The Relativistic Vlasov Equation with Current and Charge Density Given by by Alec Johnson, January 2011 X X J = Qpvpδxp Σ = Qpδxp I

The Relativistic Vlasov Equation with current and charge density given by by Alec Johnson, January 2011 X X J = qpvpδxp σ = qpδxp I. Recitation of basic electrodynamics p p

X δxp X δxp = q p , = q γ ; A. Classical electrodynamics p p γ p p γ p p p p Classical electrodynamics is governed by Newton’s sec- here p (i.e. γ v ) is the proper velocity, where γ is the ond law (for particle motion), p p p p rate at which time t elapses with respect to the proper time τ of a clock moving with particle p. Dropping the particle mpdtvp = Fp, vp := dtxp, p index, (where p is particle index, t is time, mp is particle mass, γ := d t, and xp(t) is particle position, and Fp(t) is the force on the τ particle), the Lorentz force law, p := dτ x = (dτ t)dtx = γv.

Fp = qp (E + v × B) A concrete expression relating γ to (proper) velocity is (where E(t, x) = electric ﬁeld, B(t, x) = magnetic ﬁeld, and qp is particle charge), and Maxwell’s equations, γ2 = 1 + (p/c)2, i.e. (dividing by γ2), −2 2 ∂tB + ∇ × E = 0, ∇ · B = 0, 1 = γ + (v/c) , i.e. (solving for γ), 2 −1/2 ∂tE − c ∇ × B = −J/, ∇ · E = σ/, γ = 1 − (v/c)2 . with current and charge density given by II. Vlasov equation X X J := qpvpδxp , σ := qpδxp , The Vlasov equation asserts that particles are conserved p p in phase space and that the only force acting on particles is the electromagnetic force. where c is light speed, 0 is electric permittivity, J is net current density, σ is net charge density, δ (t, x) := δ(x − xp The Vlasov equation writes xp(t)) is the particle density function, and δ is the Dirac delta function (unit spike). ∂tfs + ∇x · (vfs) + ∇p · (asfs) = 0, (1)

B. Relativistic electrodynamics where fs(t, x, p) is the particle density function of species s and as = dtp = (qs/ms)(E+v ×B) is particle acceleration Newton’s second law is invariant under Galilean transfor- and we recall that v = dtx is velocity and p = γv is proper mations, but Maxwell’s equations are not (because they velocity. imply the existence of a light speed c). The Maxwell source terms are provided by the relations Einstein (and previously Lorentz) characterized the set of aﬃne coordinate transformations, called Lorentz transformations, which leave light speed constant and which satisfy X Z X Z J = q f vdp σ := q f dp the property that if system A has velocity v in system B s s s s then system B has velocity −v in system A. s s X Z dp X Z dp = q f p , = q f γ . s s γ s s γ Einstein then modiﬁed the laws of electrodynamics to make s s them Lorentz-invariant. To do so we replace velocity with proper velocity in Newton’s second law. So the laws of relativistic electrodynamics are: If we regard fs as a linear superposition of spike functions then the Vlasov system (1) with Maxwell’s equations is Newton’s second law, equivalent to the fundamental equations of electrodynamics written in terms of individual particles. mpdtpp = Fp, vp := dtxp, A. Lorentz-invariant form the Lorentz force law,

1. fs is a scalar. Fp = qp (E + v × B) , (In this section without loss of generality we take c = 1.) and Maxwell’s equations, The particle density function fs(t, x, p) is a scalar. This means that it remains invariant under the transforma- ∂tB + ∇ × E = 0, ∇ · B = 0, 2 tion of phase space that is implied by a Lorentz trans- ∂tE − c ∇ × B = −J/, ∇ · E = σ/, formation (i.e. a relativistic inertial transformation). It 2 takes a little work to see this. By definition, the quantity This confirms that the Vlasov equation (1) is indeed phys- 3 3 dfs := fs(t, x, p)d x ∧ d p counts the number of particles ical and Lorentz-invariant. 3 3 in a region (x0 +d x, p0 +d p) in phase space. (Note that d3x ∧ d3p denotes an infinitesimal surface of simultaneous B. Moments points in (t, x, p)-space.) We argue that dfs is a scalar, then argue that d3x ∧ d3p is a scalar, and then conclude Let χ(p) be a generic moment to evaluate: that fs is a scalar. The essence of the argument is to con- Z 3 µ k d p sider a Lorentz boost which transforms into the reference χ ∂xµ (p fs) + γ∇pk (as fs) = Cs . 3 3 3 3 0 γ frame of d x∧d p (i.e. which transforms p0 +d p to d p , p so that the transformed range of velocities is centered on Integrate by parts to get zero). Z 3 Z 3 Z 3 µ d p k d p d p ∂ µ χp f = f a ∇ k χ + C χ . x s γ ses p γ s γ To see that such a boost leaves dfs invariant, consider p p p the particle event paths (“world lines”) that intersect 3 3 d x ∧ d p. The transformed region of phase space will Use p⊗n := ⊗np to denote the nth tensor power. In be intersected by exactly the same particles. Although case χ = p⊗n = Sym (p⊗n), we have (at least for spa- the transformed region of phase space is not simultane- tial indices, and then by tensoriality of the expression for ous, orthogonal projection to make it simultaneous does k ⊗n−1 temporal indices as well): aes ∇pk χ = n Sym (p aes) = not change the number of particles intersecting it because qs n Sym (F•p⊗n), where I adopt the notation (F•p)µ = ms their velocity is approximately zero. µν qs F pν and use that as = F• p, where F is the electro- e ms magnetic four-tensor with components To see that the same boost leaves d3x ∧ d3p invariant, 3 3 0 −ET /c note that orthogonality of d x and d p is invariant under F µν := . the boost, that the simultaneous region d3x is mapped E/c B × I 3 to a (nonsimultaneous) region of size γ0d x (where γ0 is 3 computed from the boost velocity), and that d p is shrunk So for χ = p⊗n (n ≥ 1) we have by a factor of γ because it is transformed to a region where 2 Z 3 Z 3 dγ = 0 (because γdγ = p · dp (because γ = 1 + p · p) ⊗n µ d p qs ⊗n d p 0 ∂xµ fsp p = n Sym F• fsp and p0 = 0 post-transformation). Again, projecting onto p γ ms p γ 3 simultaneous points does not change d x. In summary, Z 3 3 3 ⊗n d p we have argued that γ0d x and d p/γ0 are scalars, and + Csp . therefore their product d3x ∧ d3p is a scalar. p γ

2. Lorentz-invariant form For χ = 1 we get balance of particle four-flow: Z 3 Z 3 µ µ d p d p Observe that ∂tfs + ∇x · (vfs) = ∂xµ (v fs), which would ∂xµ p fs = Cs . be Lorentz-invariant if we replaced vµ with γvµ = pµ. p γ p γ This indicates that to put the Vlasov equation (1) in a Separating time and space derivatives, manifestly Lorentz-invariant form we need to multiply it Z Z Z d3p by γ. Then we have ∂ f + ∇ · f v = C . ct s s s γ µ k p p p ∂xµ (p fs) + γ∇pk (as fs) = 0; the first term is manifestly Lorentz-invariant; for the sec- More generally, for χ = p⊗n, separating time and space ond term we rewrite ∇pk in terms of derivatives with re- derivatives reveals a moment hierarchy: µ 0 k spect to the four-vector p by regarding γ = p and p as Z Z independent quantities (that is, we extend the definition of ∂ct fsχ + ∇· fsvχ all quantities (arbitrarily) beyond the quadratic manifold p p 2 2 Z Z γ = 1 + p ). Then the chain rule for partial derivatives qs χ χ = n Sym F• fs + Cs . says ms p γ p γ ∂γ pk ∂ k 7→ ∂ + ∂ k = ∂ + ∂ k , p ∂pk γ p γ γ p Explicitly, omitting the collision term, the moment hierar- so chy is: Z Z Z k qs χ k p k k χ :∂ct fsχ + ∇· fsvχ = n Sym F• fs γ∇ k (a f ) 7→ γ ∂ (a f ) + ∇ k (a f ) . ms γ p s s γ γ s s p s s p p p Z Z Let a := d p = γa be the proper acceleration. Then 1 : ∂ct fs + ∇· fsv = 0, es τ s p p p · a = p · d p = γd γ = γa0, that is, p · a = a0, so the Z Z Z e τ τ e s e qs acceleration term can indeed be cast into the form appear- p : ∂ct fsp + ∇· fsvp = F• fsv , p p ms p ing in the following manifestly Lorentz-invariant version of Z Z Z qs the Vlasov equation, pp : ∂ct fspp + ∇· fsvpp = 2 Sym F• fsvp , p p ms p µ µ ∂xµ (p fs) + ∂pµ (aes fs) = 0; ... 3

sary for all the flux moments and not just the highest flux moment). Note that (1) the evolved moments are not tensorial (be- dp To separate space and time components of vectors, use that cause γ , not dp, is tensorial) and (2) the fluxes are not identical to the evolved moments (so closures are neces- v = (1, v) and that p = (γ, p); then the moment hierarchy is:

⊗n R ⊗n R ⊗n qs −1 R ⊗n−1 R ⊗n−1 p : ∂ct p fs + ∇ · vp fs = n Sym c E p fs + vp fs × B p p ms p p ⊗n−1 R ⊗n−1 R ⊗n qs −1 R ⊗n−1 γp : ∂ct γp fs + ∇ · p fs = n Sym c E · vp fs p p ms p R R 1 : ∂ct pfs + ∇· pvfs = 0, R R qs −1 R R p : ∂ct pfs + ∇· vpfs = c E fs + vfs × B , p p ms p p

R R qs −1 R γ : ∂ct γfs + ∇· pfs = c E · vfs, p p ms p R R qs −1 R R pp : ∂ct fspp + ∇· fsvpp = 2 Sym c E · p + vpfs × B , p p ms p p R R qs −1 R γp : ∂ct γpfs + ∇ · ppfs = 2 Sym c E · vpfs p p ms p ...

C. Moment closure Moveover, by scalability in ρ, the equation of state reduces P E to a function of a single variable: ρ = Pe( ρ ). For this section, assume that c = 1. For isotropic gas dynamics, one posits that there exists a reference frame Determining values of ρ, E, and the three velocity compo- in which the distribution fs is isotropic. In this reference nents of the Lorentz boost so as to match the five evolved frame, the invariant flux tensor for χ = 1 has a single moments is a nonlinear problem that is usually solved it- nonzero component: eratively. To determine the form of this system, we apply a Lorentz R boost Λ that transforms from quantities in the reference [1, v]fs = [ρ, 0] p frame of the fluid to a laboratory frame moving at proper velocity U and the invariant flux tensor for χ = p is diagonal: Γ U Λ = , U + UU Z γ p E 0 I Γ+1 f = ; p vp s 0 P p I where Γ2 = 1 + |U|2. the pressure P is related to the energy E and density ρ Applying this boost to the tensor for the density and flux by an equation of state that is determined by the assumed of mass says that distribution of particle velocities; usually the assumed dis- ρ Γρ tribution is defined to maximize some notion of entropy. Λ · = , i.e., To see how the assumed distribution relates pressure to 0 Uρ energy, observe that pressure as well as energy is given by ρ0 = Γρ, integrating the assumed distribution against a function of γ: where ρ0 is mass density in laboratory coordinates.

Applying this boost to the tensor for the density and ﬂux R vpf = R γ−1ppf p s I p s of energy and momentum says that R −1 1 2 = I γ |p| fs p 3 E 0 Γ2E + |U|2P (E + P)ΓU Z 2 Λ · ; ·Λ = ; γ − 1 0 P (E + P)ΓU P( + UU) = I fs. I I p 3γ