where ʃ Dr denotes a functional integration. The potential energy is now a functional
of the space curve r(s) and partition function is calculated as a functional integral
over all the possible space curves. One approach to defining the functional integration
is to discretize the continuous function r (s) by a set of Ns + 1 equally spaces contour
points (r0, rɪ, ..., γλγJ. Hence,
Z0
dvj exp
(8.10)
where b2∆s is the mean-squared length of one of the Ns bonds, and ∆s = N∕Ns is the
spacing between the contour points. The quality of the approximation improves as Ns
→ ∞. Again, the Chapman-Kolmogorov equation is used to obtain the probability
density distribution of the segments.
p0(r, s + ∆s) = Z d(Δr)Φ(Δr)p0(r — ∆r, s), (8.11)
where
(8.12)
The Chapman-Kolmogorov integral equation 8.11 can be reduced to partial differen-
tial equations, which are referred to as Fokker-Planck equations by Taylor-expanding
both sides of the equation in powers of ∆s and ∆r and taking ∆s → O.
∂ . b2 2 / ʌ
—p0(r,s) = —V Po(r,s).
(8.13)
OS O
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