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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‐Erdos problem, namely, the calculation of the average number of different lattice sites visited in an N‐step random walk on a d‐dimensional lattice.