Ensemble distribution 67pqpqdx 0dxtFig. 2.3 Illustration of phase space volume conservation prescribed by Liouville’s theorem.
regions of phase space and less densely in others. We, therefore,need to describe
quantitatively how the systems in an ensemble are distributed in the phase space at
any point in time. To do this, we introduce theensemble distribution functionor
phase space distribution functionf(x,t). The phase space distribution function of an
ensemble has the property thatf(x,t)dx is the fraction of the total ensemble members
contained in the phase space volume element dx at timet. From this definition, it is
clear thatf(x,t) satisfies the following properties:
f(x,t)≥ 0
∫
dxf(x,t) = 1. (2.5.1)Therefore,f(x,t) is a probability density.
When the phase space distribution is expressed asf(x,t), we imagine ourselves
sitting at afixedlocation x in the phase space and observing the ensemble distribution
evolve around us as a function of time. In order to determine the number of ensemble
members in a small element dx at our location, we could simply “count” the number
of microstates belonging to the ensemble in dx at any timet, determine the fraction
f(x,t)dx, and build up a picture of the distribution. On the other hand, the ensemble
consists of a collection of systems all evolving in time according to Hamilton’s equations
of motion. Thus, we can also let the ensemble distribution function describe how a
bundle of trajectories in a volume element dxtcentered on a trajectory xtis distributed
at timet. This will be given by f(xt,t)dxt. The latter view more closely fits the
originally stated definition of an ensemble and will, therefore, be employed to determine
an equation satisfied byfin the phase space.
The fact thatf has a constant normalization means that there can be neither
sources of new ensemble members nor sinks that reduce the number of ensemble mem-
bers – the number of members remains constant. This also means that any volume Ω
in the phase space with a surfaceS(see Fig. 2.4) contains no sources or sinks. Thus,
the rate of decrease (or increase) of ensemble members in Ω must equal the rate at
which ensemble members leave (or enter) Ω through the surfaceS. The fraction of
ensemble members in Ω at timetcan be written as