The Jüttner distribution is the relativistically generalized classical isotropic Maxwell–Boltzmann distribution, whether written in its dependence on relativistic particle energy ϵp=mγ(p)c2 or momentum p, with γ=1+p2/m2c2. The phase-volume element dxdp is covariant (a consequence of its scalar nature). Hence the Jüttner distribution is as well covariant but not manifestly covariant. In anisotropic relativistic gases or plasmas for application to anisotropic plasmas cf. and in drifting plasmas for recent examples cf. e.g. the form of the Jüttner distribution is usually assumed. Below we provide its simple analytical derivation and manifestly covariant version.

Thermally relativistic implies thermal speeds ve/c≳10-2. Avoiding creation, annihilation, and Compton interactions requires T<mc2, with T temperature in energy units, hence weakly relativistic thermal electrons of some 10eV<T<0.5 MeV, covering most hot classical plasmas.

Maxwell–Boltzmann distributions are solutions of the stationary one-particle Boltzmann equation with the argument of the ratio of the single particle to average thermal energies, viz. ϵp/T. Properly normalized they give the probability at temperature T for finding all particles of given momentum p (or energy ϵp) in the interval dp (or dϵp) in momentum-space volume dp. With three-momentum vector p=(p⟂cos⁡ϕ,p⟂sin⁡ϕ,p‖) in index notation ϵp2=c2piδjipj+m2c4 suggests introduction of temperature anisotropy guided by the diagonal anisotropy of pressure P=N[T⟂δij+(T‖-T⟂)δ33] (as for instance in magnetized plasma), with anisotropy in direction 3 (in plasma the direction of the magnetic field b=B/B, for instance). The inverse pressure / temperature tensor is P-1=T⟂N-1Θ, Θ=Θji=δij+(A-1)δ33,withA=T⟂/T‖. Replacing δji in Eq. () with Θji, valid in the four-velocity frame Uj=(ϵp/mc,0), defining β⟂=mc2/T⟂, β‖=mc2/T‖, putting p⟂=psin⁡θ,p‖=pcos⁡θ, and defining p/mc⟶p yields ϵp2T⟂2=β⟂21+p2sin⁡2θ+Acos⁡2θ.

The square root of Eq. () enters the Boltzmann factor. Up to normalization C, the anisotropic Jüttner distribution function of the ideal gas becomes FJ(p,A)=Cexp⁡-β⟂1+p⟂2+Ap‖2. With A=1,β⟂=β‖=β this is the ordinary Jüttner function. Expanding the root in the limit c→∞ reproduces the ordinary nonrelativistic anisotropic Maxwell–Boltzmann distribution. Extensions to drifting or non-ideal gases are straightforward.

This in principle trivial result was anticipated first without proof by in application to the whistler instability in weakly relativistic anisotropic plasmas

We thank P. H. Yoon for bringing this early reference to our attention.

The distribution Eq. () is covariant, valid in time-like slices of Minkowski space. Explicit manifestly covariant isotropic versions have been provided numerically as well cf.. Since F(p) is a scalar phase space density, its manifestly covariant version is F(xν,pν)-g for both isotropic and anisotropic cases. g<0 is the determinant of the metric tensor gμν in (+---) metric, a version to be applied in curvilinear coordinates. In general relativistic four-space, μ,ν=0,1,2,3, and p→pν is the four-momentum. Operator interpretation of the three-momentum p=ℏk relates any of these versions to quantum field theory.

Jüttner's anisotropic distribution is useful for analytical or numerical as in calculations. In particle-in-cell simulations the initial distribution is prescribed. In practice there is little need to choose it in the simulations to satisfy the Jüttner equilibrium requirement. Solving for all relativistic particle orbits in their self-consistent fields, the initial distribution readily adjusts itself to the physical distribution that evolves under the mutual interactions.

As a side product, this straightforward rigorous derivation indicates that in relativistic media the isotropic temperature T and its inverse β=1/T should be understood as vectors confirmingwho suggested it for different reasons. In presence of anisotropy they become tensors. Including particle spins requires a slightly different treatment.