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1.1 The relativistic Boltzmann equation

Consider a collection of neutral particles in Minkowski spacetime. Let the signature of the metric be (− ,+, +,+ ), let all the particles have rest mass m = 1, and normalize the speed of light c to one. The four-momentum of a particle is denoted by pa, a = 0,1, 2,3. Since all particles have equal rest mass, the four-momentum for each particle is restricted to the mass shell, a 2 p pa = − m = − 1. Thus, by denoting the three-momentum by 3 p ∈ ℝ, a p may be written a 0 p = (− p ,p), where |p| is the usual Euclidean length of p and ∘ -------- p0 = 1 + |p|2 is the energy of a particle with three-momentum p. The relativistic velocity of a particle with momentum p is denoted by ˆp and is given by
ˆp = ∘---p-----. (1 ) 1 + |p|2
Note that |ˆp| < 1 = c. The relativistic Boltzmann equation models the spacetime behaviour of the one-particle distribution function f = f(t,x,p), and it has the form
( ) p-- ∂t + p0 ⋅ ∇x f = Q (f,f), (2 )
where the relativistic collision operator Q (f,g) is defined by
∫ ∫ Q (f, g) = k(p,q,ω )[f(p + a(p,q,ω )ω)g (q − a (p, q,ω)ω) − f(p)g(q)]dp dω. (3 ) ℝ3 𝕊2
(Note that g = f in Equation (2View Equation)). Here dω is the element of surface area on 𝕊2 and k(p,q,ω ) is the scattering kernel, which depends on the scattering cross-section in the interaction process. See [34Jump To The Next Citation Point] for a discussion about the scattering kernel. The function a(p,q,ω) results from the collision mechanics. If two particles, with momentum p and q respectively, collide elastically (no energy loss) with scattering angle ω ∈ 𝕊2, their momenta will change, p → p′ and q → q′. The relation between p, q and p′,q′ is
′ ′ p = p + a(p,q,ω)ω, q = q − a(p,q,ω )ω, (4 )
where
2(p0 + q0)p0q0(ω ⋅ (qˆ− ˆp)) a(p,q,ω ) = --0----0-2--------------2 . (5 ) (p + q ) − (ω ⋅ (p + q))
This relation is a consequence of four-momentum conservation,
a a a′ a′ p + q = p + q ,

or equivalently

0 0 0′ 0′ p + q = p + q , (6 ) p + q = p′ + q′. (7 )
These are the conservation equations for relativistic particle dynamics. In the classical case these equations read
|p |2 + |q|2 = |p′|2 + |q′|2, (8 ) ′ ′ p + q = p + q. (9 )
The function a(p,q,ω) is the distance between p and p′ (q and q′), and the analogue function in the Newtonian case has the form
acl(p,q,ω) = ω ⋅ (q − p). (10 )
By inserting acl in place of a in Equation (3View Equation) we obtain the classical Boltzmann collision operator (disregarding the scattering kernel, which is also different). In [20] classical solutions to the relativistic Boltzmann equations are studied as c → ∞, and it is proved that the limit as c → ∞ of these solutions satisfy the classical Boltzmann equation.

The main result concerning the existence of solutions to the classical Boltzmann equation is a theorem by DiPerna and Lions [36] that proves existence, but not uniqueness, of renormalized solutions (i.e. solutions in a weak sense, which are even more general than distributional solutions). An analogous result holds in the relativistic case, as was shown by Dudyńsky and Ekiel-Jezewska [37]. Regarding classical solutions, Illner and Shinbrot [58] have shown global existence of solutions to the nonrelativistic Boltzmann equation for small initial data (close to vacuum). At present there is no analogous result for the relativistic Boltzmann equation and this must be regarded as an interesting open problem. There is however a recent result [74] for the homogeneous relativistic Boltzmann equation where global existence is shown for small initial data and bounded scattering kernel. When the data are close to equilibrium (see below), global existence of classical solutions has been proved by Glassey and Strauss [48Jump To The Next Citation Point] in the relativistic case and by Ukai [115] in the nonrelativistic case (see also [108] and [69]).

The collision operator Q (f, g) may be written in an obvious way as

+ − Q (f, g) = Q (f,g) − Q (f,g),

where Q+ and Q − are called the gain and loss term, respectively. If the loss term is deleted the gain-term-only Boltzmann equation is obtained. It is interesting to note that the methods of proof for the small data results mentioned above concentrate on gain-term-only equations, and once that is solved it is easy to include the loss term. In [5] it is shown that the gain-term-only classical and relativistic Boltzmann equations blow up for initial data not restricted to a small neighbourhood of trivial data. Thus, if a global existence proof for unrestricted data will be given it will necessarily use the full collision operator.

The gain term has a nice regularizing property in the momentum variable. In [2Jump To The Next Citation Point] it is proved that given 2 3 f ∈ L (ℝ ) and 1 3 g ∈ L (ℝ ) with f,g ≥ 0, then

+ ∥Q (f,g)∥H1(ℝ3p) ≤ C ∥f ∥L2(ℝ3p)∥g∥L1(ℝ3p), (11 )
under some technical requirements on the scattering kernel. Here s H is the usual Sobolev space. This regularizing result was first proved by P.L. Lions [65] in the classical situation. The proof relies on the theory of Fourier integral operators and on the method of stationary phase, and requires a careful analysis of the collision geometry, which is very different in the relativistic case. See also [119] and [120] for a simplified proof in the classical and relativistic case respectively.

The regularizing theorem has many applications. An important application is to prove that solutions tend to equilibrium for large times. More precisely, Lions used the regularizing theorem to prove that solutions to the (classical) Boltzmann equation, with periodic boundary conditions, converge in L1 to a global Maxwellian,

M = e−α|p|2+β⋅p+ γ with α, γ ∈ R, α > 0, β ∈ ℝ3,

as time goes to infinity. This result had first been obtained by Arkeryd [10] by using non-standard analysis. It should be pointed out that the convergence takes place through a sequence of times tending to infinity and it is not known whether the limit is unique or depends on the sequence. In the relativistic situation, the analogous question of convergence to a relativistic Maxwellian, or a Jüttner equilibrium solution,

----- −α√ 1+|p|2+β⋅p+ γ J = e , α, β, and γ as above, with α > |β|,

had been studied by Glassey and Strauss [4849]. In the periodic case they proved convergence in a variety of function spaces for initial data close to a Jüttner solution. Having obtained the regularizing theorem for the relativistic gain term, it is a straightforward task to follow the method of Lions and prove convergence to a local Jüttner solution for arbitrary data (satisfying the natural bounds of finite energy and entropy) that is periodic in the space variables. In [2] it is next proved that the local Jüttner solution must be a global one, due to the periodicity of the solution.

For more information on the relativistic Boltzmann equation on Minkowski space we refer to [41Jump To The Next Citation Point34110] and in the nonrelativistic case we refer to the excellent review paper by Villani [116] and the books [41Jump To The Next Citation Point24].


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