Table 5
Useful averages for the Maxwell-Jüttner distribution f(x,p) = ng(p) (Eq. (13)) of temperature Θ = 1/μ = T/(mc2).
| Without drift velocity: U0 = 0 | |||
| Parameter | Value | NR | UR |
|
|
|||
| ⟨ γ ⟩ | κ32(μ) − μ-1 | 1 + 3Θ/2 | 3Θ |
| ⟨ (γβ)2 ⟩ | 3Θκ32(μ) | 3Θ | 12Θ2 |
| ⟨ β2 ⟩ | 3Θκ12(μ) | 3Θ | 3/2 |
| ⟨ γβ2 ⟩ | 3Θ | ||
| Larmor radius |
|
||
| Pressure P | nT | ||
| Enthalpy h | κ32(μ) | 1 + 5Θ/2 | 4Θ |
| Adiabatic | |||
exponent  |
1 + (μκ32(μ) − μ − 1)-1 | 5/3 | 4/3 |
With a drift velocity  |
|||
| Parameter | Value | NR | UR |
|
|
|||
| ⟨ v ⟩ | U 0 | ||
| ⟨ γv ⟩ | κ32(μ)Γ0U0 | Γ0U0 | 4ΘΓ0U0 |
| ⟨ γ ⟩ |
|
Γ0 |
|
| ⟨ pxvx ⟩ = ⟨ pzvz ⟩ | Θ/Γ0 | ||
| ⟨ pyvy ⟩ |
|
||
| ⟨(pi − ⟨ pi ⟩ )(vj − ⟨ vj ⟩ )⟩ | δij Θ/Γ0 | ||
Notes. We define κij(μ) = Ki(μ)/Kj(μ), with Kn the modified Bessel function of the nth kind. The Larmor radius is defined by ⟨ rce ⟩ = ⟨ (γv ⊥ )2 ⟩ 1/2/ωce, the pressure by P = (1/3)n ⟨ v·(γmv) ⟩ , the enthalpy by h = (n ⟨ γmc2 ⟩ + P)/(nmc2), and the adiabatic exponent by 
. NR means non-relativistic limit (Θ → 0, κ32(μ) ~ 1 + 5Θ/2), and UR ultra-relativistic limit (Θ → + ∞, κ32(μ) ~ 4Θ), with in both cases no constraints on Γ0.
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