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
3 - The dynamics of random impulsive forces
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
The slings and arrows of outrageous fortune.
ShakespearePrevious sections have made it plausible that an object in a gravitating system near equilibrium can be considered to be immersed in a bath of fluctuating forces, along with an average mean field force. We now consider a simple mathematical model for the time evolution of orbits. We use this intuitive physical picture to try to capture the essence of the problem in a fairly simple way. An advantage of this procedure is that it readily suggests modifications of the description for an improved physical picture. The results can always be checked against N-body computer experiments, and we will discuss their more exact derivation in Section 10.
At first sight, the simplest model might seem to represent the motion of each star by Newton's equation of motion with a stochastic force β(t) which fluctuates in time, i.e.,. But this turns out to be too simple. It makes the velocity undergo Brownian motion (for a Gaussian distribution of fluctuations) with an everincreasing root mean square value vrms ∝t½. Correspondingly, the root mean square position of an average star also departs monotonically from its initial value. These two properties are inconsistent with conservation of total energy, for the increase in kinetic energy must be compensated by a contraction of the system to decrease the potential energy. But the Brownian increase of every star's root mean square position from its initial value prevents the system from becoming very small.
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- Information
- Gravitational Physics of Stellar and Galactic Systems , pp. 14 - 20Publisher: Cambridge University PressPrint publication year: 1985