, i.e. each degree of freedom has to have the same translational kinetic energy.
Links:
- distribution_function¶
Lorentz factor distribution function.
- Symbol:
f(gamma)- Latex:
\(f(\gamma)\)
- lorentz_factor¶
Lorentz factor of relativistic particles.
- Symbol:
gamma- Latex:
\(\gamma\)
- reduced_temperature¶
Reduced temperature of the system.
- Symbol:
theta- Latex:
\(\theta\)
- law¶
f(gamma) = (gamma * sqrt(gamma^2 - 1)) / (theta * K_2(1 / theta)) * exp(-1 * gamma / theta)- Latex:
- \[f(\gamma) = \frac{\gamma \sqrt{\gamma^2 - 1}} {\theta K_2 \left( \frac{1}{\theta} \right)} \exp{\left( -\frac{\gamma}{\theta} \right)}\]