Consider once again the path integral expression for the
one-dimensional canonical partition function (for a finite but large
value of *P*):

(the condition is understood).
Recall that, according to the classical isomorphism, the path integral
expression for the canonical partition function is isomorphic to
the classical configuration integral for a certain *P*-particle
system. We can carry this analogy one step further by introducing
into the above expression a set of *P* momentum integrations:

Note that these momentum integrations are completely uncoupled from
the position integrations, and if we were to carry out these
momentum integrations, we would reproduce Eq. (1)
apart from trivial constants. Written in the form Eq. (2),
however, the path integral looks exactly like a phase space integral
for a *P*-particle system. We know from our work in classical
statistical mechanics that dynamical equations of motion can
be constructed that will generate this partition function.
In principle, one would start with the classical Hamiltonian

derive the corresponding classical equations of motion and
then couple in thermostats. Such an approach has certainly been
attempted with only limited success. The difficulty with this
straightforward approach is that the more ``quantum'' a system is, the
large the paramester *P* must be chosen in order to converge
the path integral. However, if *P* is large, the above Hamiltonian
describes a system with extremely stiff nearest-neighbor harmonic
bonds interacting with a very weak potential *U*/*P*. It is, therefore,
almost impossible for the system to deviate far harmonic
oscillator solutions and explore the entire available phase space.
The use of thermostats can help this problem, however, it is
also exacerbated by the fact that all the harmonic interactions
are coupled, leading to a wide variety of time scales associated
with the motion of each variable in the Hamiltonian. In order
to separate out all these time scales, one must somehow
diagonalize this harmonic interaction. One way to do this is
to use normal mode variables, and this is a perfectly valid
approach. However, we will explore another, simpler approach
here. It involves the use of a variable transformation of the
formed used in previous lectures to do the path integral for
the free-particle density matrix.

Consider a change of variables:

where

The inverse of this transformation can be worked out in closed form:

and can also be expressed as a *recursive inverse*:

The term *k*=*P* here can be used to start the recursion.
We have already seen that this transformation diagonalized the
harmonic interaction. Thus, substituting the transformation into the
path integral gives:

The parameters are given by

Note also that the momentum integrations have been changed slightly to involve a set of parameters . Introducing these parameters, again, only changes the partition function by trivial constant factors. How these should be chosen will become clear later in the discussion. The notation indicates that each variable is a generally a function of all the new variables .

A dynamics scheme can now be derived using as an effective Hamiltonian:

which, when coupled to thermostats, yields a set of equations of motion

These equations have a conserved energy (which is not a Hamiltonian):

Notice that each variable is given its *own* thermostat. This is
done to produce maximum ergodicity in the trajectories. In fact, in practice,
the chain thermostats you have used in the computer labs are employed.
Notice also that the time scale of each variable is now clear. It is
just determined by the parameters . Since the object of using
such dynamical equations is not to produce real dynamics but to sample
the phase space, we would really like each variable to move on the
*same* time scale, so that there are no slow beads trailing behind
the fast ones. This effect can be produced by choosing each parameter
to be proportional to : . Finally, the
forces on the *u* variables can be determined easily from the
chain rule and the recursive inverse given above. The result is

where the first (*i*=1) of these expressions starts the recursion in the
second equation.

Later on, when we discuss applications of path integrals, we will see why a formulation such as this for evaluating path integrals is advantageous.

Mon Mar 29 18:17:04 EST 1999