Thus, the distribution probability density for a continuous Gaussian chain takes
the form of a conventional diffusion equation with a diffusion coefficient given by
b2∕6. This equation is solved to obtain the probability density, instead of the integral
Chapman-Kolmogorov equations with the initial condition, po(r,O) = J(r). The final
distribution is given by
z ʌ / 3 V/2 f 3∣r∣2∖
(8.14)
= expΓ^J∙
In SCFT, the interaction potential of the surrounding polymer segments self con-
sistently generate the potential field on the the single polymer chain. Hence, the
theoretical formulation of a Gaussian chain in an external field is presented here. The
microscopic density of a Gaussian chain is given by

dsδ(r — r(s)).
(8.15)
Thus, the potential due to the external field (w(r) acting on the chain is given by
∕3t7χ[r, w] = / dr'w(r')p(r).
(8.16)
The partition function of the chain is
Z[w∖ — Drexp(-βU0[r] — βUι[r,w[). (8.17)
213
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