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Vlasov equation
- The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation. This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved unless the collision term in f is known.
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The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation. This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved unless the collision term in f is known.
- Collisionless Dynamics
- Distribution Function
- Collisionless Boltzmann Equation
- Gravity
- Conservation of Phase Space Density
A typical galaxy has stars but is only crossing times old, so the cumulative effects of encounters between stars are not significant. This justifies the next step, which is to idealize a galaxy as a continuous mass distribution. In this limit, each star moves in the smooth gravitational field of the galaxy. Thus instead of thinking about motion in ...
Rather than keeping track of individual stars, a galaxy may be described by the one-body distribution function; let be the mass of stars in the phase-space volume at (,) and time . This provides a complete description if stars are uncorrelated, as assumed above.
The motion of matter in phase space is governed by the phase space flow, How does this affect the total amount of mass in the phase space volume ? The rate of change of the mass is just the inflow minus the outflow; that is, the flow obeys a continuity equation in 6 dimensions: where the derivatives with respect to , and are understood to be partia...
The gravitational field is given self-consistently by Poisson's equation, Eqs. (4,5) may be viewed as a pair of coupled PDEs which together completely describe the evolution of a galaxy.
Let be the orbit of a star. What is the rate of change of along the star's orbit? The answer is zero, where the first equality is just the definition of the convective derivative in phase-space, the second equality follows on substituting the phase-flow (Eq. 2), and the last equality follows from the CBE (Eq. 4). Thus, phase-space density is conser...
contains a collisionless part df=dt, which deals with the e ects of gravity on the photon distribution function f, and collision terms C[f], which account for its interactions with other species in the universe. The collision terms in the Boltzmann equation have several important e ects.
Today we derive the collisionless Boltzmann equation in the context of galaxies, formulate the self-consistent problem and outline a few analytic approaches to solving it.
- 114KB
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a 6-dimensional equation of continuity that is analogous to the familiar 3-dimensional equation of continuity of fluid mechanics. Each point in phase space is described by a 6-D vector wξ = (ξx,ξv).
Collisionless Boltzmann Equation. Let's consider a probability distribution function (pdf) of a single particle, f, in a phase space described by canonical coordinates (~q ; ~p ). That is to say that f(~q ; ~p ; t)d3~q d3~p is the probability of that particle having ~q 2 [~q ; ~q + d3~q ] and ~p 2 [~p ; ~p + d3~p ] at time t.
Equation 2 (or 3) is known as the collisionless Boltzmann equation. It is used to study the kinetic theory of gases, atomic nuclei and for stellar dynamical systems such as galaxies and globular clusters. The collisionless Boltzmann equation is sufficiently complex that it is usually difficult to solve. Equation 2 is sometimes written Df Dt = 0
