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Unformatted text preview: PHYS 445 Lecture 4 - Chains 4 - 1 © 2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited. Lecture 4 - Chains What's Important: • random walk in higher dimensions • self-avoiding walk • central limit theorem Text : Reif Further reading: Secs. 1.7 to 1.11 of Reif Random walk in higher dimensions The one-dimensional random walk can be generalized to higher dimensions without difficulty. Consider a d-dimensional vector s i describing the i th step of a walk. On a lattice, the walk would have the appearance In terms of its elementary steps, the tail-to-tip vector m is m = s 1 + s 2 + s 3 ... = Σ i s i . As in one dimension, let's assume that all elementary steps have the same length l . Then the squared length of the walk is m 2 = m 2 = Σ i s i • Σ j s j = Σ i s i 2 + Σ i Σ j ≠ i s i • s j = N l 2 + Σ i Σ j ≠ i s i • s j . This gives the length of a specific walk. The mean square length for a given N for all walks is < m 2 > = N l 2 + < Σ i Σ j ≠ i s i • s j >....
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This note was uploaded on 09/07/2009 for the course PHYS 445 taught by Professor Davidboal during the Spring '08 term at Simon Fraser.
- Spring '08