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[tex]\frac{1}{2N^{2}}\int_{s} \int_{p} \left\langle (\textbf{R}_{s} - \textbf{R}_{p} )^{2} \right\rangle[/tex]

which describes the radius of gyration of a polymer. (the term being integrated is the average position between beads p and s)

This is shown to be equivalent to:

[tex]\frac{1}{2N^{2}}\int_{s} \int_{p} \left| s - p \right| b^{2} [/tex]

where b is the step size.

All my text book says is that the second equation has been found by using the propagator (probability density function) and two integrals:

[tex]G(\textbf{R}, N)=\left( \frac{3}{2\pi Nb^{2}}\right)^{3/2}e^{- \frac{3R^{2}}{2Nb^{2}}[/tex]

[tex]\int^{\infty}_{- \infty} e^{-ax^{2}}=\sqrt{\frac{\pi}{a}}[/tex]

[tex]\int^{\infty}_{- \infty} x^{2} e^{-ax^{2}}=\frac{1}{2a} \sqrt{\frac{\pi}{a}}[/tex]

I have no idea how this "propagator" is used to reach the second equation. Is anyone able to shed any light on what's happened? Many thanks in advance!