Span of a Polymer Chain

The x span of a polymer chain is defined as the difference between the largest x coordinate and the smallest x coordinate of any segments in the chain. The polymer chain is represented by a nearest-neighbor lattice-model random walk in which the mean square displacement of each component of a single step is 13. The distribution function of the x span of an N step random walk is obtained; and the following asymptotic formulas are obtained for its first and second moments, respectively, 2(2N/3π)1/2 and 4 ln2 N/3. The corresponding moments of the magnitude of the x component of the end-to-end distance are (2N/3π)1/2 and N/3. Excluded volume effects are not considered. It is noted that the problem of calculating the first moment of the x span is identical with the one-dimensional case of the Dvoretzky-Erdös problem, namely, the calculation of the average number of different lattice sites visited in an N-step random walk on a d-dimensional lattice.