CHASSIS - Inverse Modelling of Relaxed Dynamical Systems

TL;DR

CHASSIS is a Bayesian non-parametric algorithm that infers the phase space distribution function and gravitational potential of relaxed spherical systems using incomplete velocity data.

Contribution

It introduces a novel MCMC-based method to recover phase space and potential from limited data, applicable to relaxed gravitational systems with minimal input requirements.

Findings

Successfully infers phase space distribution and potential.

Works with incomplete, single-component velocity data.

Utilizes MCMC optimization for maximum likelihood estimation.

Abstract

The state of a non-relativistic gravitational dynamical system is known at any time $t$ if the dynamical rule, i.e. Newton's equations of motion, can be solved; this requires specification of the gravitational potential. The evolution of a bunch of phase space coordinates ${\bf w}$ is deterministic, though generally non-linear. We discuss the novel Bayesian non-parametric algorithm CHASSIS that gives phase space $pdf$ $f({\bf w})$ and potential $\Phi({\bf x})$ of a relaxed gravitational system. CHASSIS is undemanding in terms of input requirements in that it is viable given incomplete, single-component velocity information of system members. Here ${\bf x}$ is the 3-D spatial coordinate and ${\bf w}={\bf x+v}$ where ${\bf v}$ is the 3-D velocity vector. CHASSIS works with a 2-integral $f=f(E, L)$ where energy $E=\Phi + v^2/2, \: v^2 = \sum_{i=1}^{3}{v_i^2}$ and the angular momentum is $L = |{\bf r}\times{\bf v}|$, where ${\bf r}$ is the spherical spatial vector. Also, we assume spherical symmetry. CHASSIS obtains the $f(\cdot)$ from which the kinematic data is most likely to have been drawn, in the best choice for $\Phi(\cdot)$, using an MCMC optimiser (Metropolis-Hastings). The likelihood function ${\cal{L}}$ is defined in terms of the projections of $f(\cdot)$ into the space of observables and the maximum in ${\cal{L}}$ is sought by the optimiser.