Why Do Cross Terms Arise in Polymer Random Walk Calculations?

[raintrek]

I understand the mean square end-to-end distance of a polymer of stepsize b and number of units N is given by

$$\left\langle R^{2}\right\rangle = \sum_{i,j=1}^{N} \left\langle r_{i} r_{j}\right\rangle + \left\langle \sum_{i \neq j}^{N} r_{i} r_{j}\right\rangle$$

And that the cross terms drop out to produce the solution of Nb^2. However I don't understand the origin of these cross terms in the first place - what do they physically represent? And how would one evaluate the sum in the cross terms??

 

[NateD]

The cross terms in the mean square end-to-end distance of a polymer represent the interactions between the different units in the polymer. Specifically, they represent the correlations between the positions of different units that arise due to their interactions. To evaluate the sum in the cross terms, one can use the correlation function of the polymer, which describes how the positions of the different units are correlated.