The following is a brief description of initial results in examining critical phenomena in the Einstein-Vlasov system. Evidence for static, type I critical solutions have so far been found, with the associated logarithmic scaling times. For information about critical phenomena, first discovered by Matt Choptuik, see: Carsten Gundlach's Review. Please report comments or errors to Roland Stevenson.
The Einstein-Vlasov system couples Einstein's equations with the Liouville equation. The Liouville equation determines the trajectories in phase-space of collisionless particles. Therefore, in order to solve the system one would prefer to have an accurate representation of the phase space picture.
I've implemented two codes: a finite-difference code and a finite volume code. The finite difference code is both parallel and adaptive mesh refinement-capable (using Frans Pretorius's PAMR libraries). The finite volume code makes use of high-resolution shock capturing and flux-conservative methods to solve the same set of equations, and produces the same results.
The following animations show evidence for Type I critical solutions in the Einstein-Vlasov system, in that a boundary between dispersal and collapse is found by varying one parameter, namely the amplitude of a gaussian in phase-space.
In each animation, the radial momentum axis of phase space is the vertical axis, while the radial position is represented on the horizontal axis. The system is in spherical symmetry, but has an angular momentum term. Spherical symmetry is maintained by the ansatz that each shell of thickness dr at radius r is comprised of an infinite number of shells each with angular momentum equal and opposite to each other in every possible direction. Therefore the net angular momentum is zero and spherical symmetry is preserved, but the particles still experience a "centrifugal force".
I use the phrase "critical solutions" loosely, since the following evolutions are obviously just approximations to the actual critical solution, and asymptotically in time either collapse or disperse.
Here are the two scenarios played out explicitly. First an arbitrary slightly sub-critical evolution: L12_ycm0.1_h3_DV_MPG.mpg (1.2MB) and a supercritical evolution: super_L12_ycm0.1_h3_DV_MPG.mpg (1.1MB).
A couple things to notice: both sub- and super-critical evolutions approach the same intermediate attractor before "deciding" to disperse or collapse, respectively. The dispersal is indicated by a shift of the distribution function to the positive radial momentum portion of the grid (the upper half), while the collapse is indicated by both a move towards the negative radial momentum portion of the grid and the quantity 2M/r going to 1 (ie. the metric approaches Schwarzchild).
The code is second order convergent. Low resolution runs show an ingoing oscillating distribution function, but that disappears with better resolution. For example at one resolution (1.1MB) we see an oscillation, at twice the resolution (1.2MB)we see much less. The issue of whether or not a static solution is produced in the continuum is still up for debate, but the time derivatives of the metric functions converge in a second-order way to zero.
The following are animations of various initial conditions, all approaching the same type of intermediate attractor. It has yet to be confirmed whether this intermediate attractor is indeed universal up to a rescaling of the geometry and/or coordinates. Only subcritical evolutions are shown, supercritical is left to the reader's imagination.
Critical phenomena also exist in the case where two distributions of arbitrary angular momenta interact. Here are videos of two distribution functions that are allowed to interact through their combined gravitational potential. After tuning, we see behavior similar to that observed in the case of one distribution function, including staticity and logarithmic scaling. In the following scenario our initial conditions involve one distribution at L=12 under ISIF conditions, and a second distribution that is at L=13, also falling inward (though not as quickly). Distribution function specific parameters are shown in the titles of each video. The first distribution: u1 (4.0MB), and the second: u2 (4.0MB). A composite .avi file of both distributions as they reside in a third phase-space dimension representing varying angular momentum: composite (0.5MB).
Notice how everything's at L=12 so far. This is because I was trying to reproduce the results of Inaki Olabarrieta, who deserves the credit for starting me on this project. I'd like to 1.) see if the critical solution is universal for a range of angular momenta 2.) see if it applies to the massless case 3.) see if the critical solution can be written in closed form (it seems to resemble a polytrope with angular momentum).