For a
-dimensional
system with Hamiltonian
, at a given
fixed parameter value the surfaces of
form shells in phase-space
.
At an energy , the phase-space volume
enclosed by the shell is given by ,
and the weight of the shell
(2.1) |
Ott[153] showed that in the adiabatic limit
of slow time-dependent , an initial
phase-space distribution on an energy shell which
encloses phase-space volume
remains on an energy
shell for all future times.
What energy the shell has at a future time
(when the parameter has value )
is given
by the condition that the enclosed phase-space volume remains constant:
Now this time-dependence of the shell's energy can
be written
in terms of the time-evolution
of a probability density function.
This is an effective equation for timescales longer than
the ergodic (mixing) time,
so that the distribution across the surface of an
energy shell is assumed to have
equilibrated (when viewed on a coarse-grained scale)
between time-steps.
This means that the distribution can be completely
characterized by a function of energy alone:
it is reduced to a one-dimensional time-dependent distribution
.
An equivalent representation is the distribution
in the phase space volume variable .
Keep in mind that there is a time-dependent (one-to-one) relationship
between and : this can be visualized
as constant- slices
of a surface shown in Fig. 2.1b, at the value .
Likewise the probability densities are related through the
time-dependent Jacobean,
(2.4) |
There is no evolution in the adiabatic limit
when written in terms of
: