  Almost 25 years ago, Jarzynski published a paper in which it was asserted:
the work done, W, in driving a system from state A to state B, characterized by
the Helmholtz free energies FA and FB, satisfies an equality in which an
average over an ensemble of measurements for W determines the difference in
Free energy. Several features of this result require more detailed description,
to be given in the text. The equality is significant and unexpected. So is the
statement that the equality is independent of the rate of change from state A
to state B. A few years ago, I had presented three papers in which the
contraction of the description from full phase space to coordinate space only
was made. This was motivated by the large difference in time scales for momenta
relaxation and coordinate relaxation. The Jarzynski equality (JE) will be shown
here to be correct only in the limit of slow processes. The proposed example is
for a macromolecular harmonic oscillator with a Hooke's spring constant, k,
that during the time interval linearly changes from k1 to k2. When performed
slowly, we obtain JE in a form in which the free energy is related to the ratio
of the k's. However, when the transition is performed more rapidly the equality
is lost. Jarzynski has suggested to me that Hummer and Szabo have shown that JE
follows directly from the Feynman-Kac formula. We show here that F-K does not
reproduce the direct calculation executed in this paper and that it has been
misapplied in earlier papers.
