# Lecture 5 Collisionless Boltzmann Equation

**Physics 141/241 **

## Collisionless dynamics

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 a phase space of
dimensions, we can think about motion in a phase space of just
dimensions. This is a very important simplification!

## Distribution function

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.

## Collisionless Boltzmann equation

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 partial derivatives. Using the expression for the
phase-flow yields the collisionless Boltzmann
equation:

The collisionless Boltzmann equation or CBE describes the evolution of the
distribution function
.
It serves as the fundamental equation of galactic dynamics.

## Fluid continuity equation

*(See explanation with added class note here)*

## Gravity

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.*

## Conservation of phase space density

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. 5)*.
Thus, phase-space density is conserved along every orbit.

This fundamental and completely general result shows that the CBE has a much
greater level of symmetry than the N-body equations of motion; whereas the
latter conserves a fairly small set of parameters, the CBE conserves f(r,v,t)
along an infinite number of stellar orbits. We can take advantage of this
infinite array of conservation laws to obtain some important results even when
we can't explicitly solve the CBE.