Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Part I Idealized homogeneous systems – basic ideas and gentle relaxation
- 1 The average and fluctuating gravitational fields
- 2 Gentle relaxation: timescales
- 3 The dynamics of random impulsive forces
- 4 General properties of Fokker–Planck evolution
- 5 Fokker–Planck description of gravitating systems
- 6 Dynamics with a memory: non-Markovian evolution
- 7 The Boltzmann equation
- 8 Some properties of the Boltzmann equation
- 9 The virial theorem
- 10 The grand description – Liouville's equation and entropy
- 11 Extracting knowledge: the BBGKY hierarchy
- 12 Extracting knowledge: the Fourier development
- 13 Collective effects – grexons
- 14 Collective scattering
- 15 Linear response and dispersion relations
- 16 Damping and decay
- 17 Star-gas interactions
- 18 Problems and extensions
- 19 Bibliography
- Part II Infinite inhomogeneous systems – galaxy clustering
- Part III Finite spherical systems – clusters of galaxies, galactic nuclei, globular clusters
- Part IV Finite flattened systems – galaxies
- Index
10 - The grand description – Liouville's equation and entropy
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- Contents
- Preface
- Introduction
- Part I Idealized homogeneous systems – basic ideas and gentle relaxation
- 1 The average and fluctuating gravitational fields
- 2 Gentle relaxation: timescales
- 3 The dynamics of random impulsive forces
- 4 General properties of Fokker–Planck evolution
- 5 Fokker–Planck description of gravitating systems
- 6 Dynamics with a memory: non-Markovian evolution
- 7 The Boltzmann equation
- 8 Some properties of the Boltzmann equation
- 9 The virial theorem
- 10 The grand description – Liouville's equation and entropy
- 11 Extracting knowledge: the BBGKY hierarchy
- 12 Extracting knowledge: the Fourier development
- 13 Collective effects – grexons
- 14 Collective scattering
- 15 Linear response and dispersion relations
- 16 Damping and decay
- 17 Star-gas interactions
- 18 Problems and extensions
- 19 Bibliography
- Part II Infinite inhomogeneous systems – galaxy clustering
- Part III Finite spherical systems – clusters of galaxies, galactic nuclei, globular clusters
- Part IV Finite flattened systems – galaxies
- Index
Summary
All for one, one for all, that is our device.
Alexandre Dumas, ElderLangevin's equation, the Fokker–Planck equation, the master equation, and Boltzmann's equation are all just partial descriptions of gravitating systems. Each is based on different assumptions, suited to different conditions. They all arise from physical, rather intuitive, approaches to the problem. But there is also a more general description from which our previous ones emerge as special cases. We know this must be true because Newton's equations of motion provide a complete description of all the orbits. The trouble with Newton's equations is that they are not very compact: N objects generate 6N equations. True, the total angular and linear momenta, and energy, are conserved, at least for isolated systems, but this is not usually a great simplification.
By extending our imagination, we can cope with the problem. We previously imagined a six-dimensional phase space for the collisionless Boltzmann equation. Each point in this phase space represented the three position and three velocity (or momentum) coordinates of a single particle. It was a slight generalization of the twodimensional phase plane whose coordinates are values of a quantity and its first derivative resulting from a second order differential equation for that quantity. The terminology probably arose from the case of the harmonic oscillator where this plane gave the particular stage or phase in the recurring sequence of movement of the oscillator.
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- Information
- Gravitational Physics of Stellar and Galactic Systems , pp. 54 - 63Publisher: Cambridge University PressPrint publication year: 1985