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9 - The Collisionless Boltzmann Equation and the Jeans Theorem
from Part III - Collisionless Systems
Luca Ciotti, Università degli Studi, Bologna, Italy
Book:

Introduction to Stellar Dynamics

Published online:

28 May 2021

Print publication:

10 June 2021, pp 151-171
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Summary

With this chapter, the final part of the book, dedicated to collisionless stellar systems, begins. As should be clear, in order to extract information from the N-body problem, we need to move to a different approach than direct integration of the differential equations of motion, and a first (unfruitful) attempt will be based on the Liouville equation. In fact, the basic reason for the “failure” of the Liouville approach is that, despite its apparent statistical nature, the dimensionality of the phase space Γ where the function f(6N) is defined is the same as that of the original N-body problem. Suppose instead we find a way to replace the 6N-dimensional R6N phase space Γ with the six-dimensional one-particle phase space γ: we can reasonably expect that the problem would be simplified enormously, and in fact along these lines we will finally obtain the collisionless Boltzmann equation, one of the conceptual pillars of stellar dynamics.

A derivation of the Liouville equation for hard particle dynamics with non-conservative interactions
Part of
- Two-phase and multiphase flows
- Dynamical problems
Benjamin D. Goddard, Tim D. Hurst, Mark Wilkinson
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 3 / June 2021

Published online by Cambridge University Press:

28 December 2020, pp. 1040-1074

Print publication:

June 2021
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The Liouville equation is of fundamental importance in the derivation of continuum models for physical systems which are approximated by interacting particles. However, when particles undergo instantaneous interactions such as collisions, the derivation of the Liouville equation must be adapted to exclude non-physical particle positions, and include the effect of instantaneous interactions. We present the weak formulation of the Liouville equation for interacting particles with general particle dynamics and interactions, and discuss the results using two examples.

Conservative Finite-Difference Scheme for High-Frequency Acoustic Waves Propagating at an Interface Between Two Media
J. Staudacher, É Savin
Journal:

Communications in Computational Physics / Volume 11 / Issue 2 / February 2012

Published online by Cambridge University Press:

20 August 2015, pp. 351-366

Print publication:

February 2012
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This paper is an introduction to a conservative, positive numerical scheme which takes into account the phenomena of reflection and transmission of high frequency acoustic waves at a straight interface between two homogeneous media. Explicit forms of the interpolation coefficients for reflected and transmitted wave vectors on a two-dimensional uniform grid are derived. The propagation model is a Liouville transport equation solved in phase space.