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CLASSICAL SYSTEMS
.sp 2
TRAJECTORY OF A SYSTEM
.sp
time series:
.sp
.in +.5i
$q sub i (t), ~~ p sub i (t) ~~~~~~~~~~~ i=1,3N$
.in -.5i
.sp
where the conjugate momenta 
$~~p sub i ~=~ m sub i q dot sub i~$ 
and the $q sub i$ are the coordinates for the system.  
.sp 4
PHASE SPACE
.sp
6N-dimensional hyperspace
.sp
variables: \ \ \ \  $3N ~ q sub i, ~~ 3N ~ p sub i$
.sp
phase space POINT\ \ =\ \  STATE of system
phase space PATH \ \ = \ \ TRAJECTORY 
.sp 2
.SK
PROBABILITY DENSITY FUNCTION
.sp
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$rho ~=~ f( GAMMA $)
.in -.5i
.sp
where $GAMMA$ denotes the phase space or 
a set of phase space variables $(p,q)$.
.sp 
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$rho ( GAMMA ) d GAMMA$
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.sp
is the number of state points in a small region $d GAMMA$ 
about the state point $GAMMA$
.sp
The density function is normalized
.sp
.in +.5i
$int sub GAMMA d GAMMA rho ( GAMMA ) ~=~ 1$
.in -.5i
.SK
For the contraints (N, V, T), i.e., for the canonical ensemble,
omitting factors $h sup 3N$,
.in +.5i
.sp
$rho ( GAMMA ) ~=~ {e sup {- {H( GAMMA )} / kT} over Q }$
.in +.5i
.sp
$Q ~=~ int sub GAMMA d GAMMA e sup {- {H( GAMMA )} / kT}$
.sp
$H( GAMMA )$ = total energy at phase point
.br
$~~~~~~$ = $V sub p ~+~ T sub k$
.in -1.0i
.sp
The normailizing factor $Q$ is the partition function.
.SK
PROPERTIES OF A SYSTEM: AVERAGING
.sp 2
The density function $rho$ tells us all we can know about a system.
.sp
The properties of a system are obtained by averaging, e.g., 
the thermodynamic variable $E$, the energy of a system,
is the expectation value
.sp
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$E ~=~ <E> ~=~ int sub GAMMA d GAMMA rho ( GAMMA ) E ( GAMMA )$
.sp
.in -.5i
For the free energy and entropy of a system,
.sp
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$F ~=~ -kT log Q$
.sp
$S ~=~ -k < log rho >$
.in -1i
.SK
ERGODIC HYPOTHESIS
.sp
The density function $rho$ can be obtained by constructing very many copies
of a system, each evolving independently and having different starting
phase. $rho$ is obtained by tallying for this ENSEMBLE of systems, at some
instant, the number of systems (states) within $d GAMMA$ of a
phase point $GAMMA$.
.sp 2
The density function $rho$ also can be obtained by following the trajectory
of one system for a VERY long time
(although no phase point can be revisited, a region of phase space can
be revisited). $rho$ is obtained by tallying the number of
"snapshots" (trajectory points) within $d GAMMA$ of a
phase point $GAMMA$.
.sp 2
.pl +2
The ERGODIC HYPOTHESIS asserts that the same density function $rho$ is found
from both the above methods.
Thus the trajectory TIME average = the ENSEMBLE average for any variable.
.pl -2
.SK
GENERALIZED VARIABLES
.sp
The Cartesian coordinates and their conjugate momenta are just one set of
allowed variables. 
.sp
If appropriate rules are followed, other sets can be
constructed. 
.sp
For example, the set of normal coordinates, each of which is
independent of the others, 
and each is a function of one or more of the Cartesian
coordinates.
.sp 3
.SK
SIMPLIFICATION
.sp 
It is conceptually and
mathematically difficult to handle  
3N degrees of freedom (6N-dimensional phase space).
.sp
The description can be reduced to one of
fewer variables by PROJECTION. 
.sp
E.g., to describe the system with a single
coordinate, such as a REACTION COORDINATE, $x$,
.in +.5i
.sp
$rho (x) ~=~ int sub GAMMA d GAMMA rho ( GAMMA ) delta (x( GAMMA )~-~x)$
.sp
$rho (x) dx$ = probability system near point $x$
.sp
$int sub x d x rho ( x )$ = 1
.in
.sp
Information about the system has been lost.  How do we account for it?
In the ENTROPY, and through use of the FREE ENERGY for description of the
reduced-dimensional system.
.SK
TIME DEPENDENCE OF THE PROBABILITY DENSITY
.sp
For an equilibrium system, $rho$ is constant
and $d rho /dt = 0$.
For a non-equilibrium system, $rho$ evolves in time,
as it must after an external force is applied to a system
or after a perturbation is removed from a system.
For a classical system,
.sp
.in +.5i
${d rho (p,q,t)} over dt ~=~ i italic L rho (p,q,t)$
.in
.sp
where $i italic L$ is a differential operator.
