Archive for March, 2012

The Einstein-Boltzmann system

March 13, 2012

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.

Do you know these matrices?

March 9, 2012

I have come across a class of matrices with some interesting properties. I feel that they must be known but I have not been able to find anything written about them. This is probably just because I do not know the right place to look. I will describe these matrices here and I hope that somebody will be able to point out a source where I can find more information about them. Consider an $n\times n$ matrix $A$ with elements $a_{ij}$ having the following properties. The elements with $i=j$ (call them $b_i$) are negative. The elements with $j=i+1\ {\rm mod}\ n$ (call them $c_i$) are positive. All other elements are zero. The determinant of a matrix of this type is $\prod_i b_i+(-1)^{n+1}\prod_i c_i$. Notice that the two terms in this sum always have opposite signs. A property of these matrices which I found surprising is that $B=(-1)^{n+1}(\det A)A^{-1}$ is a positive matrix, i.e. all its entries $b_{ij}$ are positive. In proving this it is useful to note that the definition of the class is invariant under cyclic permutation of the indices. Therefore it is enough to show that the entries in the first row of $B$ are non-zero. Removing the first row and the first column from $A$ leaves a matrix belonging to the class originally considered. Removing the first row and a column other than the first from $A$ leaves a matrix where $a_{n1}$ is alone in its column. Thus the determinant can be expanded about that element. The result is that we are left to compute the determinant of an $(n-2)\times (n-2)$matrix which is block diagonal with the first diagonal block belonging to the class originally considered and the second diagonal block being the transpose of a matrix of that class. With these remarks it is then easy to compute the determinant of the $(n-1)\times (n-1)$ matrix resulting in each of these cases. In more detail $b_{11}=(-1)^{n+1}b_2b_3\ldots b_n$ and $b_{1j}=(-1)^{n-j}b_2b_3\ldots b_{j-1}c_j\ldots c_n$ for $j>1$.

Knowing the positivity of $(-1)^{n+1}(\det A)A^{-1}$ means that it is possible to apply the Perron-Frobenius theorem to this matrix. In the case that $\det A$ has the same sign as $(-1)^{n+1}$ it follows that $A^{-1}$ has an eigenvector all of whose entries are positive. The corresponding eigenvalue is positive and larger in magnitude than any other eigenvalue of $A^{-1}$. This vector is also an eigenvalue of $A$ with a positive eigenvalue. Looking at the characteristic polynomial it is easy to see that if $(-1)^n(b_1b_2\ldots b_n+(-1)^{n+1}c_1c_2\ldots c_n)<0$ the matrix $A$ has exactly one positive eigenvalue and that none of its eigenvalues is zero.