Analytical Treatment of Excluded Volume. II. Quasi Random-Flight Behavior

Abstract

A ``quasi'' random‐flight behavior of real chains of r segments is described by the following: The distribution of long statistical elements in typical chain configurations can be expressed by a product of functions, each describing an element independently of the others. This random property differs from true random flight in three aspects: (a) The elements' length depends on r, being as stretched as the most probable end‐to‐end distance of the chain. (b) The randomness holds only for elements constituting a large enough fraction of r. Its employment is restricted therefore to describing distances of correspondingly long enough sequences of j segments, or j/r> (j/r)_min. (c) The random behavior is characteristic of typical configurations, corresponding to a given mean state of the chain. It will lead therefore to a Gaussian distribution of distances h_i only if the specification of various h_i's does not vary the shape, and the mean state, of the chain to any appreciable extent. This implies however that j is not too large, or j/r< (j/r)_max. This description, it is argued, appears to be self‐consistent. For: assuming that the mean density of contacts in the bounded range of (j/r) can be evaluated in terms of the random elements, computing on this basis (as fluctuations) the microscopic variation of the interaction energy, and, from it, the typical variation of the configurational probability, and judging from the results, it is concluded that the mean density of contacts can be indeed evaluated with random elements—as assumed to begin with. The contacts contributing to long‐range interactions in 3‐dimensional chains fall within the range bounded by (j/r)_min to (j/r)_max; their evaluation with Gaussian probabilities (Pt. I) will therefore be justified if the above description is correct. Examination of two‐dimensional chains shows that (j/r)_max is violated; a relatively simple modification of the treatment however suffices, leading to α_n²∼n^0.434. The behavior of four‐, and more‐, dimensional chains is also considered.