The name is absent

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 N_s + 1 equally spaces contour
points (r₀, rɪ, ..., γ_λγJ. Hence,

Z₀

dvj exp

where b²∆s is the mean-squared length of one of the N_s bonds, and ∆s = N∕N_s is the
spacing between the contour points. The quality of the approximation improves as N_s
→ ∞. 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

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).

OS          O

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