1.2 The Vlasov-Maxwell and Vlasov-Poisson 1 Introduction to Kinetic Theory1 Introduction to Kinetic Theory

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 tex2html_wrap_inline1344, a =0,1,2,3. Since all particles have equal rest mass, the four-momentum for each particle is restricted to the mass shell tex2html_wrap_inline1348 . Thus, by denoting the three-momentum by tex2html_wrap_inline1332, tex2html_wrap_inline1344 may be written tex2html_wrap_inline1354, where | p | is the usual Euclidean length of p and tex2html_wrap_inline1360 is the energy of a particle with three-momentum p . The relativistic velocity of a particle with momentum p is denoted by tex2html_wrap_inline1366 and is given by


Note that tex2html_wrap_inline1368 . The relativistic Boltzmann equation models the spacetime behaviour of the one-particle distribution function f = f (t, x, p), and it has the form


where the relativistic collision operator Q (f, g) is defined by


(Note that g = f in (2Popup Equation)). Here tex2html_wrap_inline1376 is the element of surface area on tex2html_wrap_inline1378 and tex2html_wrap_inline1380 is the scattering kernel, which depends on the scattering cross-section in the interaction process. See [23Jump To The Next Citation Point In The Article] for a discussion about the scattering kernel. The function tex2html_wrap_inline1382 results from the collision mechanics. If two particles, with momentum p and q respectively, collide elastically (no energy loss) with scattering angle tex2html_wrap_inline1388, their momenta will change, tex2html_wrap_inline1390 and tex2html_wrap_inline1392 . The relation between p, q and p ', q ' is




This relation is a consequence of four-momentum conservation,


or equivalently


These are the conservation equations for relativistic particle dynamics. In the classical case these equations read


The function tex2html_wrap_inline1382 is the distance between p and p ' (q and q '), and the analogue function in the Newtonian case has the form


By inserting tex2html_wrap_inline1408 in place of a in (2Popup Equation) we obtain the classical Boltzmann collision operator (disregarding the scattering kernel, which is also different).

The main result concerning the existence of solutions to the classical Boltzmann equation is a theorem by DiPerna and Lions [25] 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 Dudynsky and Ekiel-Jezewska [26]. Regarding classical solutions, Illner and Shinbrot [46] 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. When the data are close to equilibrium (see below), global existence of classical solutions has been proved by Glassey and Strauss [36Jump To The Next Citation Point In The Article] in the relativistic case and by Ukai [87] in the nonrelativistic case (see also [84]).

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


where tex2html_wrap_inline1414 and tex2html_wrap_inline1416 are called the gain and loss term respectively. In [2Jump To The Next Citation Point In The Article] it is proved that given tex2html_wrap_inline1418 and tex2html_wrap_inline1420 with tex2html_wrap_inline1422, then


under some technical requirements on the scattering kernel. Here tex2html_wrap_inline1424 is the usual Sobolev space. This regularizing result was first proved by P.L. Lions [49] 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.

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 tex2html_wrap_inline1426 to a global Maxwellian,


as time goes to infinity. This result had first been obtained by Arkeryd [8] 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,


had been studied by Glassey and Strauss [36, 37]. 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 [29Jump To The Next Citation Point In The Article, 23, 86] and in the nonrelativistic case we refer to the excellent review paper by Villani [88] and the books [29Jump To The Next Citation Point In The Article, 16].

1.2 The Vlasov-Maxwell and Vlasov-Poisson 1 Introduction to Kinetic Theory1 Introduction to Kinetic Theory

image The Einstein-Vlasov System/Kinetic Theory
Håkan Andréasson
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