Maxwell-Jüttner distribution

The Maxwell-Jüttner distribution is a probability distribution of statistical thermodynamics . It describes the velocity distribution of an ideal gas with relativistic particle velocities. The Maxwell-Jüttner distribution is a generalization of the Maxwell-Boltzmann distribution . In contrast to the latter, it takes into account the effects of special relativity .

Comparable to the Maxwell-Boltzmann distribution , the Maxwell-Jüttner distribution requires a classic, ideal gas . Similar to the ideal gas, ideal dilution and the absence of forces between the gas particles are assumed. In the limit case of low temperatures, at which is significantly smaller than , the Maxwell-Jüttner distribution changes into the classic Maxwell-Boltzmann distribution ( is the mass of a gas particle, the speed of light and is the Boltzmann constant ). ${\displaystyle T}$ ${\displaystyle mc^{2}/k}$ ${\displaystyle m}$ ${\displaystyle c}$ ${\displaystyle k}$

Naming

The distribution function was first derived by Ferencz Jüttner (1878–1958) in 1911. Since it is a generalization of the Maxwell-Boltzmann distribution found by James Clerk Maxwell and Ludwig Boltzmann , it is therefore referred to as the Maxwell-Jüttner distribution.

The distribution function

Maxwell-Jüttner velocity distribution as a function of the relativistic velocity for different temperatures.

If a gas becomes so hot that its thermal energy exceeds the range or exceeds it, then its relativistic Maxwellian velocity distribution can be with ${\displaystyle kT}$ ${\displaystyle mc^{2}}$

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

to be discribed. In this equation means

• ${\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}$

the particle velocity in relativistic units ( Lorentz factor ) and

• ${\displaystyle \beta ={\frac {v}{c}}={\sqrt {1-1/\gamma ^{2}}}}$,
• ${\displaystyle \theta ={\frac {kT}{mc^{2}}}}$,

• ${\displaystyle K_{2}}$is the modified armchair function of the second kind .

As an alternative to the speed distribution, the momentum distribution can also be specified:

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

with . ${\displaystyle \gamma (p)={\sqrt {1+\left({\frac {p}{mc}}\right)^{2}}}}$

The Maxwell-Jüttner distribution is covariant , but not manifestly covariant (see Minkowski space ). For this reason the temperature does not vary with the mean gross velocity of the gas particles.

boundary conditions

In principle, the Maxwell-Jüttner distribution has the same boundary conditions as the Maxwell-Boltzmann distribution :

• An ideal gas is assumed.
• Interactions between the gas particles are neglected.
• Quantum effects are neglected.

In addition to these conditions, the following boundary conditions must be observed for the Maxwell-Jüttner distribution:

• The pair formation or the formation of antiparticles is excluded.

Pair formation has to be taken into account if the kinetic particle energy is in the order of magnitude of . Since the number of gas particles is not a maintenance factor , it can increase as required. For reasons of symmetry, only the number of newly formed particles to their antiparticles has to be retained. The distribution function thus newly obtained contains the chemical potential of the corresponding pair formation as a new quantity . ${\displaystyle kT}$ ${\displaystyle mc^{2}}$

Individual evidence

1. ↑ ^{a b} Ferencz Jüttner: The dynamics of a moving gas in relative theory . In: Annals of Physics . tape 340 , no. 6 , 1911, pp. 145-161 , doi : 10.1002 / andp.19113400608 .
2. ↑ ^{a b} Ferencz Jüttner: Maxwell's law of velocity distribution in relative theory . In: Annals of Physics . tape 339 , no. 5 , 1911, pp. 856-882 , doi : 10.1002 / andp.19113390503 .
3. ^ John Lighton Synge: The relativistic gas . North-Holland publishing company, Amsterdam 1957.
4. ↑ Guillermo Chacón-Acosta, Leonardo Dagdug, Hugo A. Morales-Técotl: On the Manifestly Covariant Jüttner Distribution and Equipartition Theorem . In: arXiv . October 8, 2009, arxiv : 0910.1625 .
5. ↑ Guillermo Chacón-Acosta, Leonardo Dagdug, Hugo A. Morales-Técotl: Manifestly covariant Jüttner distribution and equipartition theorem . In: Physical Review E . tape 81 , no. 2 , February 22, 2010, p. 021126 , doi : 10.1103 / PhysRevE.81.021126 .