A Monte Carlo Method for Calculating Lynden-Bell Equilibrium in Self-Gravitating Systems.

We present a Monte Carlo approach that allows us to easily implement Lynden-Bell (LB) entropy maximization for an arbitrary initial particle distribution. The direct maximization of LB entropy for an arbitrary initial distribution requires an infinite number of Lagrange multipliers to account for the Casimir invariants. This has restricted studies of Lynden-Bell's violent relaxation theory to only a very small class of initial conditions of a very simple waterbag form, for which the entropy maximization can be performed numerically. In the present approach, an arbitrary initial distribution is discretized into density levels which are then evolved using an efficient Monte Carlo algorithm towards the final equilibrium state. A comparison is also made between the LB equilibrium and explicit Molecular Dynamics simulations. We find that for most initial distributions, relaxation is incomplete and the system is not able to reach the state of maximum LB entropy. In particular, we see that the tail of the stationary particle distribution is very different from the one predicted by the theory of violent relaxation, with a hard cutoff instead of an algebraic decay predicted by LB's theory.

Euler fluid in two dimensions: Statistical approach.

We use Kirchhoff's vortex formulation of 2D Euler fluid equations to explore the equilibrium state to which a 2D incompressible fluid relaxes from an arbitrary initial flow. The vortex dynamics obeys Hamilton's equations of motion with x and y coordinates of the vortex position forming a conjugate pair. A state of fluid can, therefore, be expressed in terms of an infinite number of infinitesimal vortices. If the vortex dynamics is mixing, the final equilibrium state of the fluid should correspond to the maximum of Boltzmann entropy, with the constraint that all the Casimir invariants of the fluid must be preserved. This is the fundamental assumption of Lynden-Bell's theory of collisionless relaxation. In this paper, we will present a Monte Carlo method which allows us to find the maximum entropy state of the fluid starting from an arbitrary initial condition. We will then compare this prediction with the results of molecular dynamics simulation and demonstrate that the final state to which the fluid evolves is, actually, very different from that corresponding to the maximum of entropy. This indicates that the mixing assumption is not correct. We will then present a different approach based on core-halo distribution which allows us to accurately predict the final state to which the fluid will relax, starting from an arbitrary initial condition.