The Einstein-Boltzmann system

The Boltzmann equation provides a description of the dynamics of a large number of particles undergoing collisions, such as the molecules of a gas. The classical Boltzmann equation belongs to Newtonian physics. It has a natural relativistic generalization. The Boltzmann model is adapted to capture the effects of short-range forces acting on short time scales during collisions. The model can be extended to also include the effects of long-range forces generated collectively by the particles. If the forces are gravitational and the description is made fully relativistic then the system of equations obtained is the Einstein-Boltzmann system. In any of the cases mentioned up to now the Boltzmann equation is schematically of the form Xf=Q(f). The term on the left is a transport term giving the rate of change of the function f, the density of particles, along a vector field X on phase space. The vector field X is in general determined by the long-range forces. The term on the right is the collision term which, as its name suggests, models the effect of collisions. It is an integral term which is quadratic in f. The function f itself is a function of variables (t,x,p) representing time, position and velocity (or momentum).

How is the collision term obtained? It is important to realize that it is in no sense universal – it contains information about the particular interaction between the particles due to collisions. This can be encoded in what is called the scattering kernel. In the classical case it is possible to do the following. Fix a type of interaction between individual particles and solve the corresponding scattering problem. Each specific choice of interaction gives a scattering kernel. Once various scattering kernels have been obtained in this way it is possible to abstract from the form of the kernels obtained to define a wider class. A similar process can be carried out in special relativity although it is more complicated. Any scattering kernel which has been identified as being of interest in special relativity can be taken over directly to general relativity using the principle of equivalence. Concretely this means that if the Boltzmann collision term is expressed in terms of the components of the momenta in an orthonormal frame
then the resulting expression also applies in general relativity.

For a system of evolution equations like the Einstein-Boltzmann system one of the most basic mathematical questions is the local well-posedness of the initial value problem. For the EB system this problem was solved in 1973 by Daniel Bancel and Yvonne Choquet-Bruhat (Commun. Math. Phys. 33, 83) for a certain class of collision terms. The physical interpretation of the unknown f in the Boltzmann equation as a number density means that it should be non-negative. In the context of the initial value problem this means that it should be assumed that f is initially non-negative and that it should then be proved that the corresponding solution is non-negative. In the existence proofs for many cases of the Boltzmann equation the solution is obtained as the limit of a sequence of iterates, each of which are by construction non-negative. The convergence to the limit is strong enough that that the non-negativity of the iterates is inherited by the solution. In the theorem of Bancel and Choquet-Bruhat the solution is also constructed as the limit of a sequence of iterates but no attention is paid to non-negativity. In fact that issue is not mentioned at all in their paper. To prove non-negativity of solutions of the EB system it is enough to prove the corresponding statement for solutions of the Boltzmann equation on a given spacetime background. The latter question has been addressed in papers of Bichteler and Tadmon. On the other hand it is not easy to see how their results relate to those of Bancel and Choquet Bruhat. This question has now been investigated in a paper by Ho Lee and myself . The result is that with extra work the desired posivity result can be obtained under the assumptions of the theorem of Bancel and Choquet-Bruhat. While working on this we obtained some other insights about the EB system. One is that the assumptions of the existence theorem appear to be very restrictive and that treating physically motivated scattering kernels will probably require more refined approaches. In the almost forty years since the local existence theorem there have been very few results on the initial value problem for the EB system (with non-vanishing collision term). We hope that our paper will set the stage for further progress on this subject.

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2 Responses to “The Einstein-Boltzmann system”

  1. Mathblogging.org Weekly Picks « Mathblogging.org — the Blog Says:

    [...] Hydrobates, writes about the Einstein-Boltzmann system and a recent paper by Ho Lee and [...]

  2. The relativistic Boltzmann equation « Hydrobates Says:

    [...] a previous post I wrote about the Einstein-Boltzmann system and some recent work on that subject by Ho Lee and [...]

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