Pc(x,t)=∫∞04π3vf0dp
Uc(x,t)=∫∞04πp2ϵf0dp,
where
ϵ=(γ−1)mc2v=pγmγ=√p2m2c2+1.
a. The lower limit (0) should be the injection momentum pinj.
b. The upper limit (∞) may cause divergence.
c. Imposing cutoffs at high and low momentum is necessary.
d. Generally, the definition for pressure tensor is ∫m(vi−¯vi)(vj−¯vj)f0(v,x)d3v. For isotropic distribution function ¯vi is zero and ¯vivj=0. We recover the above definition.
e.g.
∫vxvyf0d3v=∫vsin(θ)cos(ϕ)vsin(θ)sin(ϕ)2πv2sin(θ)dθdϕ=0.
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