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

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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 ε 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 δ 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 ε). It is demonstrated that (i) the distribution function depends on ε in a regular way, while its dependence on δ is singular, and (ii) the leading term in the expression for the mean square end-to-end distance linearly grows with ε and remains independent of δ.