The end-to-end distribution function for a flexible chain with weak excluded-volume interactions

An explicit expression is derived for the distribution function of end-to-end vectors and for the mean square end-to-end distance of a flexible chain with excluded-volume interactions. The Hamiltonian for a flexible chain with weak intra-chain interactions is determined by two small parameters: the ratio $\epsilon$ of the energy of interaction between segments (within a sphere whose radius coincides with the cut-off length for the potential) to the thermal energy, and the ratio $\delta$ of the cut-off length to the radius of gyration for a Gaussian chain. Unlike conventional approaches grounded on the mean-field evaluation of the end-to-end distance, the Green function is found explicitly (in the first approximation with respect to $\epsilon$). It is demonstrated that (i) the distribution function depends on $\epsilon$ in a regular way, while its dependence on $\delta$ is singular, and (ii) the leading term in the expression for the mean square end-to-end distance linearly grows with $\epsilon$ and remains independent of $\delta$.